Chi-Square Table Calculator

A chi-square test is a statistical method used to determine whether there is a significant difference between observed frequencies and expected frequencies in one or more categories. It is widely used for testing hypotheses about categorical data, such as goodness-of-fit tests and tests of independence in contingency tables.

The critical chi-square value is the threshold from the chi-square distribution table that your test statistic must exceed (in a right-tailed test) for the result to be statistically significant. It is determined by your chosen significance level (α) and the degrees of freedom. If your calculated χ² is greater than the critical value, you reject the null hypothesis.

For a goodness-of-fit test, degrees of freedom = number of categories − 1. For a contingency table test of independence, degrees of freedom = (number of rows − 1) × (number of columns − 1). For example, a 3×4 contingency table has (3−1)×(4−1) = 6 degrees of freedom.

The most common significance level is α = 0.05, meaning there is a 5% chance of rejecting a true null hypothesis. In fields requiring stricter evidence — such as medicine or physics — α = 0.01 or even α = 0.001 is preferred. The choice should be made before collecting data, not after seeing the results.

The chi-square distribution is a family of probability distributions parameterized by degrees of freedom. It is always non-negative and positively skewed, with the skew decreasing as degrees of freedom increase. It arises naturally when summing squared standard normal random variables and forms the basis of many statistical tests.

Expected frequencies depend on your null hypothesis. For a goodness-of-fit test, the expected frequency for each category is the total sample size multiplied by the hypothesized proportion. For a contingency table, the expected frequency for each cell is (row total × column total) / grand total. All expected frequencies should ideally be at least 5 for the chi-square approximation to be reliable.

Avoid chi-square tests when expected cell frequencies are below 5, when data are not counts (e.g. percentages or rates), or when observations are not independent. In cases with small expected counts, Fisher's exact test is often a better alternative for 2×2 tables.

The p-value represents the probability of observing a chi-square statistic as extreme as yours, assuming the null hypothesis is true. A p-value less than your chosen α (e.g. 0.05) means the result is statistically significant and you reject the null hypothesis. A p-value greater than α means you fail to reject the null hypothesis — it does not prove the null is true, only that there is insufficient evidence against it.