Chi-Square Test Calculator (χ²)

The chi-square (χ²) test is a non-parametric statistical test that checks whether observed frequency data differ significantly from expected frequencies. It is used with categorical data to assess goodness of fit or independence between variables.

A χ² value of 0 means the observed data perfectly match the expected frequencies. The larger the χ², the greater the discrepancy. To assess statistical significance, compare the calculated χ² to the critical value for the given degrees of freedom and significance level α (typically 0.05 or 0.01).

A result of p < 0.05 means we reject the null hypothesis at the 5% significance level. The difference between observed and expected frequencies is statistically significant — it is unlikely to have occurred by chance alone. This is the conventional threshold in most empirical sciences.

Degrees of freedom df = k − 1, where k is the number of categories. For 3 categories df = 2, for 4 categories df = 3, and so on. The degrees of freedom determine which critical value to use when assessing whether to reject the null hypothesis.

The chi-square test is unreliable when expected frequencies in any category are less than 5. It is also not appropriate for continuous data or very small samples (n < 20). In such cases use Fisher's exact test or Yates' continuity correction instead.

The chi-square test is used in survey analysis, genetics (verifying Mendel's laws), quality control in manufacturing, marketing (A/B testing), epidemiology, and social sciences — anywhere you have categorical data and want to check agreement with an expected distribution.

Enter numbers separated by commas, semicolons or spaces. Example: "10,20,30" as observed and "15,20,25" as expected. Both series must have at least 2 positive values. The calculator will automatically pair corresponding elements.

For one degree of freedom (df=1) and significance level α=0.05 the critical value is 3.84. If the calculated χ² > 3.84 we reject H₀ at the 5% level. For α=0.01 the critical value at df=1 is 6.63.

The null hypothesis H₀ in a goodness-of-fit chi-square test states that the observed distribution matches the expected one — any deviations are due purely to random sampling variation. The alternative hypothesis H₁ posits a real difference between the distributions.

The goodness-of-fit test checks whether a single categorical variable follows a specified theoretical distribution. The independence test analyses a cross-tabulation of two categorical variables to determine whether they are associated. This calculator performs the goodness-of-fit version.