# Chi-square test

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A **chi-square test** is any statistical hypothesis test in which the test statistic has a chi-square distribution if the null hypothesis is true. These include:

- Pearson's chi-square test
- Yates' chi-square test also known as Yates' correction for continuity
- Mantel-Haenszel chi-square test
- linear-by-linear association chi-square test
- Chi-square goodness-of-fit test

The most common form of the test statistic is:

where the word "expected" often does not denote an expected value, but an observable estimate of an expected value. However, likelihood ratio tests do not have this form.

The **chi-square test** is a statistical tool to separate real effects from random variation. It can be used on data that is:

- randomly drawn from the population
- reported in raw counts of frequency (not percentages or rates)
- measured variables must be independent
- values on independent and dependent variables must be mutually exclusive
- observed frequencies cannot be too small

The chi-square test determines the probability of obtaining the observed results by chance, under a specific hypothesis. It tests independence as well as goodness of fit for a set of data.

## Contents

[show]## See also

- General likelihood-ratio tests, which are approximately chi-square tests.
- McNemar's test, related to a chi-square test
- The Wald test, which can be evaluated against a chi-square distribution

p value (e.g., *p* = 0.05). It is an indication of the likelihood of obtaining a result (0.05 = 5%). As such, it is relatively uninformative. A more helpful accompanying statistic is phi (or Cramer's phi, or Cramer's V).^{[1]} Phi is a measure of association that reports a value for the correlation between the two dichotomous variables compared in a chi-square test (2 × 2). This value gives you an indication of the extent of the relationship between the two variables. Cramer's phi can be used for even larger comparisons. It is a more meaningful measure of the practical significance of the chi-square test and is reported as the effect size.

## Chi-square test for contingency table

A chi-square test may be applied on a contingency table for testing a null hypothesis of independence of rows and columns.

## Example

A fair coin is one where heads and tails are equally likely to turn up after it is flipped. You are given a coin and asked to test if it is fair. After 100 trials, heads turn up 53 times and tails result 47 times. Here is a Chi-square analysis, where the null hypothesis is that the coin is fair:

Heads | Tails | Total | |

Observed | 53 | 47 | 100 |

Expected | 50 | 50 | 100 |

(O − E)^{2}
| 9 | 9 | |

χ^{2}
| 0.360 | 0.360 |

Since there is one(1) degree of freedom, *p* = 0.5485. There is thus a 54.85% chance of seeing this data if the coin is fair, which is not considered statistically significant evidence that the coin is NOT fair.

## See also

- General likelihood-ratio tests, which are approximately chi-square tests
- McNemar's test, related to a chi-square test
- The Wald test, which can be evaluated against a chi-square distribution

## External links

## References

- ↑ Aaron, B., Kromrey, J. D., & Ferron, J. M. (1998, November). Equating r-based and d-based effect-size indices: Problems with a commonly recommended formula. Paper presented at the annual meeting of the Florida Educational Research Association, Orlando, FL. (ERIC Document Reproduction Service No. ED433353)

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