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Without other qualification, ‘chi-squared test’ often is used as short for Pearson’s chi-squared test. The chi-squared test is used to determine whether there is a significant difference between the expected frequencies and the observed frequencies in one or more categories.

In the standard applications of the test, the observations are classified into mutually exclusive classes, and there is some theory, or say null hypothesis, which gives the probability that any observation falls into the corresponding class. The purpose of the test is to evaluate how likely it is between the observations and the null hypothesis. Chi-squared tests are often constructed from a sum of squared errors, or through the sample variance.

Test statistics that follow a chi-squared distribution arise from an assumption of independent normally distributed data, which is valid in many cases due to the central limit theorem. A chi-squared test can be used to attempt rejection of the null hypothesis that the data are independent. In the 19th century, statistical analytic methods were mainly applied in biological data analysis and it was customary for researchers to assume that observations followed a normal distribution, such as Sir George Airy and Professor Merriman, whose works were criticized by Karl Pearson in his 1900 paper. Until the end of 19th century, Pearson noticed the existence of significant skewness within some biological observations.