Carrying Out a Chi-Square Test for Goodness of Fit

Long before modern computing made statistical tests routine, scientists needed a principled way to decide whether observed frequencies in categorical data were consistent with a theoretical model. Gregor Mendel, for instance, reported ratios of plant traits that were suspiciously close to his predicted 3:1 ratios—so close, in fact, that later statisticians questioned whether the data had been selectively reported. The fundamental question is deceptively simple: when observed counts deviate from expected counts, is the discrepancy large enough to reject the proposed model, or could random sampling variability alone explain the difference? The chi-square goodness-of-fit test was developed precisely to answer this question, providing a single numerical summary—the chi-square statistic—that quantifies the overall departure of observed data from a hypothesized distribution.

The chi-square goodness-of-fit test remains one of the most widely used procedures in statistics because it addresses a question that arises constantly in practice: does an observed categorical distribution match a specific theoretical or claimed model? Whether you are testing whether a die is fair, whether genetics ratios conform to Mendelian predictions, or whether market share proportions have changed, the mechanics of the test are the same. In this lesson, we focus on carrying out the test—computing the test statistic, finding the p-value, and drawing a conclusion in context—after the hypotheses have been stated and conditions verified.

Before computing anything, it helps to have a clear mental model of what the chi-square goodness-of-fit test is doing. You begin with a single categorical variable whose possible outcomes define k categories. You have a sample of n observations, and you count how many fall in each category—these are the observed counts. Under the null hypothesis, you claim that the population follows a specific probability distribution; multiplying each hypothesized proportion by n yields the expected counts. The test statistic aggregates the squared deviations between observed and expected counts, each standardized by the expected count, producing a single nonnegative number that measures total discrepancy.

The following diagram illustrates the complete flow of a chi-square goodness-of-fit test, from raw data to conclusion. Each category contributes its own component to the overall χ² statistic, and these components are summed to produce the test statistic. Notice how categories with large deviations between observed and expected counts contribute disproportionately to the total—this is by design, since the squaring in the numerator penalizes large discrepancies more heavily.

In the diagram above, notice that every category contributes to the total χ² statistic, but categories where the observed count diverges substantially from the expected count drive the statistic upward. Category 2, with an observed count of 50 versus an expected count of 40, contributes 2.500—more than half the total χ² of 4.450. This characteristic makes the chi-square statistic sensitive to individual categories that are far from their expected values. When interpreting your results, examining which categories contribute the most to χ² provides valuable insight into where the model breaks down if H₀ is ultimately rejected.

The mathematical machinery of the chi-square goodness-of-fit test centers on three formulas. The first defines the expected counts under H₀, the second computes the test statistic, and the third identifies the reference distribution used to obtain the p-value. Together, these three components form the complete computational pipeline for carrying out the test.

A few important details about the test statistic deserve emphasis. First, the chi-square statistic is always nonnegative because every component involves a squared difference divided by a positive expected count. Second, the test is always one-sided (right-tailed): large values of χ² provide evidence against H₀, while small values suggest the data are consistent with the null model. There is no concept of a 'left-tailed' chi-square goodness-of-fit test. Third, the chi-square distribution with df = k − 1 serves as the reference distribution only when the conditions are met—specifically, the sample is random and all expected counts are at least 5. When you use a TI-83/84 calculator, the χ²-Test or χ² GOF-Test function computes both the test statistic and the p-value directly.

The chi-square distribution is a family of right-skewed distributions, each characterized by its degrees of freedom. As df increases, the distribution becomes more symmetric and shifts to the right—its mean equals df and its standard deviation equals √(2 × df). For a goodness-of-fit test with k categories, the reference distribution has df = k − 1. The p-value corresponds to the area under this curve to the right of the computed χ² statistic. A small p-value indicates that the observed data would be very unlikely if H₀ were true, providing evidence for rejecting the null hypothesis.

When interpreting the p-value, compare it to your predetermined significance level α (commonly 0.05 on the AP Exam). If the p-value is less than α, you reject H₀ and conclude that there is convincing evidence that the distribution of the categorical variable differs from the hypothesized distribution. If the p-value is greater than or equal to α, you fail to reject H₀, meaning the data do not provide sufficient evidence to conclude the distribution differs from the claimed model. Always state your conclusion in context of the problem—never simply say 'reject H₀' without explaining what that means for the specific situation.

A statistics student suspects that the color distribution of M&M's in a large bag does not match the company's published percentages. The company claims the following distribution: 24% blue, 20% orange, 16% green, 14% yellow, 13% red, and 13% brown. The student takes a random sample of 200 M&M's and records the following observed counts: Blue = 53, Orange = 38, Green = 35, Yellow = 27, Red = 22, Brown = 25. Carry out a chi-square goodness-of-fit test at the α = 0.05 significance level. (Assume the hypotheses have been stated and conditions verified.)

Observed color counts from one purchased bag can be compared with the stated model, but this design does not establish company-wide inference.

Five-category, 150-candy illustration. This does not replace the separate six-category, 200-candy worked example.

Students frequently lose points on AP free-response questions not because they cannot calculate χ², but because they make procedural or interpretive mistakes. The table below contrasts common errors with the corresponding best practices. Internalizing these distinctions can make the difference between partial and full credit.

The chi-square goodness-of-fit test is one member of a family of chi-square procedures. All three share the same test statistic formula—Σ(O − E)²/E—but they differ in the research question, the data structure, and the degrees of freedom. Understanding these distinctions helps you select the correct test on the AP Exam and in practice.

As you progress through the chi-square unit, you will see that mastering the goodness-of-fit procedure gives you a strong foundation for the other two tests. The computational machinery is identical; only the source of expected counts and the interpretation of the hypotheses change. In more advanced coursework, you may encounter likelihood-ratio tests (G-tests), which use a different formula to measure discrepancy but follow the same logical framework. The Pearson chi-square statistic remains the standard approach in introductory statistics courses and on the AP Exam because it is intuitive, computationally accessible, and performs well under the conditions typically encountered in practice.