## Quant Analysis 101: Inferential Statistics

Everything You Need To Get Started (With Examples)

By: Derek Jansen (MBA) | Reviewers: Kerryn Warren (PhD) | October 2023

If you’re new to quantitative data analysis , one of the many terms you’re likely to hear being thrown around is inferential statistics. In this post, we’ll provide an introduction to inferential stats, using straightforward language and loads of examples .

## Overview: Inferential Statistics

What are inferential statistics.

- Descriptive vs inferential statistics

## Correlation

- Key takeaways

At the simplest level, inferential statistics allow you to test whether the patterns you observe in a sample are likely to be present in the population – or whether they’re just a product of chance.

In stats-speak, this “Is it real or just by chance?” assessment is known as statistical significance . We won’t go down that rabbit hole in this post, but this ability to assess statistical significance means that inferential statistics can be used to test hypotheses and in some cases, they can even be used to make predictions .

That probably sounds rather conceptual – let’s look at a practical example.

Let’s say you surveyed 100 people (this would be your sample) in a specific city about their favourite type of food. Reviewing the data, you found that 70 people selected pizza (i.e., 70% of the sample). You could then use inferential statistics to test whether that number is just due to chance , or whether it is likely representative of preferences across the entire city (this would be your population).

PS – you’d use a chi-square test for this example, but we’ll get to that a little later.

## Inferential vs Descriptive

At this point, you might be wondering how inferentials differ from descriptive statistics. At the simplest level, descriptive statistics summarise and organise the data you already have (your sample), making it easier to understand.

Inferential statistics, on the other hand, allow you to use your sample data to assess whether the patterns contained within it are likely to be present in the broader population , and potentially, to make predictions about that population.

It’s example time again…

Let’s imagine you’re undertaking a study that explores shoe brand preferences among men and women. If you just wanted to identify the proportions of those who prefer different brands, you’d only require descriptive statistics .

However, if you wanted to assess whether those proportions differ between genders in the broader population (and that the difference is not just down to chance), you’d need to utilise inferential statistics .

In short, descriptive statistics describe your sample, while inferential statistics help you understand whether the patterns in your sample are likely to reflect within the population .

## Let’s look at some inferential tests

Now that we’ve defined inferential statistics and explained how it differs from descriptive statistics, let’s take a look at some of the most common tests within the inferential realm . It’s worth highlighting upfront that there are many different types of inferential tests and this is most certainly not a comprehensive list – just an introductory list to get you started.

A t-test is a way to compare the means (averages) of two groups to see if they are meaningfully different, or if the difference is just by chance. In other words, to assess whether the difference is statistically significant . This is important because comparing two means side-by-side can be very misleading if one has a high variance and the other doesn’t (if this sounds like gibberish, check out our descriptive statistics post here ).

As an example, you might use a t-test to see if there’s a statistically significant difference between the exam scores of two mathematics classes taught by different teachers . This might then lead you to infer that one teacher’s teaching method is more effective than the other.

It’s worth noting that there are a few different types of t-tests . In this example, we’re referring to the independent t-test , which compares the means of two groups, as opposed to the mean of one group at different times (i.e., a paired t-test). Each of these tests has its own set of assumptions and requirements, as do all of the tests we’ll discuss here – but we’ll save assumptions for another post!

While a t-test compares the means of just two groups, an ANOVA (which stands for Analysis of Variance) can compare the means of more than two groups at once . Again, this helps you assess whether the differences in the means are statistically significant or simply a product of chance.

For example, if you want to know whether students’ test scores vary based on the type of school they attend – public, private, or homeschool – you could use ANOVA to compare the average standardised test scores of the three groups .

Similarly, you could use ANOVA to compare the average sales of a product across multiple stores. Based on this data, you could make an inference as to whether location is related to (affects) sales.

In these examples, we’re specifically referring to what’s called a one-way ANOVA , but as always, there are multiple types of ANOVAs for different applications. So, be sure to do your research before opting for any specific test.

While t-tests and ANOVAs test for differences in the means across groups, the Chi-square test is used to see if there’s a difference in the proportions of various categories . In stats speak, the Chi-square test assesses whether there’s a statistically significant relationship between two categorical variables (i.e., nominal or ordinal data). If you’re not familiar with these terms, check out our explainer video here .

As an example, you could use a Chi-square test to check if there’s a link between gender (e.g., male and female) and preference for a certain category of car (e.g., sedans or SUVs). Similarly, you could use this type of test to see if there’s a relationship between the type of breakfast people eat (cereal, toast, or nothing) and their university major (business, math or engineering).

Correlation analysis looks at the relationship between two numerical variables (like height or weight) to assess whether they “move together” in some way. In stats-speak, correlation assesses whether a statistically significant relationship exists between two variables that are interval or ratio in nature .

For example, you might find a correlation between hours spent studying and exam scores. This would suggest that generally, the more hours people spend studying, the higher their scores are likely to be.

Similarly, a correlation analysis may reveal a negative relationship between time spent watching TV and physical fitness (represented by VO2 max levels), where the more time spent in front of the television, the lower the physical fitness level.

When running a correlation analysis, you’ll be presented with a correlation coefficient (also known as an r-value), which is a number between -1 and 1. A value close to 1 means that the two variables move in the same direction , while a number close to -1 means that they move in opposite directions . A correlation value of zero means there’s no clear relationship between the two variables.

What’s important to highlight here is that while correlation analysis can help you understand how two variables are related, it doesn’t prove that one causes the other . As the adage goes, correlation is not causation.

While correlation allows you to see whether there’s a relationship between two numerical variables, regression takes it a step further by allowing you to make predictions about the value of one variable (called the dependent variable) based on the value of one or more other variables (called the independent variables).

For example, you could use regression analysis to predict house prices based on the number of bedrooms, location, and age of the house. The analysis would give you an equation that lets you plug in these factors to estimate a house’s price. Similarly, you could potentially use regression analysis to predict a person’s weight based on their height, age, and daily calorie intake.

It’s worth noting that in these examples, we’ve been talking about multiple regression , as there are multiple independent variables. While this is a popular form of regression, there are many others, including simple linear, logistic and multivariate. As always, be sure to do your research before selecting a specific statistical test.

As with correlation, keep in mind that regression analysis alone doesn’t prove causation . While it can show that variables are related and help you make predictions, it can’t prove that one variable causes another to change. Other factors that you haven’t included in your model could be influencing the results. To establish causation, you’d typically need a very specific research design that allows you to control all (or at least most) variables.

## Let’s Recap

We’ve covered quite a bit of ground. Here’s a quick recap of the key takeaways:

- Inferential stats allow you to assess whether patterns in your sample are likely to be present in your population
- Some common inferential statistical tests include t-tests, ANOVA, chi-square, correlation and regression .
- Inferential statistics alone do not prove causation . To identify and measure causal relationships, you need a very specific research design.

If you’d like 1-on-1 help with your inferential statistics, check out our private coaching service , where we hold your hand throughout the quantitative research process.

## Psst… there’s more!

This post is an extract from our bestselling Udemy Course, Methodology Bootcamp . If you want to work smart, you don't want to miss this .

## You Might Also Like:

## Submit a Comment Cancel reply

Your email address will not be published. Required fields are marked *

Save my name, email, and website in this browser for the next time I comment.

- Print Friendly

- Privacy Policy

Buy Me a Coffee

Home » Inferential Statistics – Types, Methods and Examples

## Inferential Statistics – Types, Methods and Examples

Table of Contents

## Inferential Statistics

Inferential statistics is a branch of statistics that involves making predictions or inferences about a population based on a sample of data taken from that population. It is used to analyze the probabilities, assumptions, and outcomes of a hypothesis .

The basic steps of inferential statistics typically involve the following:

- Define a Hypothesis: This is often a statement about a parameter of a population, such as the population mean or population proportion.
- Select a Sample: In order to test the hypothesis, you’ll select a sample from the population. This should be done randomly and should be representative of the larger population in order to avoid bias.
- Collect Data: Once you have your sample, you’ll need to collect data. This data will be used to calculate statistics that will help you test your hypothesis.
- Perform Analysis: The collected data is then analyzed using statistical tests such as the t-test, chi-square test, or ANOVA, to name a few. These tests help to determine the likelihood that the results of your analysis occurred by chance.
- Interpret Results: The analysis can provide a probability, called a p-value, which represents the likelihood that the results occurred by chance. If this probability is below a certain level (commonly 0.05), you may reject the null hypothesis (the statement that there is no effect or relationship) in favor of the alternative hypothesis (the statement that there is an effect or relationship).

## Inferential Statistics Types

Inferential statistics can be broadly categorized into two types: parametric and nonparametric. The selection of type depends on the nature of the data and the purpose of the analysis.

## Parametric Inferential Statistics

These are statistical methods that assume data comes from a type of probability distribution and makes inferences about the parameters of the distribution. Common parametric methods include:

- T-tests : Used when comparing the means of two groups to see if they’re significantly different.
- Analysis of Variance (ANOVA) : Used to compare the means of more than two groups.
- Regression Analysis : Used to predict the value of one variable (dependent) based on the value of another variable (independent).
- Chi-square test for independence : Used to test if there is a significant association between two categorical variables.
- Pearson’s correlation : Used to test if there is a significant linear relationship between two continuous variables.

## Nonparametric Inferential Statistics

These are methods used when the data does not meet the requirements necessary to use parametric statistics, such as when data is not normally distributed. Common nonparametric methods include:

- Mann-Whitney U Test : Non-parametric equivalent to the independent samples t-test.
- Wilcoxon Signed-Rank Test : Non-parametric equivalent to the paired samples t-test.
- Kruskal-Wallis Test : Non-parametric equivalent to the one-way ANOVA.
- Spearman’s rank correlation : Non-parametric equivalent to the Pearson correlation.
- Chi-square test for goodness of fit : Used to test if the observed frequencies for a categorical variable match the expected frequencies.

## Inferential Statistics Formulas

Inferential statistics use various formulas and statistical tests to draw conclusions or make predictions about a population based on a sample from that population. Here are a few key formulas commonly used:

Confidence Interval for a Mean:

When you have a sample and want to make an inference about the population mean (µ), you might use a confidence interval.

The formula for a confidence interval around a mean is:

[Sample Mean] ± [Z-score or T-score] * (Standard Deviation / sqrt[n]) where:

- Sample Mean is the mean of your sample data
- Z-score or T-score is the value from the Z or T distribution corresponding to the desired confidence level (Z is used when the population standard deviation is known or the sample size is large, otherwise T is used)
- Standard Deviation is the standard deviation of the sample
- sqrt[n] is the square root of the sample size

Hypothesis Testing:

Hypothesis testing often involves calculating a test statistic, which is then compared to a critical value to decide whether to reject the null hypothesis.

A common test statistic for a test about a mean is the Z-score:

Z = (Sample Mean - Hypothesized Population Mean) / (Standard Deviation / sqrt[n])

where all variables are as defined above.

Chi-Square Test:

The Chi-Square Test is used when dealing with categorical data.

The formula is:

χ² = Σ [ (Observed-Expected)² / Expected ]

- Observed is the actual observed frequency
- Expected is the frequency we would expect if the null hypothesis were true

The t-test is used to compare the means of two groups. The formula for the independent samples t-test is:

t = (mean1 - mean2) / sqrt [ (sd1²/n1) + (sd2²/n2) ] where:

- mean1 and mean2 are the sample means
- sd1 and sd2 are the sample standard deviations
- n1 and n2 are the sample sizes

## Inferential Statistics Examples

Sure, inferential statistics are used when making predictions or inferences about a population from a sample of data. Here are a few real-time examples:

- Medical Research: Suppose a pharmaceutical company is developing a new drug and they’re currently in the testing phase. They gather a sample of 1,000 volunteers to participate in a clinical trial. They find that 700 out of these 1,000 volunteers reported a significant reduction in their symptoms after taking the drug. Using inferential statistics, they can infer that the drug would likely be effective for the larger population.
- Customer Satisfaction: Suppose a restaurant wants to know if its customers are satisfied with their food. They could survey a sample of their customers and ask them to rate their satisfaction on a scale of 1 to 10. If the average rating was 8.5 from a sample of 200 customers, they could use inferential statistics to infer that the overall customer population is likely satisfied with the food.
- Political Polling: A polling company wants to predict who will win an upcoming presidential election. They poll a sample of 10,000 eligible voters and find that 55% prefer Candidate A, while 45% prefer Candidate B. Using inferential statistics, they infer that Candidate A has a higher likelihood of winning the election.
- E-commerce Trends: An e-commerce company wants to improve its recommendation engine. They analyze a sample of customers’ purchase history and notice a trend that customers who buy kitchen appliances also frequently buy cookbooks. They use inferential statistics to infer that recommending cookbooks to customers who buy kitchen appliances would likely increase sales.
- Public Health: A health department wants to assess the impact of a health awareness campaign on smoking rates. They survey a sample of residents before and after the campaign. If they find a significant reduction in smoking rates among the surveyed group, they can use inferential statistics to infer that the campaign likely had an impact on the larger population’s smoking habits.

## Applications of Inferential Statistics

Inferential statistics are extensively used in various fields and industries to make decisions or predictions based on data. Here are some applications of inferential statistics:

- Healthcare: Inferential statistics are used in clinical trials to analyze the effect of a treatment or a drug on a sample population and then infer the likely effect on the general population. This helps in the development and approval of new treatments and drugs.
- Business: Companies use inferential statistics to understand customer behavior and preferences, market trends, and to make strategic decisions. For example, a business might sample customer satisfaction levels to infer the overall satisfaction of their customer base.
- Finance: Banks and financial institutions use inferential statistics to evaluate the risk associated with loans and investments. For example, inferential statistics can help in determining the risk of default by a borrower based on the analysis of a sample of previous borrowers with similar credit characteristics.
- Quality Control: In manufacturing, inferential statistics can be used to maintain quality standards. By analyzing a sample of the products, companies can infer the quality of all products and decide whether the manufacturing process needs adjustments.
- Social Sciences: In fields like psychology, sociology, and education, researchers use inferential statistics to draw conclusions about populations based on studies conducted on samples. For instance, a psychologist might use a survey of a sample of people to infer the prevalence of a particular psychological trait or disorder in a larger population.
- Environment Studies: Inferential statistics are also used to study and predict environmental changes and their impact. For instance, researchers might measure pollution levels in a sample of locations to infer overall pollution levels in a wider area.
- Government Policies: Governments use inferential statistics in policy-making. By analyzing sample data, they can infer the potential impacts of policies on the broader population and thus make informed decisions.

## Purpose of Inferential Statistics

The purposes of inferential statistics include:

- Estimation of Population Parameters: Inferential statistics allows for the estimation of population parameters. This means that it can provide estimates about population characteristics based on sample data. For example, you might want to estimate the average weight of all men in a country by sampling a smaller group of men.
- Hypothesis Testing: Inferential statistics provides a framework for testing hypotheses. This involves making an assumption (the null hypothesis) and then testing this assumption to see if it should be rejected or not. This process enables researchers to draw conclusions about population parameters based on their sample data.
- Prediction: Inferential statistics can be used to make predictions about future outcomes. For instance, a researcher might use inferential statistics to predict the outcomes of an election or forecast sales for a company based on past data.
- Relationships Between Variables: Inferential statistics can also be used to identify relationships between variables, such as correlation or regression analysis. This can provide insights into how different factors are related to each other.
- Generalization: Inferential statistics allows researchers to generalize their findings from the sample to the larger population. It helps in making broad conclusions, given that the sample is representative of the population.
- Variability and Uncertainty: Inferential statistics also deal with the idea of uncertainty and variability in estimates and predictions. Through concepts like confidence intervals and margins of error, it provides a measure of how confident we can be in our estimations and predictions.
- Error Estimation : It provides measures of possible errors (known as margins of error), which allow us to know how much our sample results may differ from the population parameters.

## Limitations of Inferential Statistics

Inferential statistics, despite its many benefits, does have some limitations. Here are some of them:

- Sampling Error : Inferential statistics are often based on the concept of sampling, where a subset of the population is used to infer about the population. There’s always a chance that the sample might not perfectly represent the population, leading to sampling errors.
- Misleading Conclusions : If assumptions for statistical tests are not met, it could lead to misleading results. This includes assumptions about the distribution of data, homogeneity of variances, independence, etc.
- False Positives and Negatives : There’s always a chance of a Type I error (rejecting a true null hypothesis, or a false positive) or a Type II error (not rejecting a false null hypothesis, or a false negative).
- Dependence on Quality of Data : The accuracy and validity of inferential statistics depend heavily on the quality of data collected. If data are biased, inaccurate, or collected using flawed methods, the results won’t be reliable.
- Limited Predictive Power : While inferential statistics can provide estimates and predictions, these are based on the current data and may not fully account for future changes or variables not included in the model.
- Complexity : Some inferential statistical methods can be quite complex and require a solid understanding of statistical principles to implement and interpret correctly.
- Influenced by Outliers : Inferential statistics can be heavily influenced by outliers. If these extreme values aren’t handled properly, they can lead to misleading results.
- Over-reliance on P-values : There’s a tendency in some fields to overly rely on p-values to determine significance, even though p-values have several limitations and are often misunderstood.

## About the author

## Muhammad Hassan

Researcher, Academic Writer, Web developer

## You may also like

## Cluster Analysis – Types, Methods and Examples

## Discriminant Analysis – Methods, Types and...

## MANOVA (Multivariate Analysis of Variance) –...

## Documentary Analysis – Methods, Applications and...

## ANOVA (Analysis of variance) – Formulas, Types...

## Graphical Methods – Types, Examples and Guide

Purdue Online Writing Lab Purdue OWL® College of Liberal Arts

## Basic Inferential Statistics: Theory and Application

## Welcome to the Purdue OWL

This page is brought to you by the OWL at Purdue University. When printing this page, you must include the entire legal notice.

Copyright ©1995-2018 by The Writing Lab & The OWL at Purdue and Purdue University. All rights reserved. This material may not be published, reproduced, broadcast, rewritten, or redistributed without permission. Use of this site constitutes acceptance of our terms and conditions of fair use.

This handout explains how to write with statistics including quick tips, writing descriptive statistics, writing inferential statistics, and using visuals with statistics.

The heart of statistics is inferential statistics. Descriptive statistics are typically straightforward and easy to interpret. Unlike descriptive statistics, inferential statistics are often complex and may have several different interpretations.

The goal of inferential statistics is to discover some property or general pattern about a large group by studying a smaller group of people in the hopes that the results will generalize to the larger group. For example, we may ask residents of New York City their opinion about their mayor. We would probably poll a few thousand individuals in New York City in an attempt to find out how the city as a whole views their mayor. The following section examines how this is done.

A population is the entire group of people you would like to know something about. In our previous example of New York City, the population is all of the people living in New York City. It should not include people from England, visitors in New York, or even people who know a lot about New York City.

A sample is a subset of the population. Just like you may sample different types of ice cream at the grocery store, a sample of a population should be just a smaller version of the population.

It is extremely important to understand how the sample being studied was drawn from the population. The sample should be as representative of the population as possible. There are several valid ways of creating a sample from a population, but inferential statistics works best when the sample is drawn at random from the population. Given a large enough sample, drawing at random ensures a fair and representative sample of a population.

## Comparing two or more groups

Much of statistics, especially in medicine and psychology, is used to compare two or more groups and attempts to figure out if the two groups are different from one another.

Example: Drug X

Let us say that a drug company has developed a pill, which they think increases the recovery time from the common cold. How would they actually find out if the pill works or not? What they might do is get two groups of people from the same population (say, people from a small town in Indiana who had just caught a cold) and administer the pill to one group, and give the other group a placebo. They could then measure how many days each group took to recover (typically, one would calculate the mean of each group). Let's say that the mean recovery time for the group with the new drug was 5.4 days, and the mean recovery time for the group with the placebo was 5.8 days.

The question becomes, is this difference due to random chance, or does taking the pill actually help you recover from the cold faster? The means of the two groups alone does not help us determine the answer to this question. We need additional information.

## Sample Size

If our example study only consisted of two people (one from the drug group and one from the placebo group) there would be so few participants that we would not have much confidence that there is a difference between the two groups. That is to say, there is a high probability that chance explains our results (any number of explanations might account for this, for example, one person might be younger, and thus have a better immune system). However, if our sample consisted of 1,000 people in each group, then the results become much more robust (while it might be easy to say that one person is younger than another, it is hard to say that 1,000 random people are younger than another 1,000 random people). If the sample is drawn at random from the population, then these 'random' variations in participants should be approximately equal in the two groups, given that the two groups are large. This is why inferential statistics works best when there are lots of people involved.

Be wary of statistics that have small sample sizes, unless they are in a peer-reviewed journal. Professional statisticians can interpret results correctly from small sample sizes, and often do, but not everyone is a professional, and novice statisticians often incorrectly interpret results. Also, if your author has an agenda, they may knowingly misinterpret results. If your author does not give a sample size, then he or she is probably not a professional, and you should be wary of the results. Sample sizes are required information in almost all peer-reviewed journals, and therefore, should be included in anything you write as well.

## Variability

Even if we have a large enough sample size, we still need more information to reach a conclusion. What we need is some measure of variability. We know that the typical person takes about 5-6 days to recover from a cold, but does everyone recover around 5-6 days, or do some people recover in 1 day, and others recover in 10 days? Understanding the spread of the data will tell us how effective the pill is. If everyone in the placebo group takes exactly 5.8 days to recover, then it is clear that the pill has a positive effect, but if people have a wide variability in their length of recovery (and they probably do) then the picture becomes a little fuzzy. Only when the mean, sample size, and variability have been calculated can a proper conclusion be made. In our case, if the sample size is large, and the variability is small, then we would receive a small p-value (probability-value). Small p-values are good, and this term is prominent enough to warrant further discussion.

In classic inferential statistics, we make two hypotheses before we start our study, the null hypothesis, and the alternative hypothesis.

Null Hypothesis: States that the two groups we are studying are the same.

Alternative Hypothesis: States that the two groups we are studying are different.

The goal in classic inferential statistics is to prove the null hypothesis wrong. The logic says that if the two groups aren't the same, then they must be different. A low p-value indicates a low probability that the null hypothesis is correct (thus, providing evidence for the alternative hypothesis).

Remember: It's good to have low p-values.

## Introduction to Inferential Statistics

Current as of 2024-01-05

Lecture : MW 12-1:30pm (MCNB 309)

Dr. Marc Trussler

Fox-Fels Hall 32 (3814 Walnut Street)

Office Hours: M 9-11am

TA: Dylan Radley

Fox-Fels Hall 35 (3814 Walnut Street)

Office Hours:

Tuesday 11-12

Tuesday 3-4

Thursday 12-1

## Course Description

The first step of many data science sequences is to learn a great deal about how to work with individual data sets: cleaning, tidying, merging, describing and visualizing data. These are crucial skills in data analytics, but describing a data set is not our ultimate goal. The ultimate goal of data science is to make inferences about the world based on the small sample of data that we have.

PSCI 1801 shifts focus to this goal of inference. Using a methodology that emphasizes intuition and simulation over mathematics, this course will cover the key statistical concepts of probability, sampling, distributions, hypothesis testing, and covariance. The goal of the class is for students to ultimately have the knowledge and ability to perform, customize, and explain bivariate and multivariate regression. Students who have not taken PSCI-1800 should have basic familiarity with R, including working with vectors and matrices, basic summary statistics, visualizations, and for() loops.

## Expectations and policies

Prerequisite knowledge.

PSCI 1800 (formerly 107) or similar R course. To help us better understand the nature of inferential statistics, we will be running quite a lot of simulations in R . Students entering the class should have a working knowledge of the R programming language, and in particular know how to use square brackets to index vectors and to run for() loops. We will be doing a short refresher on these concepts in the first two weeks of class.

## Course Slack Channel

We will use Slack to communicate with the class. You will receive an invitation to join the our channel shortly after the start of class. One of the better things to come through the pandemic is the use of Slack for classroom communications. It is a really good tool to allow us to send quick and informal messages to individual students or groups (or for you to message us). Similarly, it allows you to collaborate with other students in the class, and is a great place to get simple questions answered. Because we will be making announcements via Slack, it is extremely important you get this set up.

## Format/Attendance

The lectures will be in person. While this is not a discussion-based class, there is an expectation of some amount of participation and feedback. Attendance will not be recorded, though do note you are scored on participation.

The course will require students to have access to a personal computer in order to run the statistics software. If this is not possible, please consult with one of the instructors as soon as possible. Support to cover course costs is available through ( https://srfs.upenn.edu/sfs )[Student Financial Services].

## Academic integrity

We expect all students to abide by the rules of the University and to follow the Code of Academic Integrity. 1

For Problem Sets : Collaboration on problem sets is permitted. Ultimately, however, the write-up and code that you turn in must be your own creation. Please write the names of any students you worked with at the top of each problem set. 2

For Exams : Collaboration on the take home exams is cheating. Anyone caught collaborating (and I have caught many) will be immediately referred to the University’s disciplinary system.

## Re-grading of assignments

All student work will be assessed using fair criteria that are uniform across the class. If, however, you are unsatisfied with the grade you received on a particular assignment (beyond simple clerical errors), you can request a re-grade using the following protocol. First, you may not send any grade complaints or requests for re-grades until at least 24 hours after the graded assignment was returned to you. After that, you must document your specific grievances in writing by submitting a PDF or Word Document to the teaching staff. In this document you should explain exactly which parts of the assignment you believe were mis-graded, and provide documentation for why your answers were correct.We will then re-score the entire assignment (including portions for which you did not have grievances), and the new score will be the one you receive on the assignment (even if it is lower than your original score).

## Late policy

Notwithstanding everything below: exceptions to all of these policies will be made for health reasons, extraordinary family circumstances, and religious holidays. The teaching staff are extremely reasonable and lenient, as long as you discuss with us potential issues *before} the deadline.

For problem sets: You are granted 5 ``grace days’’ throughout the semester. Over the course of the semester you can use these when you need to turn problem sets in late. You can only use 3 grace days on any given assignment. You do not have to ask to use these days. This is counted in whole days, so if a problem set is turned in at 5:01pm the day it is due (i.e. 1 minute late) you will have used 1 grace day. If you turn the problem set in at 5:01pm the day after it is due (i.e. 24 hours and 1 minute late) you will have used 2 grace days etc. Choosing to not complete a problem set (see policy below) does not affect your grace days. Once you are out of grace days subsequently late problem sets will be graded as incomplete.

The nature of the two exams (timed exams completed during a certain window) does not allow for any extensions.

## Assessment and grading

All assignments will be graded anonymously. Please hand assignments in on Canvas with your student number, not your name.

Participation (5%)

This portion of your grade mixes two components:

Traditional participation including: asking and answering questions in lecture and in recitations, asking and answering questions on the course Slack, or attending office hours.

The completion of weekly ``check-in’’ quizzes on Canvas. These will be available each week, will take less than 5 minutes, and will be graded by completion (not correctness).

Problem sets (45%)

Five problem sets (roughly every two weeks)

Scored out of 100.

You are free to do as many of the problem sets as you like. If you do not complete a problem set, the percentage points for that assignment will be transferred to the first exam (for PS1 and PS2), or the second exam (for PS3, PS4, & PS5). For example if you don’t complete PS2, the first exam would then be worth 34% of your final grade. If you don’t complete PS4 & PS5, the second exam would be worth 43% of your final grade.

First Exam 25%

- This will be an open-book 24 hour take-home test. The test will open on Monday, October 16 at 3:00pm and close on Friday, October 20 at 11:59pm. You can select any 24 hour period to do the test during this window. The latest you can open the test and still have 24 hours to complete it is therefore October 19th at 11:59pm. You may not work with other students on this exam. It will take a similar form as the problem sets.

Second Exam 25%

- This will be a 3 hour open-book take-home test completed on December 20th. You may not work with other students on this exam. Because of the shortened time frame this exam will be less coding intensive and focus more on theoretic concepts.

We will use R in this class, which you can download for free at https://www.r-project.org/ . R is completely open source and has an almost endless set of resources online. Virtually any data science job you could apply nowadays to will require some background in R programming.

While R is the language we will use, RStudio is a free program that makes it considerably easier to work with R. After installing R, you should install RStudio https://www.rstudio.com . Please have both R and RStudio installed by the end of the first week of classes.

If you’re having trouble installing either program, there are more detailed installation instructions on the course Canvas page.

There is one mandatory textbook for this course and two optional:

Data Analysis for Social Science: A Friendly and Practical Introduction. Elena Llaudet & Kosuke Imai. (Mandatory).

- I have chosen this book because it does a really good job of weaving in the basics of statistics with the use of R. Generally speaking the assigned readings from this book will be slightly less technical than what is in the class notes. This book is available at the bookstore and from Amazon. There is only one addition, but be sure to get the (way cheaper) paperback version.

Quantitative Social Science: an Introduction. Kosuke Imai.

- This is the original, graduate level, textbook the Llaudet and Imai textbook is based on. The chapters are largely the same, but this textbook is much more math intensive. I have included below the equivalent readings (labeled QSS) if you want to go into greater detail. These readings are completely optional.

Statistics: Fourth Edition. Freedman, Pisani, Purves. (Optional).

- This textbook has a slightly more conversational and intuitive approach, but does not incorporate those lessons with R. While having this book is not mandatory I really like the style and common-sense explanations of this book. It’s a great companion to have around.

## Class Schedule

Week 1: august 30.

The population is the point.

Excerpt from Mlodinow (on Canvas).

## Week 2: (No Monday class) - September 6

Llaudet & Imai 1

## Week 3: September 11 - September 13

R Review/Start probability

Llaudet & Imai 6.1,6.2,6.7

(QSS 4.11, 6.1)

September 12: course selection period ends

## Week 4: September 18 - September 20

Conditional probability and independence

## Week 5: September 25 - September 27

Random Variables I: Discrete

Llaudet & Imai 6.4.1

Problem Set 1 Due Wednesday 7pm .

## Week 6: October 2 - October 4

Random Variables II: Continuous

Llaudet & Imai 6.4.2-6.4.4

## Week 7:October 9- October 11

Sampling and confidence intervals

Llaudet & Imai 6.5.1,6.5.2

October 9: Drop period ends

Problem Set 2 Due Wednesday 7pm .

## Week 8: October 16 - October 18

Wednesday class will be a drop-in review session in our usual classroom.

First Midterm Exam period Monday 3:00pm to Friday 11:59pm .

## Week 9: October 23 - October 25

Standard error of the mean/Field Trip

Llaudet & Imai 6.5.3

On October 25th we will take a class field trip to the NBC News Decision Desk.

October 27: Grade type change deadline.

## Week 10: October 30 - November 1

Hypothesis Tests and Power

Llaudet & Imai 7.1 7.3 7.4

Problem Set 3 Due Wednesday 7pm .

## Week 11: November 6 - November 8

Two continuous variables and covariation

Llaudet & Imai 3.5

November 6: Withdrawal deadline

## Week 12: November 13 - November 15

Correlation and bivariate regressionn

Llaudet & Imai 4.3

Problem Set 4 Due Wednesday 7pm .

## Week 13: November 20 – (No Wednesday Class)

Multivariate Regression I

Llaudet & Imai 2.1-2.4

## Week 14: November 27 - November 29

Multivariate regression II

Llaudet & Imai 5.1-5.5

## Week 15: December 4- December 6

Interaction with regression

Excerpt from Kam and Franzese (Canvas)

Problem Set 5 Due Wednesday 7pm .

## Week 16: December 11

Prediction with regression

Llaudet & Imai 4.5-4.6

(QSS 7.3.1,7.3.2)

If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

To log in and use all the features of Khan Academy, please enable JavaScript in your browser.

## Statistics and probability

Unit 1: analyzing categorical data, unit 2: displaying and comparing quantitative data, unit 3: summarizing quantitative data, unit 4: modeling data distributions, unit 5: exploring bivariate numerical data, unit 6: study design, unit 7: probability, unit 8: counting, permutations, and combinations, unit 9: random variables, unit 10: sampling distributions, unit 11: confidence intervals, unit 12: significance tests (hypothesis testing), unit 13: two-sample inference for the difference between groups, unit 14: inference for categorical data (chi-square tests), unit 15: advanced regression (inference and transforming), unit 16: analysis of variance (anova).

- school Campus Bookshelves
- menu_book Bookshelves
- perm_media Learning Objects
- login Login
- how_to_reg Request Instructor Account
- hub Instructor Commons
- Download Page (PDF)
- Download Full Book (PDF)
- Periodic Table
- Physics Constants
- Scientific Calculator
- Reference & Cite
- Tools expand_more
- Readability

selected template will load here

This action is not available.

## 7: Inferential Statistics and Hypothesis Testing

- Last updated
- Save as PDF
- Page ID 22063

- Michelle Oja
- Taft College
- 7.1: Growth Mindset What's growth mindset?
- 7.2.1: Can Samples Predict Populations?
- 7.2.2: Descriptive versus Inferential Statistics
- 7.3: The Research Hypothesis and the Null Hypothesis It's confusing, but we don't statistically test the Research Hypothesis. We test the Null Hypothesis.
- 7.4: Null Hypothesis Significance Testing What do we do with the Research Hypothesis and the Null Hypothesis?
- 7.5.1: Critical Values
- 7.5.2: Summary of p-values and NHST
- 7.6: Steps of the Hypothesis Testing Process Four easy steps!
- 7.7.1: Power and Sample Size
- 7.7.2: The p-value of a Test

## Have a language expert improve your writing

Run a free plagiarism check in 10 minutes, generate accurate citations for free.

- Knowledge Base

## The Beginner's Guide to Statistical Analysis | 5 Steps & Examples

Statistical analysis means investigating trends, patterns, and relationships using quantitative data . It is an important research tool used by scientists, governments, businesses, and other organizations.

To draw valid conclusions, statistical analysis requires careful planning from the very start of the research process . You need to specify your hypotheses and make decisions about your research design, sample size, and sampling procedure.

After collecting data from your sample, you can organize and summarize the data using descriptive statistics . Then, you can use inferential statistics to formally test hypotheses and make estimates about the population. Finally, you can interpret and generalize your findings.

This article is a practical introduction to statistical analysis for students and researchers. We’ll walk you through the steps using two research examples. The first investigates a potential cause-and-effect relationship, while the second investigates a potential correlation between variables.

## Table of contents

Step 1: write your hypotheses and plan your research design, step 2: collect data from a sample, step 3: summarize your data with descriptive statistics, step 4: test hypotheses or make estimates with inferential statistics, step 5: interpret your results, other interesting articles.

To collect valid data for statistical analysis, you first need to specify your hypotheses and plan out your research design.

## Writing statistical hypotheses

The goal of research is often to investigate a relationship between variables within a population . You start with a prediction, and use statistical analysis to test that prediction.

A statistical hypothesis is a formal way of writing a prediction about a population. Every research prediction is rephrased into null and alternative hypotheses that can be tested using sample data.

While the null hypothesis always predicts no effect or no relationship between variables, the alternative hypothesis states your research prediction of an effect or relationship.

- Null hypothesis: A 5-minute meditation exercise will have no effect on math test scores in teenagers.
- Alternative hypothesis: A 5-minute meditation exercise will improve math test scores in teenagers.
- Null hypothesis: Parental income and GPA have no relationship with each other in college students.
- Alternative hypothesis: Parental income and GPA are positively correlated in college students.

## Planning your research design

A research design is your overall strategy for data collection and analysis. It determines the statistical tests you can use to test your hypothesis later on.

First, decide whether your research will use a descriptive, correlational, or experimental design. Experiments directly influence variables, whereas descriptive and correlational studies only measure variables.

- In an experimental design , you can assess a cause-and-effect relationship (e.g., the effect of meditation on test scores) using statistical tests of comparison or regression.
- In a correlational design , you can explore relationships between variables (e.g., parental income and GPA) without any assumption of causality using correlation coefficients and significance tests.
- In a descriptive design , you can study the characteristics of a population or phenomenon (e.g., the prevalence of anxiety in U.S. college students) using statistical tests to draw inferences from sample data.

Your research design also concerns whether you’ll compare participants at the group level or individual level, or both.

- In a between-subjects design , you compare the group-level outcomes of participants who have been exposed to different treatments (e.g., those who performed a meditation exercise vs those who didn’t).
- In a within-subjects design , you compare repeated measures from participants who have participated in all treatments of a study (e.g., scores from before and after performing a meditation exercise).
- In a mixed (factorial) design , one variable is altered between subjects and another is altered within subjects (e.g., pretest and posttest scores from participants who either did or didn’t do a meditation exercise).
- Experimental
- Correlational

First, you’ll take baseline test scores from participants. Then, your participants will undergo a 5-minute meditation exercise. Finally, you’ll record participants’ scores from a second math test.

In this experiment, the independent variable is the 5-minute meditation exercise, and the dependent variable is the math test score from before and after the intervention. Example: Correlational research design In a correlational study, you test whether there is a relationship between parental income and GPA in graduating college students. To collect your data, you will ask participants to fill in a survey and self-report their parents’ incomes and their own GPA.

## Measuring variables

When planning a research design, you should operationalize your variables and decide exactly how you will measure them.

For statistical analysis, it’s important to consider the level of measurement of your variables, which tells you what kind of data they contain:

- Categorical data represents groupings. These may be nominal (e.g., gender) or ordinal (e.g. level of language ability).
- Quantitative data represents amounts. These may be on an interval scale (e.g. test score) or a ratio scale (e.g. age).

Many variables can be measured at different levels of precision. For example, age data can be quantitative (8 years old) or categorical (young). If a variable is coded numerically (e.g., level of agreement from 1–5), it doesn’t automatically mean that it’s quantitative instead of categorical.

Identifying the measurement level is important for choosing appropriate statistics and hypothesis tests. For example, you can calculate a mean score with quantitative data, but not with categorical data.

In a research study, along with measures of your variables of interest, you’ll often collect data on relevant participant characteristics.

## Prevent plagiarism. Run a free check.

In most cases, it’s too difficult or expensive to collect data from every member of the population you’re interested in studying. Instead, you’ll collect data from a sample.

Statistical analysis allows you to apply your findings beyond your own sample as long as you use appropriate sampling procedures . You should aim for a sample that is representative of the population.

## Sampling for statistical analysis

There are two main approaches to selecting a sample.

- Probability sampling: every member of the population has a chance of being selected for the study through random selection.
- Non-probability sampling: some members of the population are more likely than others to be selected for the study because of criteria such as convenience or voluntary self-selection.

In theory, for highly generalizable findings, you should use a probability sampling method. Random selection reduces several types of research bias , like sampling bias , and ensures that data from your sample is actually typical of the population. Parametric tests can be used to make strong statistical inferences when data are collected using probability sampling.

But in practice, it’s rarely possible to gather the ideal sample. While non-probability samples are more likely to at risk for biases like self-selection bias , they are much easier to recruit and collect data from. Non-parametric tests are more appropriate for non-probability samples, but they result in weaker inferences about the population.

If you want to use parametric tests for non-probability samples, you have to make the case that:

- your sample is representative of the population you’re generalizing your findings to.
- your sample lacks systematic bias.

Keep in mind that external validity means that you can only generalize your conclusions to others who share the characteristics of your sample. For instance, results from Western, Educated, Industrialized, Rich and Democratic samples (e.g., college students in the US) aren’t automatically applicable to all non-WEIRD populations.

If you apply parametric tests to data from non-probability samples, be sure to elaborate on the limitations of how far your results can be generalized in your discussion section .

## Create an appropriate sampling procedure

Based on the resources available for your research, decide on how you’ll recruit participants.

- Will you have resources to advertise your study widely, including outside of your university setting?
- Will you have the means to recruit a diverse sample that represents a broad population?
- Do you have time to contact and follow up with members of hard-to-reach groups?

Your participants are self-selected by their schools. Although you’re using a non-probability sample, you aim for a diverse and representative sample. Example: Sampling (correlational study) Your main population of interest is male college students in the US. Using social media advertising, you recruit senior-year male college students from a smaller subpopulation: seven universities in the Boston area.

## Calculate sufficient sample size

Before recruiting participants, decide on your sample size either by looking at other studies in your field or using statistics. A sample that’s too small may be unrepresentative of the sample, while a sample that’s too large will be more costly than necessary.

There are many sample size calculators online. Different formulas are used depending on whether you have subgroups or how rigorous your study should be (e.g., in clinical research). As a rule of thumb, a minimum of 30 units or more per subgroup is necessary.

To use these calculators, you have to understand and input these key components:

- Significance level (alpha): the risk of rejecting a true null hypothesis that you are willing to take, usually set at 5%.
- Statistical power : the probability of your study detecting an effect of a certain size if there is one, usually 80% or higher.
- Expected effect size : a standardized indication of how large the expected result of your study will be, usually based on other similar studies.
- Population standard deviation: an estimate of the population parameter based on a previous study or a pilot study of your own.

Once you’ve collected all of your data, you can inspect them and calculate descriptive statistics that summarize them.

## Inspect your data

There are various ways to inspect your data, including the following:

- Organizing data from each variable in frequency distribution tables .
- Displaying data from a key variable in a bar chart to view the distribution of responses.
- Visualizing the relationship between two variables using a scatter plot .

By visualizing your data in tables and graphs, you can assess whether your data follow a skewed or normal distribution and whether there are any outliers or missing data.

A normal distribution means that your data are symmetrically distributed around a center where most values lie, with the values tapering off at the tail ends.

In contrast, a skewed distribution is asymmetric and has more values on one end than the other. The shape of the distribution is important to keep in mind because only some descriptive statistics should be used with skewed distributions.

Extreme outliers can also produce misleading statistics, so you may need a systematic approach to dealing with these values.

## Calculate measures of central tendency

Measures of central tendency describe where most of the values in a data set lie. Three main measures of central tendency are often reported:

- Mode : the most popular response or value in the data set.
- Median : the value in the exact middle of the data set when ordered from low to high.
- Mean : the sum of all values divided by the number of values.

However, depending on the shape of the distribution and level of measurement, only one or two of these measures may be appropriate. For example, many demographic characteristics can only be described using the mode or proportions, while a variable like reaction time may not have a mode at all.

## Calculate measures of variability

Measures of variability tell you how spread out the values in a data set are. Four main measures of variability are often reported:

- Range : the highest value minus the lowest value of the data set.
- Interquartile range : the range of the middle half of the data set.
- Standard deviation : the average distance between each value in your data set and the mean.
- Variance : the square of the standard deviation.

Once again, the shape of the distribution and level of measurement should guide your choice of variability statistics. The interquartile range is the best measure for skewed distributions, while standard deviation and variance provide the best information for normal distributions.

Using your table, you should check whether the units of the descriptive statistics are comparable for pretest and posttest scores. For example, are the variance levels similar across the groups? Are there any extreme values? If there are, you may need to identify and remove extreme outliers in your data set or transform your data before performing a statistical test.

From this table, we can see that the mean score increased after the meditation exercise, and the variances of the two scores are comparable. Next, we can perform a statistical test to find out if this improvement in test scores is statistically significant in the population. Example: Descriptive statistics (correlational study) After collecting data from 653 students, you tabulate descriptive statistics for annual parental income and GPA.

It’s important to check whether you have a broad range of data points. If you don’t, your data may be skewed towards some groups more than others (e.g., high academic achievers), and only limited inferences can be made about a relationship.

A number that describes a sample is called a statistic , while a number describing a population is called a parameter . Using inferential statistics , you can make conclusions about population parameters based on sample statistics.

Researchers often use two main methods (simultaneously) to make inferences in statistics.

- Estimation: calculating population parameters based on sample statistics.
- Hypothesis testing: a formal process for testing research predictions about the population using samples.

You can make two types of estimates of population parameters from sample statistics:

- A point estimate : a value that represents your best guess of the exact parameter.
- An interval estimate : a range of values that represent your best guess of where the parameter lies.

If your aim is to infer and report population characteristics from sample data, it’s best to use both point and interval estimates in your paper.

You can consider a sample statistic a point estimate for the population parameter when you have a representative sample (e.g., in a wide public opinion poll, the proportion of a sample that supports the current government is taken as the population proportion of government supporters).

There’s always error involved in estimation, so you should also provide a confidence interval as an interval estimate to show the variability around a point estimate.

A confidence interval uses the standard error and the z score from the standard normal distribution to convey where you’d generally expect to find the population parameter most of the time.

## Hypothesis testing

Using data from a sample, you can test hypotheses about relationships between variables in the population. Hypothesis testing starts with the assumption that the null hypothesis is true in the population, and you use statistical tests to assess whether the null hypothesis can be rejected or not.

Statistical tests determine where your sample data would lie on an expected distribution of sample data if the null hypothesis were true. These tests give two main outputs:

- A test statistic tells you how much your data differs from the null hypothesis of the test.
- A p value tells you the likelihood of obtaining your results if the null hypothesis is actually true in the population.

Statistical tests come in three main varieties:

- Comparison tests assess group differences in outcomes.
- Regression tests assess cause-and-effect relationships between variables.
- Correlation tests assess relationships between variables without assuming causation.

Your choice of statistical test depends on your research questions, research design, sampling method, and data characteristics.

## Parametric tests

Parametric tests make powerful inferences about the population based on sample data. But to use them, some assumptions must be met, and only some types of variables can be used. If your data violate these assumptions, you can perform appropriate data transformations or use alternative non-parametric tests instead.

A regression models the extent to which changes in a predictor variable results in changes in outcome variable(s).

- A simple linear regression includes one predictor variable and one outcome variable.
- A multiple linear regression includes two or more predictor variables and one outcome variable.

Comparison tests usually compare the means of groups. These may be the means of different groups within a sample (e.g., a treatment and control group), the means of one sample group taken at different times (e.g., pretest and posttest scores), or a sample mean and a population mean.

- A t test is for exactly 1 or 2 groups when the sample is small (30 or less).
- A z test is for exactly 1 or 2 groups when the sample is large.
- An ANOVA is for 3 or more groups.

The z and t tests have subtypes based on the number and types of samples and the hypotheses:

- If you have only one sample that you want to compare to a population mean, use a one-sample test .
- If you have paired measurements (within-subjects design), use a dependent (paired) samples test .
- If you have completely separate measurements from two unmatched groups (between-subjects design), use an independent (unpaired) samples test .
- If you expect a difference between groups in a specific direction, use a one-tailed test .
- If you don’t have any expectations for the direction of a difference between groups, use a two-tailed test .

The only parametric correlation test is Pearson’s r . The correlation coefficient ( r ) tells you the strength of a linear relationship between two quantitative variables.

However, to test whether the correlation in the sample is strong enough to be important in the population, you also need to perform a significance test of the correlation coefficient, usually a t test, to obtain a p value. This test uses your sample size to calculate how much the correlation coefficient differs from zero in the population.

You use a dependent-samples, one-tailed t test to assess whether the meditation exercise significantly improved math test scores. The test gives you:

- a t value (test statistic) of 3.00
- a p value of 0.0028

Although Pearson’s r is a test statistic, it doesn’t tell you anything about how significant the correlation is in the population. You also need to test whether this sample correlation coefficient is large enough to demonstrate a correlation in the population.

A t test can also determine how significantly a correlation coefficient differs from zero based on sample size. Since you expect a positive correlation between parental income and GPA, you use a one-sample, one-tailed t test. The t test gives you:

- a t value of 3.08
- a p value of 0.001

The final step of statistical analysis is interpreting your results.

## Statistical significance

In hypothesis testing, statistical significance is the main criterion for forming conclusions. You compare your p value to a set significance level (usually 0.05) to decide whether your results are statistically significant or non-significant.

Statistically significant results are considered unlikely to have arisen solely due to chance. There is only a very low chance of such a result occurring if the null hypothesis is true in the population.

This means that you believe the meditation intervention, rather than random factors, directly caused the increase in test scores. Example: Interpret your results (correlational study) You compare your p value of 0.001 to your significance threshold of 0.05. With a p value under this threshold, you can reject the null hypothesis. This indicates a statistically significant correlation between parental income and GPA in male college students.

Note that correlation doesn’t always mean causation, because there are often many underlying factors contributing to a complex variable like GPA. Even if one variable is related to another, this may be because of a third variable influencing both of them, or indirect links between the two variables.

## Effect size

A statistically significant result doesn’t necessarily mean that there are important real life applications or clinical outcomes for a finding.

In contrast, the effect size indicates the practical significance of your results. It’s important to report effect sizes along with your inferential statistics for a complete picture of your results. You should also report interval estimates of effect sizes if you’re writing an APA style paper .

With a Cohen’s d of 0.72, there’s medium to high practical significance to your finding that the meditation exercise improved test scores. Example: Effect size (correlational study) To determine the effect size of the correlation coefficient, you compare your Pearson’s r value to Cohen’s effect size criteria.

## Decision errors

Type I and Type II errors are mistakes made in research conclusions. A Type I error means rejecting the null hypothesis when it’s actually true, while a Type II error means failing to reject the null hypothesis when it’s false.

You can aim to minimize the risk of these errors by selecting an optimal significance level and ensuring high power . However, there’s a trade-off between the two errors, so a fine balance is necessary.

## Frequentist versus Bayesian statistics

Traditionally, frequentist statistics emphasizes null hypothesis significance testing and always starts with the assumption of a true null hypothesis.

However, Bayesian statistics has grown in popularity as an alternative approach in the last few decades. In this approach, you use previous research to continually update your hypotheses based on your expectations and observations.

Bayes factor compares the relative strength of evidence for the null versus the alternative hypothesis rather than making a conclusion about rejecting the null hypothesis or not.

If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.

- Student’s t -distribution
- Normal distribution
- Null and Alternative Hypotheses
- Chi square tests
- Confidence interval

Methodology

- Cluster sampling
- Stratified sampling
- Data cleansing
- Reproducibility vs Replicability
- Peer review
- Likert scale

Research bias

- Implicit bias
- Framing effect
- Cognitive bias
- Placebo effect
- Hawthorne effect
- Hostile attribution bias
- Affect heuristic

## Is this article helpful?

Other students also liked.

- Descriptive Statistics | Definitions, Types, Examples
- Inferential Statistics | An Easy Introduction & Examples
- Choosing the Right Statistical Test | Types & Examples

## More interesting articles

- Akaike Information Criterion | When & How to Use It (Example)
- An Easy Introduction to Statistical Significance (With Examples)
- An Introduction to t Tests | Definitions, Formula and Examples
- ANOVA in R | A Complete Step-by-Step Guide with Examples
- Central Limit Theorem | Formula, Definition & Examples
- Central Tendency | Understanding the Mean, Median & Mode
- Chi-Square (Χ²) Distributions | Definition & Examples
- Chi-Square (Χ²) Table | Examples & Downloadable Table
- Chi-Square (Χ²) Tests | Types, Formula & Examples
- Chi-Square Goodness of Fit Test | Formula, Guide & Examples
- Chi-Square Test of Independence | Formula, Guide & Examples
- Coefficient of Determination (R²) | Calculation & Interpretation
- Correlation Coefficient | Types, Formulas & Examples
- Frequency Distribution | Tables, Types & Examples
- How to Calculate Standard Deviation (Guide) | Calculator & Examples
- How to Calculate Variance | Calculator, Analysis & Examples
- How to Find Degrees of Freedom | Definition & Formula
- How to Find Interquartile Range (IQR) | Calculator & Examples
- How to Find Outliers | 4 Ways with Examples & Explanation
- How to Find the Geometric Mean | Calculator & Formula
- How to Find the Mean | Definition, Examples & Calculator
- How to Find the Median | Definition, Examples & Calculator
- How to Find the Mode | Definition, Examples & Calculator
- How to Find the Range of a Data Set | Calculator & Formula
- Hypothesis Testing | A Step-by-Step Guide with Easy Examples
- Interval Data and How to Analyze It | Definitions & Examples
- Levels of Measurement | Nominal, Ordinal, Interval and Ratio
- Linear Regression in R | A Step-by-Step Guide & Examples
- Missing Data | Types, Explanation, & Imputation
- Multiple Linear Regression | A Quick Guide (Examples)
- Nominal Data | Definition, Examples, Data Collection & Analysis
- Normal Distribution | Examples, Formulas, & Uses
- Null and Alternative Hypotheses | Definitions & Examples
- One-way ANOVA | When and How to Use It (With Examples)
- Ordinal Data | Definition, Examples, Data Collection & Analysis
- Parameter vs Statistic | Definitions, Differences & Examples
- Pearson Correlation Coefficient (r) | Guide & Examples
- Poisson Distributions | Definition, Formula & Examples
- Probability Distribution | Formula, Types, & Examples
- Quartiles & Quantiles | Calculation, Definition & Interpretation
- Ratio Scales | Definition, Examples, & Data Analysis
- Simple Linear Regression | An Easy Introduction & Examples
- Skewness | Definition, Examples & Formula
- Statistical Power and Why It Matters | A Simple Introduction
- Student's t Table (Free Download) | Guide & Examples
- T-distribution: What it is and how to use it
- Test statistics | Definition, Interpretation, and Examples
- The Standard Normal Distribution | Calculator, Examples & Uses
- Two-Way ANOVA | Examples & When To Use It
- Type I & Type II Errors | Differences, Examples, Visualizations
- Understanding Confidence Intervals | Easy Examples & Formulas
- Understanding P values | Definition and Examples
- Variability | Calculating Range, IQR, Variance, Standard Deviation
- What is Effect Size and Why Does It Matter? (Examples)
- What Is Kurtosis? | Definition, Examples & Formula
- What Is Standard Error? | How to Calculate (Guide with Examples)

## What is your plagiarism score?

- Request new password
- Create a new account

## Basic SPSS Tutorial

Student resources, additional assignments for inferential statistics.

Suggested assignments for inferential statistics are designed to promote in-depth engagement with the material.

Download all additional assignments for inferential statistics, and the accompanying data files:

## Assignment 1

## Introduction to Statistics

(15 reviews)

David Lane, Rice University

Copyright Year: 2003

Publisher: David Lane

Language: English

## Formats Available

Conditions of use.

Learn more about reviews.

Reviewed by Terri Torres, professor, Oregon Institute of Technology on 8/17/23

This author covers all the topics that would be covered in an introductory statistics course plus some. I could imagine using it for two courses at my university, which is on the quarter system. I would rather have the problem of too many topics... read more

Comprehensiveness rating: 5 see less

This author covers all the topics that would be covered in an introductory statistics course plus some. I could imagine using it for two courses at my university, which is on the quarter system. I would rather have the problem of too many topics rather than too few.

Content Accuracy rating: 5

Yes, Lane is both thorough and accurate.

Relevance/Longevity rating: 5

What is covered is what is usually covered in an introductory statistics book. The only topic I may, given sufficient time, cover is bootstrapping.

Clarity rating: 5

The book is clear and well-written. For the trickier topics, simulations are included to help with understanding.

Consistency rating: 5

All is organized in a way that is consistent with the previous topic.

Modularity rating: 5

The text is organized in a way that easily enables navigation.

Organization/Structure/Flow rating: 5

The text is organized like most statistics texts.

Interface rating: 5

Easy navigation.

Grammatical Errors rating: 5

I didn't see any grammatical errors.

Cultural Relevance rating: 5

Nothing is included that is culturally insensitive.

The videos that accompany this text are short and easy to watch and understand. Videos should be short enough to teach, but not so long that they are tiresome. This text includes almost everything: videos, simulations, case studies---all nicely organized in one spot. In addition, Lane has promised to send an instructor's manual and slide deck.

Reviewed by Professor Sandberg, Professor, Framingham State University on 6/29/21

This text covers all the usual topics in an Introduction to Statistics for college students. In addition, it has some additional topics that are useful. read more

This text covers all the usual topics in an Introduction to Statistics for college students. In addition, it has some additional topics that are useful.

I did not find any errors.

Some of the examples are dated. And the frequent use of male/female examples need updating in terms of current gender splits.

I found it was easy to read and understand and I expect that students would also find the writing clear and the explanations accessible.

Even with different authors of chapter, the writing is consistent.

The text is well organized into sections making it easy to assign individual topics and sections.

The topics are presented in the usual order. Regression comes later in the text but there is a difference of opinions about whether to present it early with descriptive statistics for bivariate data or later with inferential statistics.

I had no problem navigating the text online.

The writing is grammatical correct.

I saw no issues that would be offensive.

I did like this text. It seems like it would be a good choice for most introductory statistics courses. I liked that the Monty Hall problem was included in the probability section. The author offers to provide an instructor's manual, PowerPoint slides and additional questions. These additional resources are very helpful and not always available with online OER texts.

Reviewed by Emilio Vazquez, Associate Professor, Trine University on 4/23/21

This appears to be an excellent textbook for an Introductory Course in Statistics. It covers subjects in enough depth to fulfill the needs of a beginner in Statistics work yet is not so complex as to be overwhelming. read more

This appears to be an excellent textbook for an Introductory Course in Statistics. It covers subjects in enough depth to fulfill the needs of a beginner in Statistics work yet is not so complex as to be overwhelming.

I found no errors in their discussions. Did not work out all of the questions and answers but my sampling did not reveal any errors.

Some of the examples may need updating depending on the times but the examples are still relevant at this time.

This is a Statistics text so a little dry. I found that the derivation of some of the formulas was not explained. However the background is there to allow the instructor to derive these in class if desired.

The text is consistent throughout using the same verbiage in various sections.

The text dose lend itself to reasonable reading assignments. For example the chapter (Chapter 3) on Summarizing Distributions covers Central Tendency and its associated components in an easy 20 pages with Measures of Variability making up most of the rest of the chapter and covering approximately another 20 pages. Exercises are available at the end of each chapter making it easy for the instructor to assign reading and exercises to be discussed in class.

The textbook flows easily from Descriptive to Inferential Statistics with chapters on Sampling and Estimation preceding chapters on hypothesis testing

I had no problems with navigation

All textbooks have a few errors but certainly nothing glaring or making text difficult

I saw no issues and I am part of a cultural minority in the US

Overall I found this to be a excellent in-depth overview of Statistical Theory, Concepts and Analysis. The length of the textbook appears to be more than adequate for a one-semester course in Introduction to Statistics. As I no longer teach a full statistics course but simply a few lectures as part of our Research Curriculum, I am recommending this book to my students as a good reference. Especially as it is available on-line and in Open Access.

Reviewed by Audrey Hickert, Assistant Professor, Southern Illinois University Carbondale on 3/29/21

All of the major topics of an introductory level statistics course for social science are covered. Background areas include levels of measurement and research design basics. Descriptive statistics include all major measures of central tendency and... read more

All of the major topics of an introductory level statistics course for social science are covered. Background areas include levels of measurement and research design basics. Descriptive statistics include all major measures of central tendency and dispersion/variation. Building blocks for inferential statistics include sampling distributions, the standard normal curve (z scores), and hypothesis testing sections. Inferential statistics include how to calculate confidence intervals, as well as conduct tests of one-sample tests of the population mean (Z- and t-tests), two-sample tests of the difference in population means (Z- and t-tests), chi square test of independence, correlation, and regression. Doesn’t include full probability distribution tables (e.g., t or Z), but those can be easily found online in many places.

I did not find any errors or issues of inaccuracy. When a particular method or practice is debated in the field, the authors acknowledge it (and provide citations in some circumstances).

Relevance/Longevity rating: 4

Basic statistics are standard, so the core information will remain relevant in perpetuity. Some of the examples are dated (e.g., salaries from 1999), but not problematic.

Clarity rating: 4

All of the key terms, formulas, and logic for statistical tests are clearly explained. The book sometimes uses different notation than other entry-level books. For example, the variance formula uses "M" for mean, rather than x-bar.

The explanations are consistent and build from and relate to corresponding sections that are listed in each unit.

Modularity is a strength of this text in both the PDF and interactive online format. Students can easily navigate to the necessary sections and each starts with a “Prerequisites” list of other sections in the book for those who need the additional background material. Instructors could easily compile concise sub-sections of the book for readings.

The presentation of topics differs somewhat from the standard introductory social science statistics textbooks I have used before. However, the modularity allows the instructor and student to work through the discrete sections in the desired order.

Interface rating: 4

For the most part the display of all images/charts is good and navigation is straightforward. One concern is that the organization of the Table of Contents does not exactly match the organizational outline at the start of each chapter in the PDF version. For example, sometimes there are more detailed sub-headings at the start of chapter and occasionally slightly different section headings/titles. There are also inconsistencies in section listings at start of chapters vs. start of sub-sections.

The text is easy to read and free from any obvious grammatical errors.

Although some of the examples are outdated, I did not review any that were offensive. One example of an outdated reference is using descriptive data on “Men per 100 Women” in U.S. cities as “useful if we are looking for an opposite-sex partner”.

This is a good introduction level statistics text book if you have a course with students who may be intimated by longer texts with more detailed information. Just the core basics are provided here and it is easy to select the sections you need. It is a good text if you plan to supplement with an array of your own materials (lectures, practice, etc.) that are specifically tailored to your discipline (e.g., criminal justice and criminology). Be advised that some formulas use different notation than other standard texts, so you will need to point that out to students if they differ from your lectures or assessment materials.

Reviewed by Shahar Boneh, Professor, Metropolitan State University of Denver on 3/26/21, updated 4/22/21

The textbook is indeed quite comprehensive. It can accommodate any style of introductory statistics course. read more

The textbook is indeed quite comprehensive. It can accommodate any style of introductory statistics course.

The text seems to be statistically accurate.

It is a little too extensive, which requires instructors to cover it selectively, and has a potential to confuse the students.

It is written clearly.

Consistency rating: 4

The terminology is fairly consistent. There is room for some improvement.

By the nature of the subject, the topics have to be presented in a sequential and coherent order. However, the book breaks things down quite effectively.

Organization/Structure/Flow rating: 3

Some of the topics are interleaved and not presented in the order I would like to cover them.

Good interface.

The grammar is ok.

The book seems to be culturally neutral, and not offensive in any way.

I really liked the simulations that go with the book. Parts of the book are a little too advanced for students who are learning statistics for the first time.

Reviewed by Julie Gray, Adjunct Assistant Professor, University of Texas at Arlington on 2/26/21

The textbook is for beginner-level students. The concept development is appropriate--there is always room to grow to high higher level, but for an introduction, the basics are what is needed. This is a well-thought-through OER textbook project by... read more

The textbook is for beginner-level students. The concept development is appropriate--there is always room to grow to high higher level, but for an introduction, the basics are what is needed. This is a well-thought-through OER textbook project by Dr. Lane and colleagues. It is obvious that several iterations have only made it better.

I found all the material accurate.

Essentially, statistical concepts at the introductory level are accepted as universal. This suggests that the relevance of this textbook will continue for a long time.

The book is well written for introducing beginners to statistical concepts. The figures, tables, and animated examples reinforce the clarity of the written text.

Yes, the information is consistent; when it is introduced in early chapters it ties in well in later chapters that build on and add more understanding for the topic.

Modularity rating: 4

The book is well-written with attention to modularity where possible. Due to the nature of statistics, that is not always possible. The content is presented in the order that I usually teach these concepts.

The organization of the book is good, I particularly like the sample lecture slide presentations and the problem set with solutions for use in quizzes and exams. These are available by writing to the author. It is wonderful to have access to these helpful resources for instructors to use in preparation.

I did not find any interface issues.

The book is well written. In my reading I did not notice grammatical errors.

For this subject and in the examples given, I did not notice any cultural issues.

For the field of social work where qualitative data is as common as quantitative, the importance of giving students the rationale or the motivation to learn the quantitative side is understated. To use this text as an introductory statistics OER textbook in a social work curriculum, the instructor will want to bring in field-relevant examples to engage and motivate students. The field needs data-driven decision making and evidence-based practices to become more ubiquitous than not. Preparing future social workers by teaching introductory statistics is essential to meet that goal.

Reviewed by Mamata Marme, Assistant Professor, Augustana College on 6/25/19

This textbook offers a fairly comprehensive summary of what should be discussed in an introductory course in Statistics. The statistical literacy exercises are particularly interesting. It would be helpful to have the statistical tables... read more

Comprehensiveness rating: 4 see less

This textbook offers a fairly comprehensive summary of what should be discussed in an introductory course in Statistics. The statistical literacy exercises are particularly interesting. It would be helpful to have the statistical tables attached in the same package, even though they are available online.

The terminology and notation used in the textbook is pretty standard. The content is accurate.

The statistical literacy example are up to date but will need to be updated fairly regularly to keep the textbook fresh. The applications within the chapter are accessible and can be used fairly easily over a couple of editions.

The textbook does not necessarily explain the derivation of some of the formulae and this will need to be augmented by the instructor in class discussion. What is beneficial is that there are multiple ways that a topic is discussed using graphs, calculations and explanations of the results. Statistics textbooks have to cover a wide variety of topics with a fair amount of depth. To do this concisely is difficult. There is a fine line between being concise and clear, which this textbook does well, and being somewhat dry. It may be up to the instructor to bring case studies into the readings we are going through the topics rather than wait until the end of the chapter.

The textbook uses standard notation and terminology. The heading section of each chapter is closely tied to topics that are covered. The end of chapter problems and the statistical literacy applications are closely tied to the material covered.

The authors have done a good job treating each chapter as if they stand alone. The lack of connection to a past reference may create a sense of disconnect between the topics discussed

The text's "modularity" does make the flow of the material a little disconnected. If would be better if there was accountability of what a student should already have learnt in a different section. The earlier material is easy to find but not consistently referred to in the text.

I had no problem with the interface. The online version is more visually interesting than the pdf version.

I did not see any grammatical errors.

Cultural Relevance rating: 4

I am not sure how to evaluate this. The examples are mostly based on the American experience and the data alluded to mostly domestic. However, I am not sure if that creates a problem in understanding the methodology.

Overall, this textbook will cover most of the topics in a survey of statistics course.

Reviewed by Alexandra Verkhovtseva, Professor, Anoka-Ramsey Community College on 6/3/19

This is a comprehensive enough text, considering that it is not easy to create a comprehensive statistics textbook. It is suitable for an introductory statistics course for non-math majors. It contains twenty-one chapters, covering the wide range... read more

This is a comprehensive enough text, considering that it is not easy to create a comprehensive statistics textbook. It is suitable for an introductory statistics course for non-math majors. It contains twenty-one chapters, covering the wide range of intro stats topics (and some more), plus the case studies and the glossary.

The content is pretty accurate, I did not find any biases or errors.

The book contains fairly recent data presented in the form of exercises, examples and applications. The topics are up-to-date, and appropriate technology is used for examples, applications, and case studies.

The language is simple and clear, which is a good thing, since students are usually scared of this class, and instructors are looking for something to put them at ease. I would, however, try to make it a little more interesting, exciting, or may be even funny.

Consistency is good, the book has a great structure. I like how each chapter has prerequisites and learner outcomes, this gives students a good idea of what to expect. Material in this book is covered in good detail.

The text can be easily divided into sub-sections, some of which can be omitted if needed. The chapter on regression is covered towards the end (chapter 14), but part of it can be covered sooner in the course.

The book contains well organized chapters that makes reading through easy and understandable. The order of chapters and sections is clear and logical.

The online version has many functions and is easy to navigate. This book also comes with a PDF version. There is no distortion of images or charts. The text is clean and clear, the examples provided contain appropriate format of data presentation.

No grammatical errors found.

The text uses simple and clear language, which is helpful for non-native speakers. I would include more culturally-relevant examples and case studies. Overall, good text.

In all, this book is a good learning experience. It contains tools and techniques that free and easy to use and also easy to modify for both, students and instructors. I very much appreciate this opportunity to use this textbook at no cost for our students.

Reviewed by Dabrina Dutcher, Assistant Professor, Bucknell University on 3/4/19

This is a reasonably thorough first-semester statistics book for most classes. It would have worked well for the general statistics courses I have taught in the past but is not as suitable for specialized introductory statistics courses for... read more

This is a reasonably thorough first-semester statistics book for most classes. It would have worked well for the general statistics courses I have taught in the past but is not as suitable for specialized introductory statistics courses for engineers or business applications. That is OK, they have separate texts for that! The only sections that feel somewhat light in terms of content are the confidence intervals and ANOVA sections. Given that these topics are often sort of crammed in at the end of many introductory classes, that might not be problematic for many instructors. It should also be pointed out that while there are a couple of chapters on probability, this book spends presents most formulas as "black boxes" rather than worry about the derivation or origin of the formulas. The probability sections do not include any significant combinatorics work, which is sometimes included at this level.

I did not find any errors in the formulas presented but I did not work many end-of-chapter problems to gauge the accuracy of their answers.

There isn't much changing in the introductory stats world, so I have no concerns about the book becoming outdated rapidly. The examples and problems still feel relevant and reasonably modern. My only concern is that the statistical tool most often referenced in the book are TI-83/84 type calculators. As students increasingly buy TI-89s or Inspires, these sections of the book may lose relevance faster than other parts.

Solid. The book gives a list of key terms and their definitions at the end of each chapter which is a nice feature. It also has a formula review at the end of each chapter. I can imagine that these are heavily used by students when studying! Formulas are easy to find and read and are well defined. There are a few areas that I might have found frustrating as a student. For example, the explanation for the difference in formulas for a population vs sample standard deviation is quite weak. Again, this is a book that focuses on sort of a "black-box" approach but you may have to supplement such sections for some students.

I did not detect any problems with inconsistent symbol use or switches in terminology.

Modularity rating: 3

This low rating should not be taken as an indicator of an issue with this book but would be true of virtually any statistics book. Different books still use different variable symbols even for basic calculated statistics. So trying to use a chapter of this book without some sort of symbol/variable cheat-sheet would likely be frustrating to the students.

However, I think it would be possible to skip some chapters or use the chapters in a different order without any loss of functionality.

This book uses a very standard order for the material. The chapter on regressions comes later than it does in some texts but it doesn't really matter since that chapter never seems to fit smoothly anywhere.

There are numerous end of chapter problems, some with answers, available in this book. I'm vacillating on whether these problems would be more useful if they were distributed after each relevant section or are better clumped at the end of the whole chapter. That might be a matter of individual preference.

I did not detect any problems.

I found no errors. However, there were several sections where the punctuation seemed non-ideal. This did not affect the over-all useability of the book though

I'm not sure how well this book would work internationally as many of the examples contain domestic (American) references. However, I did not see anything offensive or biased in the book.

Reviewed by Ilgin Sager, Assistant Professor, University of Missouri - St. Louis on 1/14/19

As the title implies, this is a brief introduction textbook. It covers the fundamental of the introductory statistics, however not a comprehensive text on the subject. A teacher can use this book as the sole text of an introductory statistics.... read more

As the title implies, this is a brief introduction textbook. It covers the fundamental of the introductory statistics, however not a comprehensive text on the subject. A teacher can use this book as the sole text of an introductory statistics. The prose format of definitions and theorems make theoretical concepts accessible to non-math major students. The textbook covers all chapters required in this level course.

It is accurate; the subject matter in the examples to be up to date, is timeless and wouldn't need to be revised in future editions; there is no error except a few typographical errors. There are no logic errors or incorrect explanations.

This text will remain up to date for a long time since it has timeless examples and exercises, it wouldn't be outdated. The information is presented clearly with a simple way and the exercises are beneficial to follow the information.

The material is presented in a clear, concise manner. The text is easy readable for the first time statistics student.

The structure of the text is very consistent. Topics are presented with examples, followed by exercises. Problem sets are appropriate for the level of learner.

When the earlier matters need to be referenced, it is easy to find; no trouble reading the book and finding results, it has a consistent scheme. This book is set very well in sections.

The text presents the information in a logical order.

The learner can easily follow up the material; there is no interface problem.

There is no logic errors and incorrect explanations, a few typographical errors is just to be ignored.

Not applicable for this textbook.

Reviewed by Suhwon Lee, Associate Teaching Professor, University of Missouri on 6/19/18

This book is pretty comprehensive for being a brief introductory book. This book covers all necessary content areas for an introduction to Statistics course for non-math majors. The text book provides an effective index, plenty of exercises,... read more

This book is pretty comprehensive for being a brief introductory book. This book covers all necessary content areas for an introduction to Statistics course for non-math majors. The text book provides an effective index, plenty of exercises, review questions, and practice tests. It provides references and case studies. The glossary and index section is very helpful for students and can be used as a great resource.

Content appears to be accurate throughout. Being an introductory book, the book is unbiased and straight to the point. The terminology is standard.

The content in textbook is up to date. It will be very easy to update it or make changes at any point in time because of the well-structured contents in the textbook.

The author does a great job of explaining nearly every new term or concept. The book is easy to follow, clear and concise. The graphics are good to follow. The language in the book is easily understandable. I found most instructions in the book to be very detailed and clear for students to follow.

Overall consistency is good. It is consistent in terms of terminology and framework. The writing is straightforward and standardized throughout the text and it makes reading easier.

The authors do a great job of partitioning the text and labeling sections with appropriate headings. The table of contents is well organized and easily divisible into reading sections and it can be assigned at different points within the course.

Organization/Structure/Flow rating: 4

Overall, the topics are arranged in an order that follows natural progression in a statistics course with some exception. They are addressed logically and given adequate coverage.

The text is free of any issues. There are no navigation problems nor any display issues.

The text contains no grammatical errors.

The text is not culturally insensitive or offensive in any way most of time. Some examples might need to consider citing the sources or use differently to reflect current inclusive teaching strategies.

Overall, it's well-written and good recourse to be an introduction to statistical methods. Some materials may not need to be covered in an one-semester course. Various examples and quizzes can be a great recourse for instructor.

Reviewed by Jenna Kowalski, Mathematics Instructor, Anoka-Ramsey Community College on 3/27/18

The text includes the introductory statistics topics covered in a college-level semester course. An effective index and glossary are included, with functional hyperlinks. read more

The text includes the introductory statistics topics covered in a college-level semester course. An effective index and glossary are included, with functional hyperlinks.

Content Accuracy rating: 3

The content of this text is accurate and error-free, based on a random sampling of various pages throughout the text. Several examples included information without formal citation, leading the reader to potential bias and discrimination. These examples should be corrected to reflect current values of inclusive teaching.

The text contains relevant information that is current and will not become outdated in the near future. The statistical formulas and calculations have been used for centuries. The examples are direct applications of the formulas and accurately assess the conceptual knowledge of the reader.

The text is very clear and direct with the language used. The jargon does require a basic mathematical and/or statistical foundation to interpret, but this foundational requirement should be met with course prerequisites and placement testing. Graphs, tables, and visual displays are clearly labeled.

The terminology and framework of the text is consistent. The hyperlinks are working effectively, and the glossary is valuable. Each chapter contains modules that begin with prerequisite information and upcoming learning objectives for mastery.

The modules are clearly defined and can be used in conjunction with other modules, or individually to exemplify a choice topic. With the prerequisite information stated, the reader understands what prior mathematical understanding is required to successfully use the module.

The topics are presented well, but I recommend placing Sampling Distributions, Advanced Graphs, and Research Design ahead of Probability in the text. I think this rearranged version of the index would better align with current Introductory Statistics texts. The structure is very organized with the prerequisite information stated and upcoming learner outcomes highlighted. Each module is well-defined.

Adding an option of returning to the previous page would be of great value to the reader. While progressing through the text systematically, this is not an issue, but when the reader chooses to skip modules and read select pages then returning to the previous state of information is not easily accessible.

No grammatical errors were found while reviewing select pages of this text at random.

Cultural Relevance rating: 3

Several examples contained data that were not formally cited. These examples need to be corrected to reflect current inclusive teaching strategies. For example, one question stated that “while men are XX times more likely to commit murder than women, …” This data should be cited, otherwise the information can be interpreted as biased and offensive.

An included solutions manual for the exercises would be valuable to educators who choose to use this text.

Reviewed by Zaki Kuruppalil, Associate Professor, Ohio University on 2/1/18

This is a comprehensive book on statistical methods, its settings and most importantly the interpretation of the results. With the advent of computers and software’s, complex statistical analysis can be done very easily. But the challenge is the... read more

This is a comprehensive book on statistical methods, its settings and most importantly the interpretation of the results. With the advent of computers and software’s, complex statistical analysis can be done very easily. But the challenge is the knowledge of how to set the case, setting parameters (for example confidence intervals) and knowing its implication on the interpretation of the results. If not done properly this could lead to deceptive inferences, inadvertently or purposely. This book does a great job in explaining the above using many examples and real world case studies. If you are looking for a book to learn and apply statistical methods, this is a great one. I think the author could consider revising the title of the book to reflect the above, as it is more than just an introduction to statistics, may be include the word such as practical guide.

The contents of the book seems accurate. Some plots and calculations were randomly selected and checked for accuracy.

The book topics are up to date and in my opinion, will not be obsolete in the near future. I think the smartest thing the author has done is, not tied the book with any particular software such as minitab or spss . No matter what the software is, standard deviation is calculated the same way as it is always. The only noticeable exception in this case was using the Java Applet for calculating Z values in page 261 and in page 416 an excerpt of SPSS analysis is provided for ANOVA calculations.

The contents and examples cited are clear and explained in simple language. Data analysis and presentation of the results including mathematical calculations, graphical explanation using charts, tables, figures etc are presented with clarity.

Terminology is consistant. Framework for each chapter seems consistent with each chapter beginning with a set of defined topics, and each of the topic divided into modules with each module having a set of learning objectives and prerequisite chapters.

The text book is divided into chapters with each chapter further divided into modules. Each of the modules have detailed learning objectives and prerequisite required. So you can extract a portion of the book and use it as a standalone to teach certain topics or as a learning guide to apply a relevant topic.

Presentation of the topics are well thought and are presented in a logical fashion as if it would be introduced to someone who is learning the contents. However, there are some issues with table of contents and page numbers, for example chapter 17 starts in page 597 not 598. Also some tables and figures does not have a number, for instance the graph shown in page 114 does not have a number. Also it would have been better if the chapter number was included in table and figure identification, for example Figure 4-5 . Also in some cases, for instance page 109, the figures and titles are in two different pages.

No major issues. Only suggestion would be, since each chapter has several modules, any means such as a header to trace back where you are currently, would certainly help.

Grammatical Errors rating: 4

Easy to read and phrased correctly in most cases. Minor grammatical errors such as missing prepositions etc. In some cases the author seems to have the habbit of using a period after the decimal. For instance page 464, 467 etc. For X = 1, Y' = (0.425)(1) + 0.785 = 1.21. For X = 2, Y' = (0.425)(2) + 0.785 = 1.64.

However it contains some statements (even though given as examples) that could be perceived as subjective, which the author could consider citing the sources. For example from page 11: Statistics include numerical facts and figures. For instance: • The largest earthquake measured 9.2 on the Richter scale. • Men are at least 10 times more likely than women to commit murder. • One in every 8 South Africans is HIV positive. • By the year 2020, there will be 15 people aged 65 and over for every new baby born.

Solutions for the exercises would be a great teaching resource to have

Reviewed by Randy Vander Wal, Professor, The Pennsylvania State University on 2/1/18

As a text for an introductory course, standard topics are covered. It was nice to see some topics such as power, sampling, research design and distribution free methods covered, as these are often omitted in abbreviated texts. Each module... read more

As a text for an introductory course, standard topics are covered. It was nice to see some topics such as power, sampling, research design and distribution free methods covered, as these are often omitted in abbreviated texts. Each module introduces the topic, has appropriate graphics, illustration or worked example(s) as appropriate and concluding with many exercises. An instructor’s manual is available by contacting the author. A comprehensive glossary provides definitions for all the major terms and concepts. The case studies give examples of practical applications of statistical analyses. Many of the case studies contain the actual raw data. To note is that the on-line e-book provides several calculators for the essential distributions and tests. These are provided in lieu of printed tables which are not included in the pdf. (Such tables are readily available on the web.)

The content is accurate and error free. Notation is standard and terminology is used accurately, as are the videos and verbal explanations therein. Online links work properly as do all the calculators. The text appears neutral and unbiased in subject and content.

The text achieves contemporary relevance by ending each section with a Statistical Literacy example, drawn from contemporary headlines and issues. Of course, the core topics are time proven. There is no obvious material that may become “dated”.

The text is very readable. While the pdf text may appear “sparse” by absence varied colored and inset boxes, pictures etc., the essential illustrations and descriptions are provided. Meanwhile for this same content the on-line version appears streamlined, uncluttered, enhancing the value of the active links. Moreover, the videos provide nice short segments of “active” instruction that are clear and concise. Despite being a mathematical text, the text is not overly burdened by formulas and numbers but rather has “readable feel”.

This terminology and symbol use are consistent throughout the text and with common use in the field. The pdf text and online version are also consistent by content, but with the online e-book offering much greater functionality.

The chapters and topics may be used in a selective manner. Certain chapters have no pre-requisite chapter and in all cases, those required are listed at the beginning of each module. It would be straightforward to select portions of the text and reorganize as needed. The online version is highly modular offering students both ease of navigation and selection of topics.

Chapter topics are arranged appropriately. In an introductory statistics course, there is a logical flow given the buildup to the normal distribution, concept of sampling distributions, confidence intervals, hypothesis testing, regression and additional parametric and non-parametric tests. The normal distribution is central to an introductory course. Necessary precursor topics are covered in this text, while its use in significance and hypothesis testing follow, and thereafter more advanced topics, including multi-factor ANOVA.

Each chapter is structured with several modules, each beginning with pre-requisite chapter(s), learning objectives and concluding with Statistical Literacy sections providing a self-check question addressing the core concept, along with answer, followed by an extensive problem set. The clear and concise learning objectives will be of benefit to students and the course instructor. No solutions or answer key is provided to students. An instructor’s manual is available by request.

The on-line interface works well. In fact, I was pleasantly surprised by its options and functionality. The pdf appears somewhat sparse by comparison to publisher texts, lacking pictures, colored boxes, etc. But the on-line version has many active links providing definitions and graphic illustrations for key terms and topics. This can really facilitate learning as making such “refreshers” integral to the new material. Most sections also have short videos that are professionally done, with narration and smooth graphics. In this way, the text is interactive and flexible, offering varied tools for students. To note is that the interactive e-book works for both IOS and OS X.

The text in pdf form appeared to free of grammatical errors, as did the on-line version, text, graphics and videos.

This text contains no culturally insensitive or offensive content. The focus of the text is on concepts and explanation.

The text would be a great resource for students. The full content would be ambitious for a 1-semester course, such use would be unlikely. The text is clearly geared towards students with no statistics background nor calculus. The text could be used in two styles of course. For 1st year students early chapters on graphs and distributions would be the starting point, omitting later chapters on Chi-square, transformations, distribution-free and size effect chapters. Alternatively, for upper level students the introductory chapters could be bypassed with the latter chapters then covered to completion.

This text adopts a descriptive style of presentation with topics well and fully explained, much like the “Dummy series”. For this, it may seem a bit “wordy”, but this can well serve students and notably it complements powerpoint slides that are generally sparse on written content. This text could be used as the primary text, for regular lectures, or as reference for a “flipped” class. The e-book videos are an enabling tool if this approach is adopted.

Reviewed by David jabon, Associate Professor, DePaul University on 8/15/17

This text covers all the standard topics in a semester long introductory course in statistics. It is particularly well indexed and very easy to navigate. There is comprehensive hyperlinked glossary. read more

This text covers all the standard topics in a semester long introductory course in statistics. It is particularly well indexed and very easy to navigate. There is comprehensive hyperlinked glossary.

The material is completely accurate. There are no errors. The terminology is standard with one exception: the book calls what most people call the interquartile range, the H-spread in a number of places. Ideally, the term "interquartile range" would be used in place of every reference to "H-spread." "Interquartile range" is simply a better, more descriptive term of the concept that it describes. It is also more commonly used nowadays.

This book came out a number of years ago, but the material is still up to date. Some more recent case studies have been added.

The writing is very clear. There are also videos for almost every section. The section on boxplots uses a lot of technical terms that I don't find are very helpful for my students (hinge, H-spread, upper adjacent value).

The text is internally consistent with one exception that I noted (the use of the synonymous words "H-spread" and "interquartile range").

The text book is brokenly into very short sections, almost to a fault. Each section is at most two pages long. However at the end of each of these sections there are a few multiple choice questions to test yourself. These questions are a very appealing feature of the text.

The organization, in particular the ordering of the topics, is rather standard with a few exceptions. Boxplots are introduced in Chapter II before the discussion of measures of center and dispersion. Most books introduce them as part of discussion of summaries of data using measure of center and dispersion. Some statistics instructors may not like the way the text lumps all of the sampling distributions in a single chapter (sampling distribution of mean, sampling distribution for the difference of means, sampling distribution of a proportion, sampling distribution of r). I have tried this approach, and I now like this approach. But it is a very challenging chapter for students.

The book's interface has no features that distracted me. Overall the text is very clean and spare, with no additional distracting visual elements.

The book contains no grammatical errors.

The book's cultural relevance comes out in the case studies. As of this writing there are 33 such case studies, and they cover a wide range of issues from health to racial, ethnic, and gender disparity.

Each chapter as a nice set of exercises with selected answers. The thirty three case studies are excellent and can be supplement with some other online case studies. An instructor's manual and PowerPoint slides can be obtained by emailing the author. There are direct links to online simulations within the text. This text is very high quality textbook in every way.

## Table of Contents

- 1. Introduction
- 2. Graphing Distributions
- 3. Summarizing Distributions
- 4. Describing Bivariate Data
- 5. Probability
- 6. Research Design
- 7. Normal Distributions
- 8. Advanced Graphs
- 9. Sampling Distributions
- 10. Estimation
- 11. Logic of Hypothesis Testing
- 12. Testing Means
- 14. Regression
- 15. Analysis of Variance
- 16. Transformations
- 17. Chi Square
- 18. Distribution-Free Tests
- 19. Effect Size
- 20. Case Studies
- 21. Glossary

## Ancillary Material

- Ancillary materials are available by contacting the author or publisher .

## About the Book

Introduction to Statistics is a resource for learning and teaching introductory statistics. This work is in the public domain. Therefore, it can be copied and reproduced without limitation. However, we would appreciate a citation where possible. Please cite as: Online Statistics Education: A Multimedia Course of Study (http://onlinestatbook.com/). Project Leader: David M. Lane, Rice University. Instructor's manual, PowerPoint Slides, and additional questions are available.

## About the Contributors

David Lane is an Associate Professor in the Departments of Psychology, Statistics, and Management at the Rice University. Lane is the principal developer of this resource although many others have made substantial contributions. This site was developed at Rice University, University of Houston-Clear Lake, and Tufts University.

## Contribute to this Page

- SPSS for Business Analytics: Student's Comprehensive Guide

## SPSS for Business Analytics: A Curriculum-Based Approach

In the ever-evolving landscape of business, the significance of data in decision-making cannot be overstated. As data volumes and intricacies burgeon, the demand for robust analytics tools becomes paramount. Among these, IBM's Statistical Package for the Social Sciences (SPSS) has emerged as a prominent player in the realm of business analytics. This blog is dedicated to equipping students with a comprehensive grasp of SPSS, emphasizing a curriculum-based approach tailored to empower them not only in tackling assignments but also in excelling in the multifaceted domain of business analytics. Whether you require help with your SPSS assignment or seek to master the intricacies of business analytics using SPSS, this blog serves as a valuable resource to enhance your understanding and proficiency in leveraging data for informed decision-making in the business world.

Understanding SPSS involves navigating its user-friendly interface and harnessing its diverse statistical techniques. Through a structured curriculum, students will gain proficiency in installing and setting up SPSS, mastering the basics of data entry and management, and advancing to exploratory data analysis (EDA). Building upon this foundation, the blog will delve into advanced analytics, covering inferential statistics, regression analysis, and multivariate analysis techniques. Real-world applications and case studies will illustrate how SPSS is applied to address business challenges, providing students with tangible insights.

This comprehensive curriculum-based approach ensures that students not only grasp the theoretical aspects of SPSS but also develop practical skills that are invaluable in solving assignments and making informed decisions in the dynamic landscape of business analytics.

## Understanding the Basics of SPSS

SPSS, or the Statistical Package for the Social Sciences, stands as a powerful ally in the world of business analytics. To embark on a fruitful journey with SPSS, it's imperative to delve into its fundamental aspects. This section will serve as a compass, guiding students through the initial steps of installing and setting up SPSS. By understanding the system requirements and navigating the licensing process, students can ensure a smooth initiation into the realm of statistical analysis.

Once the groundwork is laid, the focus shifts to the basics of data entry and management. This involves creating a structured data file, entering information accurately, and learning the intricacies of variable types and properties. A meticulous approach to data cleaning and transformation is emphasized, as clean datasets are the bedrock for meaningful analysis.

Moving beyond mere data entry, students will explore the realm of exploratory data analysis (EDA). This involves techniques to summarize and visualize data effectively, such as generating descriptive statistics and creating visual representations. This section acts as a springboard, propelling students into the intricate world of SPSS analytics with a solid understanding of its foundational elements.

## Overview of SPSS

SPSS, an acronym for the Statistical Package for the Social Sciences, stands as a testament to IBM's commitment to providing a robust statistical software package. Developed by IBM, this tool has become a cornerstone in the world of data analysis. Initially tailored for applications in the social sciences, SPSS has undergone a transformative evolution, adapting itself into a versatile analytical tool applicable across diverse disciplines, with a particular stronghold in business analytics. Its widespread adoption can be attributed not only to its origin in academia but also to its user-friendly interface and an extensive repertoire of statistical techniques.

## Installation and Setup

Before students embark on their analytical journey with SPSS, a foundational step involves acquainting themselves with the installation and setup process. This phase serves as a gateway to the myriad capabilities of SPSS, ensuring that users can seamlessly integrate the software into their systems. Navigating through this section, students will be guided through a step-by-step process, demystifying the complexities of SPSS installation. Armed with this knowledge, they can confidently initiate their statistical analyses, laying a solid groundwork for their exploration of the software's capabilities in subsequent modules. This understanding of installation and setup is pivotal, forming the bedrock for a comprehensive grasp of SPSS functionalities.

## 1: System Requirements

Understanding the system requirements is a crucial initial step in the SPSS journey. For optimal performance, students must acquaint themselves with the specific prerequisites for running SPSS on their systems. This encompasses key details about compatible operating systems, minimum memory specifications, and required disk space. By meticulously adhering to these system requirements, students pave the way for a seamless SPSS experience. This not only prevents potential technical issues but also lays the foundation for efficient data analysis, ensuring that their focus remains on deriving meaningful insights rather than grappling with software compatibility challenges.

## 2: Licensing and Activation

The prowess of SPSS comes to life through a valid license and activation. This H3 subsection guides students through the crucial process of obtaining and activating their SPSS license. Clear and concise instructions will be provided, empowering students to unlock the full spectrum of SPSS capabilities for their assignments. A valid license not only ensures compliance but also grants access to the myriad statistical tools within SPSS, enabling students to explore, analyze, and derive valuable insights from their datasets. This foundational knowledge of licensing and activation serves as a gateway to a rich and comprehensive SPSS experience, facilitating a seamless integration of this powerful tool into their analytical toolkit.

## Building a Strong Foundation in SPSS

In the intricate realm of data analytics, establishing a robust foundation is paramount, and this holds particularly true for mastering the capabilities of SPSS. As we venture into the various facets of building this foundation, the focus shifts to equipping students with the fundamental skills that serve as the building blocks for proficient SPSS utilization.

## Basics of Data Entry and Management

Before diving into advanced analytics, students need to master the basics of data entry and management in SPSS. This section becomes the cornerstone of their SPSS journey, as it covers essential topics such as creating a data file, entering data, and organizing variables. A hands-on approach will be emphasized, allowing students to actively engage with the software and develop practical skills that extend beyond theoretical understanding.

As students delve into the nuances of data entry, they will learn the importance of maintaining data integrity and cleanliness. Creating a structured data file lays the groundwork for efficient analysis, and understanding variable types and properties becomes pivotal. This hands-on exploration ensures that students not only comprehend the theoretical aspects but also gain confidence in navigating SPSS for effective data management—a skill set indispensable in the broader landscape of business analytics.

## 1: Variable Types and Properties

Understanding different variable types is fundamental to effective data management. In SPSS, variables can be numeric or categorical, each serving distinct roles in analysis. This subsection elucidates these distinctions, guiding students on how to assign variable properties such as labels and formats. This knowledge proves invaluable in shaping the data for subsequent analyses, ensuring accuracy and relevance in the interpretation of results.

## 2: Data Cleaning and Transformation

The journey towards insightful analysis begins with clean and organized data. This subsection focuses on the critical aspect of data cleaning and transformation. Students will delve into techniques for identifying and handling missing values, outliers, and other anomalies that could skew results. Furthermore, the section delves into data transformation methods, equipping students with the skills needed to prepare datasets for advanced statistical analysis. As students progress in their SPSS journey, these foundational practices become indispensable in ensuring the reliability and validity of their analyses.

## Exploratory Data Analysis (EDA) with SPSS

Exploratory Data Analysis (EDA) stands as a critical pillar in the analytics process, offering students a profound insight into the intricacies of their datasets. In this section, we will meticulously lead students through a comprehensive journey within SPSS, unraveling a diverse array of techniques designed for both data summarization and visualization. Within the realm of descriptive statistics, students will explore the nuances of mean, median, and standard deviation, gaining a nuanced understanding of their data's central tendencies and dispersion. Furthermore, this segment will delve into the realm of frequency distributions, unveiling the distribution patterns of categorical variables. The journey continues with the exploration of graphical representations, where students will harness the power of histograms, boxplots, and scatterplots to visually encapsulate the essence of their data. By fostering a deep connection between theory and practical application, this section ensures that students are well-equipped to derive meaningful insights from their datasets using the versatile tools embedded in SPSS.

## 1: Creating Descriptive Statistics

In the realm of statistical analysis, creating descriptive statistics serves as a foundational step for extracting meaningful insights from datasets. SPSS, as a robust tool, empowers students with a diverse set of tools to delve into the numerical characteristics of their data. This includes computing key metrics such as mean, median, and standard deviation, which not only provide a snapshot of central tendencies but also illuminate the variability within the data. By adeptly navigating these functions, students gain a deeper understanding of their datasets, enabling them to identify subtle patterns or trends that may inform subsequent analyses.

## 2: Data Visualization in SPSS

Moving beyond numerical summaries, the art of data visualization takes center stage in SPSS. Visualizing data is a powerful means of conveying complex information in an accessible format. Within SPSS, students can explore a plethora of visualization options, including the creation of visually compelling histograms, informative boxplots, and insightful scatterplots. This subsection will guide students on how to harness SPSS's visualization capabilities effectively, emphasizing the importance of selecting the most appropriate method based on the unique characteristics of their data and the specific analytical goals they aim to achieve. Through hands-on exploration, students will discover the transformative impact of visual representation in enhancing the interpretability of their datasets.

## Advanced Analytics with SPSS

Once students have grasped the fundamental aspects of SPSS, the journey into advanced analytics becomes both exciting and essential for a comprehensive understanding of statistical methodologies. This section delves into the sophisticated tools SPSS offers, empowering students to unravel intricate insights from complex datasets.

In this advanced analytics realm, SPSS acts as a potent ally, guiding students through inferential statistics, where they transition from merely describing data to making informed predictions about populations. Through techniques like t-tests, ANOVA, and regression analysis, students gain the prowess to draw meaningful conclusions from their data.

As the learning journey progresses, students explore multivariate analysis techniques, such as factor analysis, cluster analysis, and discriminant analysis. These tools enable them to unravel hidden patterns, classify cases, and derive nuanced insights that go beyond the capabilities of basic statistical methods.

Through a curriculum-based approach, this section equips students not just with technical skills but with a strategic mindset, preparing them to tackle complex business challenges and contribute meaningfully to the evolving landscape of data-driven decision-making. Advanced analytics with SPSS becomes a gateway to unlocking the full potential of statistical analysis in diverse professional domains.

## Inferential Statistics in SPSS

With a robust foundation established, students can seamlessly advance to the realm of inferential statistics within the SPSS environment. Inferential statistics serve as a powerful gateway, allowing students to extrapolate conclusions about broader populations based on carefully sampled data. In this comprehensive section, we will delve into the intricacies of various inferential statistical techniques that SPSS offers, providing students with a nuanced understanding of tools like t-tests, ANOVA, and regression analysis.

## 1: Conducting Hypothesis Tests

In the realm of hypothesis testing, students will learn to formulate precise hypotheses, select the appropriate test within SPSS, and interpret results effectively. Practical examples and step-by-step guidance will empower students to confidently apply hypothesis testing, a fundamental skill in drawing meaningful inferences from their data.

## 2: Regression Analysis in SPSS

The exploration of regression analysis will extend beyond the basics, encompassing advanced concepts such as multiple regression and logistical regression. Through hands-on exercises, students will gain proficiency in employing regression models within SPSS, enabling them to analyze complex relationships and make informed predictions based on their data. This in-depth coverage ensures that students not only grasp the theoretical underpinnings but also acquire practical skills for real-world applications.

## Multivariate Analysis Techniques

To tackle complex business challenges, students need to explore multivariate analysis techniques. This section will introduce methods such as factor analysis, cluster analysis, and discriminant analysis available in SPSS, providing students with a diverse toolkit for advanced analytics.

## 1: Factor Analysis in SPSS

Factor analysis is employed to identify underlying factors that explain patterns of correlations among variables. Students will learn how to perform factor analysis in SPSS, interpret factor loadings, and make informed decisions based on the results.

## 2: Cluster Analysis and Discriminant Analysis

Cluster analysis helps identify natural groupings within data, while discriminant analysis is useful for classifying cases into predefined groups. This subsection will delve into the application of these techniques using SPSS, equipping students with the skills to solve assignments involving complex datasets.

## Applying SPSS to Business Challenges

In this section, we delve into compelling real-world applications and case studies that vividly demonstrate the practical relevance of SPSS in addressing diverse business challenges. From marketing analytics optimizing customer engagement to financial analytics mitigating risks in volatile markets, each case study illuminates the strategic use of SPSS. These examples not only showcase the versatility of SPSS but also inspire students to bridge the gap between theory and application, empowering them to approach assignments with a practical mindset honed by insights from actual industry scenarios. Let's explore the transformative impact of SPSS on business analytics through these illuminating case studies.

## 1: Marketing Analytics with SPSS

This subsection will focus on how SPSS can be applied in marketing analytics, covering topics such as customer segmentation, market basket analysis, and predictive modeling. Students will gain insights into how businesses use SPSS to optimize marketing strategies and enhance customer engagement.

## 2: Financial Analytics and Risk Management

In the realm of finance, SPSS proves valuable for risk assessment, fraud detection, and financial forecasting. Case studies in this section will highlight how financial institutions utilize SPSS to make informed decisions and mitigate risks in a volatile market.

In conclusion, achieving mastery in SPSS for business analytics requires a deliberate and curriculum-focused strategy. This blog has meticulously outlined a roadmap, encompassing fundamental SPSS principles, the establishment of a robust foundation, delving into advanced analytics, and practical applications to real-world business challenges. By diligently following this comprehensive guide, students can not only navigate the intricacies of SPSS with confidence but also adeptly tackle assignments with a profound understanding of statistical methodologies. This proficiency positions them to make meaningful contributions to the evolving landscape of business analytics, where data-driven decision-making is paramount. Embracing the systematic approach outlined here empowers students to not only succeed academically but also to apply their SPSS skills pragmatically, making a lasting impact in diverse industries that increasingly rely on analytics for informed and strategic choices.

## Post a comment...

Spss for business analytics: student's comprehensive guide submit your assignment, attached files.

## IMAGES

## VIDEO

## COMMENTS

Example: Inferential statistics. You randomly select a sample of 11th graders in your state and collect data on their SAT scores and other characteristics. You can use inferential statistics to make estimates and test hypotheses about the whole population of 11th graders in the state based on your sample data.

This page titled 1.4: Inferential Statistics is shared under a Public Domain license and was authored, remixed, and/or curated by David Lane via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request. In statistics, we often rely on a sample --- that is, a small ...

Welcome to Inferential Statistics! In this course we will discuss Foundations for Inference. Check out the learning objectives, start watching the videos, and finally work on the quiz and the labs of this week. ... If you want to complete the course and earn a Course Certificate by submitting assignments for a grade, you can upgrade your ...

Inferential stats allow you to assess whether patterns in your sample are likely to be present in your population. Some common inferential statistical tests include t-tests, ANOVA, chi-square, correlation and regression. Inferential statistics alone do not prove causation. To identify and measure causal relationships, you need a very specific ...

Video. Video: Unit 4A: Introduction to Statistical Inference (15:45) Recall again the Big Picture, the four-step process that encompasses statistics: data production, exploratory data analysis, probability and inference. We are about to start the fourth and final unit of this course, where we draw on principles learned in the other units ...

Sure, inferential statistics are used when making predictions or inferences about a population from a sample of data. Here are a few real-time examples: Medical Research: Suppose a pharmaceutical company is developing a new drug and they're currently in the testing phase. They gather a sample of 1,000 volunteers to participate in a clinical ...

This is a new approach to an introductory statistical inference textbook, motivated by probability theory as logic. It is targeted to the typical Statistics 101 college student, and covers the topics typically covered in the first semester of such a course. It is freely available under the Creative Commons License, and includes a software library in Python for making some of the calculations ...

Statistical inference is the process of drawing conclusions about populations or scientific truths from data. There are many modes of performing inference including statistical modeling, data oriented strategies and explicit use of designs and randomization in analyses. Furthermore, there are broad theories (frequentists, Bayesian, likelihood ...

Assignments may require some computation in R programming language. Students in this course will learn, the principles underlying statistical methods including sample vs population; how to implement inferential tasks including testing, estimation, confidence intervals; model selection and how to use models based on a few specific distributions ...

Inferential statistics are concerned with making inferences based on relations found in the sample, to relations in the population. Inferential statistics help us decide, for example, whether the differences between groups that we see in our data are strong enough to provide support for our hypothesis that group differences exist in general, in the entire population.

The normal distribution is central to the theory of inferential statistics. This theoretical distribution is bell-shaped and symmetrical, with the mean, the median, and the mode all coinciding at its peak and frequencies gradually decreasing at both ends of the curve. In a normal distribution, a constant proportion of the area under the curve ...

Unlike descriptive statistics, inferential statistics are often complex and may have several different interpretations. The goal of inferential statistics is to discover some property or general pattern about a large group by studying a smaller group of people in the hopes that the results will generalize to the larger group.

Prerequisite knowledge. PSCI 1800 (formerly 107) or similar R course. To help us better understand the nature of inferential statistics, we will be running quite a lot of simulations in R.Students entering the class should have a working knowledge of the R programming language, and in particular know how to use square brackets to index vectors and to run for() loops.

Basic SPSS Tutorial: chapter 4 Inferential Statistics, extra assignments Of course researchers do not have to compute standard error, t-values, and probabilities. Statistical software packages, like SPSS can do this very easily. Test whether the sample mean of the variable Polorientation is significantly higher than 13.5. See BST: section 5.4

Unit 7: Probability. 0/1600 Mastery points. Basic theoretical probability Probability using sample spaces Basic set operations Experimental probability. Randomness, probability, and simulation Addition rule Multiplication rule for independent events Multiplication rule for dependent events Conditional probability and independence.

Unit 7: Assignment #2 (due before 11:59 pm Central on MON JUL 8): To become familiar with some of the ways that descriptive and inferential statistics can be used to deceive people, read Chapters 2 through 6 of (a slender!) book titled How to Lie with Statistics by Darrell Huff. NOTE: This book was published in 1954; therefore, the examples are ...

7.7.2: The p-value of a Test. This page titled 7: Inferential Statistics and Hypothesis Testing is shared under a not declared license and was authored, remixed, and/or curated by Michelle Oja. So far we've been using statistics to mostly describe a sample. But we can do so much more with what we've learned about probability and the Standard ...

Inferential statistics are based on the assumption that sampling is random. We trust a random sample to represent different segments of society in close to the appropriate proportions (provided the sample is large enough; see below). ... This random division of the sample into two groups is called random assignment. Random assignment is ...

Step 4: Test hypotheses or make estimates with inferential statistics. A number that describes a sample is called a statistic, while a number describing a population is called a parameter. Using inferential statistics, you can make conclusions about population parameters based on sample statistics.

Statistical inference is the process of using data analysis to infer properties of an underlying distribution of probability. Inferential statistical analysis infers properties of a population, for example by testing hypotheses and deriving estimates. It is assumed that the observed data set is sampled from a larger population.. Inferential statistics can be contrasted with descriptive statistics.

Suggested assignments for inferential statistics are designed to promote in-depth engagement with the material. Download all additional assignments for inferential statistics, and the accompanying data files: Assignments Inferential Statistics.zip. Assignment 1. Details: Inferential Statistics 1.pdf Data: democrats.sav. Assignment 2

Inferential statistics include how to calculate confidence intervals, as well as conduct tests of one-sample tests of the population mean (Z- and t-tests), two-sample tests of the difference in population means (Z- and t-tests), chi square test of independence, correlation, and regression.

STAT200: Assignment #3 - Inferential Statistics Analysis and Writeup. Income can influence influence expenditure because as a household earns more money that are able to spend more money. Specific to annual expenditures is where households spend the most of their income.

Meet Thomas Pearce, a distinguished statistics assignment expert who holds a master's degree in Statistics from University of Texas. With over a decade of hands-on experience in the field, Thomas has honed an exceptional skill set in statistical analysis, data interpretation, and advanced modeling techniques. ... Inferential statistics serve as ...