A Closer Look at Chi-Square and t-Tests, Explained

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A Closer Look at Chi-Square and t-Tests

Research Methods in Psychology · Chapter 15: Inferential Statistics and Hypothesis TestingSection 15.4 · lesson 90 of 949 min · free with a free account

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In this lesson

Two of the workhorse procedures of inferential statistics.

  • Match a research situation to the right test
  • Distinguish the three principal forms of t-test
  • Describe goodness-of-fit and tests of independence
  • Recognize each test's assumptions and their violations

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Inferential Statistics and Hypothesis Testing

A Closer Look at Chi-Square and t-Tests

Two families of test do most of the work in introductory psychology research. Chi squared tests, and t tests. By the end of this section you will identify research situations appropriate for each. You will distinguish the three principal types of t test. One sample, independent samples, and paired samples. You will describe the chi squared goodness of fit test, and the test of independence. You will recognize the assumptions of each test, and what happens when they fail. And you will interpret published results that report either statistic.

Two Families

Sorted by the Kind of Data

The two families are distinguished by the kind of data they handle. Chi squared tests examine relationships and patterns among categorical data. That means counts within categories, not measurements on a scale. The t test family examines differences in means, among continuous data. With variants that handle different design structures. Together these two families cover an enormous range of research questions. They form the backbone of empirical reporting at the introductory level. And each relies on its own reference distribution. The t distribution for tests of means, and the chi squared distribution for categorical data. Knowing when each is appropriate is essential to reading research, and to conducting it.

The Reference

The t-Distribution

Start with the t test, which compares means. It yields a test statistic, t, that follows the t distribution under the null hypothesis. The t distribution looks a lot like the standard normal curve. With one difference that matters at small sample sizes. It has heavier tails when samples are small. There is a reason for those heavier tails, and it is worth understanding. You are estimating the population standard deviation from the sample itself. That adds uncertainty, and the heavier tails account for it. As sample size grows, that extra uncertainty shrinks. And the t distribution converges on the normal.

Form One

The One-Sample t-Test

The first form is the one sample t test. It compares the mean of a single sample against a known or hypothesized population value. A teacher might use it to test whether her class's average differs from the national average. A clinician might test whether a patient group's anxiety scores differ from a normative value. In both cases there is only one group, compared against a fixed number. The null hypothesis says the population mean equals that reference value. The alternative says it does not. The test statistic is the sample mean minus the reference value, divided by the standard error. And the degrees of freedom equal n minus one. That n minus one is the same idea you met with the sample variance.

Form Two

The Independent-Samples t-Test

The second form is the independent samples t test. It compares the means of two separate groups. This is the standard test in two group between subjects experiments. The null hypothesis says the two population means are equal. The alternative says they differ. The statistic compares the difference between sample means against something. Against the standard error of that difference. And the degrees of freedom depend on the sample sizes of the two groups. There is a variant worth knowing by name, called Welch's t test. It does not assume equal variances across the groups. It is now the default in many statistical packages, and preferred when equal variances cannot be defended.

Form Three

The Paired-Samples t-Test

The third form is the paired samples t test. It is sometimes called the dependent samples test, or the repeated measures t test. It compares two means within the same participants, or within matched pairs. So a within subjects experiment with two conditions calls for it. And so does a matched pair between subjects design with two conditions. The procedure has two steps. First, compute the difference between the paired observations for each participant. Second, test whether the mean of those differences is zero. Notice what focusing on within pair differences accomplishes. It removes individual differences from the comparison entirely. Which makes it typically more powerful than the independent samples version.

Side by Side

Three Forms, Three Uses

Put the three forms side by side, because choosing between them is mechanical. The one sample test compares a sample mean to a known population or theoretical value. Use it when comparing a group to a normative value. The independent samples test compares the means of two separate groups. Use it for two group between subjects experiments. The paired samples test compares two measurements from the same participants. Or from matched pairs, which behave the same way statistically. Use it for within subjects designs, and for matched pair experiments. Notice that the design decides the test. Not the other way around.

Assumptions

What the t-Test Requires

T tests carry three assumptions, and they are not equally important. First, independent observations, both within groups and across them. Second, approximately normally distributed sampling distributions. Third, for the classical independent samples version, equal variances across groups. Now the differences in how much each one matters. The t test is reasonably robust to modest violations of normality. Particularly with samples larger than about thirty. Unequal variances are handled effectively by Welch's correction. So two of the three assumptions have workable remedies. Independence, by contrast, is fundamental. Violating it produces inferences that cannot be saved by post hoc adjustments.

The Other Family

Chi-Square Tests

Now the other family. Chi squared tests. They handle categorical data. That means counts of observations falling into discrete categories. Not continuous measurements on a scale. Two forms appear most often in introductory contexts. The goodness of fit test, which compares observed counts to a theoretical distribution. And the test of independence, which examines whether two categorical variables are related. The chi squared distribution behaves a bit like the t distribution. It takes different shapes depending on the degrees of freedom. But unlike the t, it is bounded at zero, and it is positively skewed. That makes sense, because it measures squared discrepancies, which cannot be negative.

Chi-Square One

The Goodness-of-Fit Test

The goodness of fit test asks a simple question. Does the distribution of observations across categories match a theoretical pattern? Here is a genetics example. Do the offspring of crossed plants appear in the predicted three to one ratio? And here is a psychology example. Do participants choose among four equally available options at equal rates? Or do they systematically prefer one of them? In both cases the mechanics are the same. The test compares observed frequencies to expected frequencies. Expected frequencies derived from the null hypothesis. And the statistic grows as the discrepancy between them grows.

Chi-Square Two

The Test of Independence

The test of independence asks whether two categorical variables are related. Counts are arranged in a contingency table. Rows for the levels of one variable, columns for the levels of the other. The null hypothesis is that the two variables are independent. Which means knowing the value of one tells you nothing about the other. The statistic compares observed cell counts against what independence would predict. A significant value indicates the two variables are associated. Though follow up analyses are needed to characterize the nature of that association. Two assumptions matter here. Observations must be independent, and expected cell counts must be large enough. Typically at least five in each cell, or the approximation breaks down. When counts are too small, use Fisher's exact test instead.

Reading Results

How These Tests Are Reported

Finally, how these results actually appear on the page. Conventional reporting includes four things. The test statistic, its degrees of freedom in parentheses, the p value, and an effect size. A t test result might read as follows. T, in parentheses fifty eight, equals two point three four. P equals point zero two three, and d equals zero point six one. The fifty eight is the degrees of freedom, and the d is Cohen's d. A chi squared result carries one extra piece of information. Chi squared, in parentheses two, with N equals two hundred and forty, equals eight point seven one. P equals point zero one three, and V equals point one nine. The two is the degrees of freedom, and V is Cramer's V, an effect size measure. The shape is the same in both. Statistic, degrees of freedom, p value, effect size.

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FAQ

Questions students ask us

About A Closer Look at Chi-Square and t-Tests

When should you use a chi-square test instead of a t test?
Use a chi-square test for categorical data, meaning counts of observations in categories, and a t test to compare means of continuous data. For example, whether college major is related to preferred study method calls for chi-square, while whether two groups differ in average exam score calls for a t test. Chi-square statistics are evaluated against the chi-square distribution and t statistics against the t distribution.
What are the three types of t tests?
The three types are the one-sample, independent-samples, and paired-samples t test. A one-sample t test compares one group's mean with a known or hypothesized value, such as a national average, using n - 1 degrees of freedom. An independent-samples t test compares two separate groups, as in a two-group between-subjects experiment. A paired-samples t test compares two measurements from the same participants or from matched pairs.
What is the difference between a chi-square goodness of fit test and a test of independence?
A goodness of fit test compares observed counts on one categorical variable with a theoretical distribution, while a test of independence checks whether two categorical variables are related. Testing whether the offspring of crossed plants follow a 3:1 ratio is goodness of fit. Arranging counts in a contingency table to see whether two variables are associated is a test of independence. For a test of independence, expected counts should typically be at least 5 per cell; with smaller counts, Fisher's exact test is an alternative.
How do you report a t test result in APA style?
Report the test statistic, its degrees of freedom in parentheses, the p value, and an effect size, for example t(58) = 2.34, p = .023, d = 0.61, where 58 is the degrees of freedom and d is Cohen's d. A chi-square result follows the same pattern and adds the sample size, for example chi-square(2, N = 240) = 8.71, p = .013, V = .19, where V is Cramer's V.

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