Chi-Square GOF Calculator

A chi-square goodness-of-fit test compares observed category counts with counts expected under a specified model. It asks whether the differences between observed and expected frequencies are larger than ordinary random variation would suggest.

Examples include testing whether a die appears fair, whether customer choices match a forecasted distribution, or whether defects are spread across categories as expected. The data are counts, not percentages or continuous measurements.

The test statistic sums squared differences between observed and expected counts, scaled by expected counts. Larger discrepancies contribute more, especially when they are large relative to expectation.

The statistic is compared with a chi-square distribution. Degrees of freedom generally equal the number of categories minus one, adjusted further if parameters were estimated from the data. The p-value indicates how unusual the observed mismatch would be if the expected model were true.

The chi-square approximation works best when expected counts are not too small. Very small expected counts can make the approximation unreliable because the continuous chi-square distribution is standing in for discrete count behavior.

A common rule of thumb is that expected counts should generally be at least 5 in most or all categories. When categories are sparse, combining categories or using exact methods may be more appropriate.

A significant result says the observed distribution does not fit the expected distribution well, but it does not automatically explain why. Residuals by category help identify where the mismatch is largest.

Good interpretation returns to the model. Were the expected proportions justified? Were observations independent? Were categories complete and mutually exclusive? The test detects mismatch; domain knowledge explains it.