Chi Squared Distribution

Hypothesis Testing

${\chi }^2$ Distribution

The ${\chi }^2$ distribution allows you to:

• test how well a given set of data fits a specified distribution (Goodness of Fit)
• test the independence of two variables

It takes a test statistic

where $O$ refers to observed frequencies, and $E$ refers to expected frequencies.

If we’re using test statistic $X^2$ with the ${\chi }^2$ distribution, we write

where $\nu$ is the number of degrees of freedom, and $\alpha$ is the level of significance.

Value of $\nu$

In a goodness of fit test, $\nu$ is the number of classes minus the number of restrictions.

In a test for independence for two variables, if your contingency table has $h$ rows and $k$ columns,

Interpretation

If the value of $X^2$ is low, then this means there’s a less significant difference between the observed and expected frequencies. The higher $X^2$ is, the more significant the differences become.

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