Making statistics intuitive
The null hypothesis in statistics states that there is no difference between groups or no relationship between variables. It is one of two mutually exclusive hypotheses about a population in a hypothesis test. [Read more…] about Null Hypothesis: Definition, Rejecting & Examples
A confidence interval (CI) is a range of values that is likely to contain the value of an unknown population parameter. These intervals represent a plausible domain for the parameter given the characteristics of your sample data. Confidence intervals are derived from sample statistics and are calculated using a specified confidence level. [Read more…] about Confidence Intervals: Interpreting, Finding & Formulas
These F-tables provide the critical values for right-tail F-tests. Your F-test results are statistically significant when its test statistic is greater than this value. [Read more…] about F-table
A sampling distribution of a statistic is a type of probability distribution created by drawing many random samples of a given size from the same population. These distributions help you understand how a sample statistic varies from sample to sample. [Read more…] about Sampling Distribution: Definition, Formula & Examples
A critical value defines regions in the sampling distribution of a test statistic. These values play a role in both hypothesis tests and confidence intervals. In hypothesis tests, critical values determine whether the results are statistically significant. For confidence intervals, they help calculate the upper and lower limits. [Read more…] about Critical Value: Definition, Finding & Calculator
This chi-square table provides the critical values for chi-square (χ2) hypothesis tests. The column and row intersections are the right-tail critical values for a given probability and degrees of freedom. [Read more…] about Chi-Square Table
A z-table, also known as the standard normal table, provides the area under the curve to the left of a z-score. This area represents the probability that z-values will fall within a region of the standard normal distribution. Use a z-table to find probabilities corresponding to ranges of z-scores and to find p-values for z-tests. [Read more…] about Z-table
This t-distribution table provides the critical t-values for both one-tailed and two-tailed t-tests, and confidence intervals. Learn how to use this t-table with the information, examples, and illustrations below the table. [Read more…] about T-Distribution Table of Critical Values
A test statistic assesses how consistent your sample data are with the null hypothesis in a hypothesis test. Test statistic calculations take your sample data and boil them down to a single number that quantifies how much your sample diverges from the null hypothesis. As a test statistic value becomes more extreme, it indicates larger differences between your sample data and the null hypothesis. [Read more…] about Test Statistic: Definition, Types & Formulas
Use a paired t-test when each subject has a pair of measurements, such as a before and after score. A paired t-test determines whether the mean change for these pairs is significantly different from zero. This test is an inferential statistics procedure because it uses samples to draw conclusions about populations.
Paired t tests are also known as a paired sample t-test or a dependent samples t test. These names reflect the fact that the two samples are paired or dependent because they contain the same subjects. Conversely, an independent samples t test contains different subjects in the two samples. [Read more…] about Paired T Test: Definition & When to Use It
Use an independent samples t test when you want to compare the means of precisely two groups—no more and no less! Typically, you perform this test to determine whether two population means are different. This procedure is an inferential statistical hypothesis test, meaning it uses samples to draw conclusions about populations. The independent samples t test is also known as the two sample t test. [Read more…] about Independent Samples T Test: Definition, Using & Interpreting
The standard error of the mean (SEM) is a bit mysterious. You’ll frequently find it in your statistical output. Is it a measure of variability? How does the standard error of the mean compare to the standard deviation? How do you interpret it?
In this post, I answer all these questions about the standard error of the mean, show how it relates to sample size considerations and statistical significance, and explain the general concept of other types of standard errors. In fact, I view standard errors as the doorway from descriptive statistics to inferential statistics. You’ll see how that works! [Read more…] about Standard Error of the Mean (SEM)
Moderna has announced encouraging preliminary results for their COVID-19 vaccine. In this post, I assess the available data and explain what the vaccine’s effectiveness really means. I also look at Moderna’s experimental design and examine how it incorporates statistical procedures and concepts that I discuss throughout my blog posts and books. [Read more…] about Assessing a COVID-19 Vaccination Experiment and Its Results
In my post about how to interpret p-values, I emphasize that p-values are not an error rate. The number one misinterpretation of p-values is that they are the probability of the null hypothesis being correct.
The correct interpretation is that p-values indicate the probability of observing your sample data, or more extreme, when you assume the null hypothesis is true. If you don’t solidly grasp that correct interpretation, please take a moment to read that post first.
Hopefully, that’s clear.
Unfortunately, one part of that blog post confuses some readers. In that post, I explain how p-values are not a probability, or error rate, of a hypothesis. I then show how that misinterpretation is dangerous because it overstates the evidence against the null hypothesis. [Read more…] about P-Values, Error Rates, and False Positives
I’m thrilled to release my new book! Hypothesis Testing: An Intuitive Guide for Making Data Driven Decisions. [Read more…] about New eBook Release! Hypothesis Testing: An Intuitive Guide
Failing to reject the null hypothesis is an odd way to state that the results of your hypothesis test are not statistically significant. Why the peculiar phrasing? “Fail to reject” sounds like one of those double negatives that writing classes taught you to avoid. What does it mean exactly? There’s an excellent reason for the odd wording!
In this post, learn what it means when you fail to reject the null hypothesis and why that’s the correct wording. While accepting the null hypothesis sounds more straightforward, it is not statistically correct! [Read more…] about Failing to Reject the Null Hypothesis
Significance levels in statistics are a crucial component of hypothesis testing. However, unlike other values in your statistical output, the significance level is not something that statistical software calculates. Instead, you choose the significance level. Have you ever wondered why?
In this post, I’ll explain the significance level conceptually, why you choose its value, and how to choose a good value. Statisticians also refer to the significance level as alpha (α). [Read more…] about Understanding Significance Levels in Statistics
Chi-squared tests of independence determine whether a relationship exists between two categorical variables. Do the values of one categorical variable depend on the value of the other categorical variable? If the two variables are independent, knowing the value of one variable provides no information about the value of the other variable.
I’ve previously written about Pearson’s chi-square test of independence using a fun Star Trek example. Are the uniform colors related to the chances of dying? You can test the notion that the infamous red shirts have a higher likelihood of dying. In that post, I focus on the purpose of the test, applied it to this example, and interpreted the results.
In this post, I’ll take a bit of a different approach. I’ll show you the nuts and bolts of how to calculate the expected values, chi-square value, and degrees of freedom. Then you’ll learn how to use the chi-squared distribution in conjunction with the degrees of freedom to calculate the p-value. [Read more…] about How the Chi-Squared Test of Independence Works
Use a variances test to determine whether the variability of two groups differs. In this post, we’ll work through a two-sample variances test that Excel provides. Even if Excel isn’t your primary statistical software, this post provides an excellent introduction to variance tests. Excel refers to this analysis as F-Test Two-Sample for Variances. [Read more…] about How to Test Variances in Excel
Excel can perform various statistical analyses, including t-tests. It is an excellent option because nearly everyone can access Excel. This post is a great introduction to performing and interpreting t-tests even if Excel isn’t your primary statistical software package.
In this post, I provide step-by-step instructions for using Excel to perform t-tests. Importantly, I also show you how to select the correct form of t-test, choose the right options, and interpret the results. I also include links to additional resources I’ve written, which present clear explanations of relevant t-test concepts that you won’t find in Excel’s documentation. And, I use an example dataset for us to work through and interpret together! [Read more…] about How to do t-Tests in Excel