## χ^{2} Calculations

**Two-way Contingency Tables**

The two-way contingency table tests for the independence of two variables.
If the two variables are truly independent, then the individual entries in the table will be distributed according to the proportions given by the row and column totals.
In this case **χ**^{2} will be zero.
A large **χ**^{2} will indicate the observed values are not close to the expected values,
hence the two variables are **not independent**.