Chi-Squared Goodness of Fit Tests

The chi-squared ($\chi^2$χ2) goodness of fit test is used to determine whether observed data follow a particular theoretical distribution. This is a key hypothesis testing technique in Further Statistics.

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The Idea Behind the Test

You have observed frequencies $O_i$Oi​ from an experiment or survey, and expected frequencies $E_i$Ei​ from a theoretical model (e.g., uniform, binomial, Poisson, normal). The test asks: are the differences between observed and expected frequencies larger than what we would expect from random variation alone?

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The Test Statistic

$X^2 = \sum_i \frac{(O_i - E_i)^2}{E_i}$X2=∑i​Ei​(Oi​−Ei​)2​

This statistic measures the overall discrepancy between observed and expected. Under the null hypothesis, $X^2$X2 approximately follows a chi-squared distribution with $\nu$ν degrees of freedom.

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Degrees of Freedom

$\nu = (\text{number of classes}) - 1 - (\text{number of estimated parameters})$ν=(number of classes)−1−(number of estimated parameters)