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Pearson's chi-squared test (''χ'') is a statistical test applied to sets of categorical data to evaluate how likely it is that any observed difference between the sets arose by chance. It is suitable for unpaired data from large samples. It is the most widely used of many chi-squared tests (e.g., Yates, likelihood ratio, portmanteau test in time series, etc.) – statistical procedures whose results are evaluated by reference to the chi-squared distribution. Its properties were first investigated by Karl Pearson in 1900. In contexts where it is important to improve a distinction between the test statistic and its distribution, names similar to ''Pearson χ-squared'' test or statistic are used. It tests a null hypothesis stating that the frequency distribution of certain events observed in a sample is consistent with a particular theoretical distribution. The events considered must be mutually exclusive and have total probability 1. A common case for this is where the events each cover an outcome of a categorical variable. A simple example is the hypothesis that an ordinary six-sided die is "fair" (i. e., all six outcomes are equally likely to occur). ==Definition== Pearson's chi-squared test is used to assess two types of comparison: tests of goodness of fit and tests of independence. * A test of goodness of fit establishes whether or not an observed frequency distribution differs from a theoretical distribution. * A test of independence assesses whether unpaired observations on two variables, expressed in a contingency table, are independent of each other (e.g. polling responses from people of different nationalities to see if one's nationality is related to the response). The procedure of the test includes the following steps: # Calculate the chi-squared test statistic, , which resembles a normalized sum of squared deviations between observed and theoretical frequencies (see below). # Determine the degrees of freedom, ''df'', of that statistic, which is essentially the number of categories reduced by the number of parameters of the fitted distribution. # Select a desired level of confidence (significance level, p-value or alpha level) for the result of the test. # Compare to the critical value from the chi-squared distribution with ''df'' degrees of freedom and the selected confidence level (one-sided since the test is only one direction, i.e. is the test value greater than the critical value?), which in many cases gives a good approximation of the distribution of . # Accept or reject the null hypothesis that the observed frequency distribution is different from the theoretical distribution based on whether the test statistic exceeds the critical value of . If the test statistic exceeds the critical value of , the null hypothesis (H0 = there is no difference between the distributions) can be rejected with the selected level of confidence and the alternative hypothesis (H1 = there is a difference between the distributions) can be accepted with the selected level of confidence. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Pearson's chi-squared test」の詳細全文を読む スポンサード リンク
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