# Chi-square (χ^{2}) Distribution — Tests, Formula, Table [Worked Examples]

by **Jack Bodeley** on *September 28, 2021*

##### The chi-square (χ^{2}) test is a test of discrepancies between observed frequencies and a corresponding set of expected frequencies derived from a null hypothesis.

## Chi-square (χ^{2}) Tests — Goodness of Fit v Test of Independence

### One-variable *Goodness of fit* χ^{2} Tests

When data is distributed along a single qualitative variable, the one-variable χ^{2} test evaluates these discrepancies as a "goodness of fit" test.

A one-variable χ^{2} test evaluates whether observed frequencies for a single qualitative variable are adequately described by hypothesized or expected frequencies.

**One-variable χ ^{2} tests are designed to evaluate the adequacy with which observed frequencies are described by hypothesized or expected frequencies i.e. goodness of fit.**

*Observations are classified in only one way.*

### Two-variables *Test of independence* χ^{2} Tests

When data are cross-classified along two qualitative variables, the two-variable χ^{2} test evaluates these discrepancies as a "test of independence" or lack of predictability between the two qualitative variables.

*Observations are classified in two ways, i.e. cross-classified according to two qualitative variables.*

## Formulae

$$\chi^{2}= \sum \frac{(observed\ frequency - expected\ frequency)^2}{expected\ frequency}$$