Pearson's chi-squared test

Pearson's chi-squared test (Chi-squared test), also known as the chi-square goodness-of-fit test or chi-square test for independence, is a statistical hypothesis test used to determine whether there is a significant difference between the expected frequencies and the observed frequencies in one or more categories of a contingency table. In the field of statistics, it is one of the most common tests for analyzing categorical data.

Overview

The test is applicable in situations where the data can be categorized into a contingency table, and the sample size is sufficiently large. The chi-squared test provides a method to gauge the discrepancy between observed and expected frequencies under the null hypothesis that no difference exists. It was developed by Karl Pearson in the early 20th century, hence the name.

Assumptions

Before applying Pearson's chi-squared test, certain assumptions must be met:

• Observations are independently drawn from the population.
• The sample size is large enough. As a rule of thumb, all expected counts should be at least 5.
• The data are categorical rather than numerical.

Calculation

The test statistic is calculated as: \[ \chi^2 = \sum \frac{(O_i - E_i)^2}{E_i} \] where \(O_i\) is the observed frequency for category \(i\), \(E_i\) is the expected frequency for category \(i\), and the summation is over all categories.

Applications

Pearson's chi-squared test is widely used in two major scenarios:

• Goodness-of-fit test: To determine how well an observed distribution fits with an expected distribution.
• Test of independence: To determine if there is a significant association between two categorical variables.

Limitations

While widely used, the test has limitations:

• It is not suitable for small sample sizes.
• It can only be used on categorical data.
• The test is sensitive to the sample size, meaning that with very large samples, even trivial differences can appear statistically significant.

Examples

An example of a goodness-of-fit test would be comparing the observed color distribution of M&Ms to the expected distribution claimed by the manufacturer. For a test of independence, one might analyze data from a survey to see if there is an association between gender and preference for a particular type of music.

See Also

• Fisher's exact test
• Likelihood-ratio test
• Yates's correction for continuity

References

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  Cumulative distribution function of the chi-square distribution