Chi-Square Goodness-of-Fit Test Calculator

How to Use This Calculator

1. Enter your categories — fill in category names and observed counts. Add or remove rows as needed.
2. Set expected proportions — enter a value between 0–1 per category (must sum to 1), or click Set Equal Proportions.
3. Click Calculate — get the χ² statistic, degrees of freedom, p-value, and critical values instantly.
4. Read the conclusion — if p < 0.05, reject H₀. Your data doesn't fit the expected distribution.

Background: The Chi-Square Goodness-of-Fit Test

What Does This Test Measure?

The Chi-Square Goodness-of-Fit test checks whether the distribution of a single categorical variable matches a hypothesized (expected) distribution. Unlike the Test of Independence, which compares two variables against each other, this test compares your observed category counts against theoretical proportions you specify — for example, testing whether a die is fair, or whether survey responses match a company's historical distribution.

The Formula

χ² = Σ [(O − E)² / E], where O is the observed count for each category and E is the expected count (total N × expected proportion). Degrees of freedom = k − 1, where k is the number of categories.

Goodness-of-Fit vs. Test of Independence: What's the Difference?

These two chi-square tests are often confused because they use the same formula and distribution, but they answer different questions. The Goodness-of-Fit test (this page) checks whether one categorical variable matches a hypothesized distribution — for example, "does this die roll each face equally often?" The Test of Independence checks whether two categorical variables are related to each other — for example, "is gender related to product preference?" If you have one variable and a theoretical expectation, use this page. If you have two variables and want to know if they're associated, use the Test of Independence instead.

Assumptions of the Chi-Square Goodness-of-Fit Test

• Random sampling: Observations should come from a random sample of the population.
• Independence: Each observation must fall into exactly one category, and observations must be independent of each other.
• Expected count ≥ 5: Each category should have an expected count of at least 5. When this isn't met, the chi-square approximation becomes unreliable — the calculator above will warn you if any category falls below this threshold.
• Categorical data: The variable being tested must be categorical (nominal or ordinal), not continuous.

What If an Expected Count Is Below 5?

When one or more categories have an expected count under 5, the chi-square distribution no longer approximates the test statistic well, and the resulting p-value may be inaccurate. Common solutions include combining adjacent categories with low expected counts into a single "Other" category, collecting more data to increase the total sample size, or using an exact test (such as a multinomial exact test) instead of the chi-square approximation when combining categories isn't appropriate for the research question.

How to Do This in R Studio

Use the built-in chisq.test() function with observed counts and expected proportions:

observed <- c(8, 12, 7, 15, 9, 9)
expected_p <- rep(1/6, 6)  # equal proportions

chisq.test(observed, p = expected_p)

How to Do This in SPSS

Analyze → Nonparametric Tests → Legacy Dialogs → Chi-square. Move your categorical variable into the Test Variable List. Under Expected Values, choose "All categories equal" for uniform proportions, or "Values" to enter custom expected proportions or counts per category, then click OK.

How to Do This in JASP

Frequencies → Multinomial Test. Drag your categorical variable into the Factor box. Under Expected Proportions, either leave the default (equal proportions) or enter your own hypothesized proportions for each category, and JASP will output the χ² statistic, df, and p-value.