Categorical › Contingency / proportions

Multinomial / χ² goodness-of-fit

The multinomial (chi-square) goodness-of-fit test asks whether one categorical column's counts match expected proportions, equal by default.

What is Multinomial / χ² goodness-of-fit?

The chi-square GoF statistic compares observed category counts to expected counts under a hypothesised proportion vector. With no proportions given, it tests the multinomial null of equal proportions.

The χ² approximation assumes expected counts ≥ ~5 per cell; the result flags when that's violated.

When should I use Multinomial / χ² goodness-of-fit?

  • Testing die fairness / uniform preference across k options.
  • Comparing category frequencies to census or historical proportions.

What data does it need?

One categorical column + optional comma-separated expected proportions (alphabetical level order).

What does it report?

χ², df, p + an observed-vs-expected table with percentages.

What does it assume?

  • Independent observations.
  • Expected counts not too small (≥ 5 guideline).

How do I interpret the result?

Small p: the distribution differs from the hypothesised proportions. Inspect the observed-vs-expected table for which categories drive it.

See also

References

  • Agresti (2013). Categorical Data Analysis, 3rd ed.