Work with category counts

Goodness of fit compares one categorical variable with specified probabilities pi, using expected counts Ei=npi. The statistic is χ²=∑(Oi-Ei)²/Ei. If k category probabilities are fully specified, df=k-1; subtract further degrees of freedom for parameters estimated from these same data. Observations should be independent and categories mutually exclusive. For the elementary chi-square approximation here, use expected counts at least 5.