Chi-square calculator

What is this test?

A chi-square test is a method for comparing categorical counts with counts expected under a null model. The data are not raw measurements like heights, scores, or times. They are counts in categories, such as survey choices, outcome groups, product types, diagnosis classes, or rows and columns in a contingency table. The test asks whether the differences between observed and expected counts are larger than ordinary sampling variation would explain under the stated model.

Statoma supports two common versions. The goodness-of-fit test uses one categorical variable and a specified expected count for each category. It answers questions such as whether observed choices match a planned distribution. The test of independence uses a two-way table. It answers whether the distribution of one categorical variable appears to change across the levels of another categorical variable.

Both versions use the same statistic. Each cell contributes a squared difference between observed and expected count, divided by the expected count. Squaring removes signs and makes larger discrepancies count more heavily. Dividing by the expected count keeps the scale comparable across cells. The final statistic is compared with a chi-square distribution whose degrees of freedom depend on the test structure.

The test is useful because it turns a categorical pattern into a transparent diagnostic number, but it is not a complete explanation. A significant result says the pattern is unlikely under the null model. It does not say which categories matter most, whether the association is strong, or whether the study design supports a causal claim. The expected counts and cell contributions are the first place to look after the p-value.

When to use it

• Use goodness of fit when you have one categorical variable and a clear expected count for each category under the null model.
• Use goodness of fit when checking whether observed frequencies follow a theoretical, historical, design-based, or planned distribution.
• Use the independence test when two categorical variables are cross tabulated and the question is whether their distributions are associated.
• Use counts from independent observations. Repeated measurements on the same unit, paired categories, or clustered sampling often need a different method.
• Use another method when expected counts are very small. Exact tests or simulation-based approaches may be more reliable in sparse tables.

The key design question is whether the categories represent one distribution or a two-way relationship. In a goodness-of-fit problem, expected counts come from outside the data, such as a known market share, a genetic ratio, a fair random mechanism, or a preregistered target. In an independence problem, expected counts are calculated from the row totals, column totals, and grand total. That difference changes the degrees of freedom and the meaning of the expected values.

It is also important that categories are mutually exclusive. If one respondent can appear in more than one category, the counts are not independent category totals in the usual sense. If categories leave some observations out, the expected distribution may no longer match the observed total. A clean chi-square test starts with a clean count table.

How it works

The chi-square statistic is the sum of cell contributions. Each contribution compares an observed count with its expected count under the null model. The more a cell differs from expectation, the larger its contribution. Cells with larger expected counts need larger raw differences to contribute the same amount as cells with smaller expected counts.

In a goodness-of-fit test, O is the observed count in a category and E is the expected count for that same category. If there are k categories and all expected counts are fixed in advance, the degrees of freedom are k minus 1. Some advanced settings estimate parameters from the data before forming expected counts; those settings reduce degrees of freedom further and are outside this calculator.

In an independence test, expected counts are calculated from the table margins. If the row variable and column variable are independent, the expected count in a cell equals its row total times its column total divided by the grand total. The degrees of freedom are based on how many row and column proportions can vary freely once the margins are fixed.

The p-value is the right-tail probability above the observed statistic. Chi-square statistics cannot be negative, and larger values mean more departure from the null model. That is why the usual test looks only to the right tail rather than splitting probability between two tails.

Worked example

Suppose a survey records three response categories with observed counts 50, 30, and 20. A planned distribution expects 40, 40, and 20. The first category contributes (50 - 40)^2 / 40 = 2.5. The second category contributes (30 - 40)^2 / 40 = 2.5. The third category contributes 0 because the observed and expected counts match. The chi-square statistic is therefore 5. With three categories, the degrees of freedom are 2.

A statistic of 5 with 2 degrees of freedom gives a p-value of about 0.0821. That is not the probability that the expected distribution is true. It is the probability of getting a chi-square statistic at least this large if the expected distribution and sampling assumptions were correct. The result suggests some mismatch, but it would usually not cross a 0.05 threshold. The cell contributions show that the mismatch is concentrated in the first two categories.

For independence, imagine a two-by-two table where one row has counts 20 and 30, and the other row has counts 30 and 20. The row totals are both 50, the column totals are both 50, and the grand total is 100. If the row and column variables were independent, each cell would have expected count 25. The statistic is 4 with 1 degree of freedom, giving a p-value of about 0.0455. That result points to association, but it still does not explain the study design or practical size of the relationship.

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