A **chi-squared test**, also written as **χ ^{2} test**, is a statistical hypothesis test that is valid to perform when the test statistic is chi-squared distributed under the null hypothesis, specifically Pearson’s chi-squared test and variants thereof. Pearson’s chi-squared test is used to determine whether there is a statistically significant difference between the expected frequencies and the observed frequencies in one or more categories of a contingency table.

In the standard applications of this test, the observations are classified into mutually exclusive classes. If the null hypothesis that there are no differences between the classes in the population is true, the test statistic computed from the observations follows a χ^{2} frequency distribution. The purpose of the test is to evaluate how likely the observed frequencies would be assuming the null hypothesis is true.

Test statistics that follow a χ^{2} distribution occur when the observations are independent and normally distributed, which are assumptions often justified under the central limit theorem. There are also χ^{2} tests for testing the null hypothesis of independence of a pair of random variables based on observations of the pairs.

Chi-squared tests often refer to tests for which the distribution of the test statistic approaches the χ^{2} distribution asymptotically, meaning that the sampling distribution (if the null hypothesis is true) of the test statistic approximates a chi-squared distribution more and more closely as sample sizes increase.

In other simpler terms, it is a statistical test used to compare the difference between relative frequency of observed events to the frequency expected based on the assumption that is to be tested.

#### References

Wikipedia. Chi Square Test. https://en.wikipedia.org/wiki/Chi-squared_test