# Chi-squared test

**chi-square testchi-squared statisticChi-squaredchi square testchi squarechi squaredchi squared testchi-squareChi-square testsχ'' 2 test statistic**

[[File:Chi-square distributionCDF-English.png|thumb|right|300px|Chi-squared distribution, showingwikipedia

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### Pearson's chi-squared test

**chi-square statisticPearson chi-squared testchi-square**

Without other qualification, 'chi-squared test' often is used as short for Pearson's chi-squared test. Using the chi-squared distribution to interpret Pearson's chi-squared statistic requires one to assume that the discrete probability of observed binomial frequencies in the table can be approximated by the continuous chi-squared distribution.

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.

### Chi-squared distribution

**chi-squaredchi-square distributionchi square distribution**

test''', is any statistical hypothesis test where the sampling distribution of the test statistic is a chi-squared distribution when the null hypothesis is true. One test statistic that follows a chi-squared distribution exactly is the test that the variance of a normally distributed population has a given value based on a sample variance. Using the chi-squared distribution to interpret Pearson's chi-squared statistic requires one to assume that the discrete probability of observed binomial frequencies in the table can be approximated by the continuous chi-squared distribution.

The chi-square distribution is used in the common chi-square tests for goodness of fit of an observed distribution to a theoretical one, the independence of two criteria of classification of qualitative data, and in confidence interval estimation for a population standard deviation of a normal distribution from a sample standard deviation.

### Statistical hypothesis testing

**hypothesis testingstatistical teststatistical tests**

test''', is any statistical hypothesis test where the sampling distribution of the test statistic is a chi-squared distribution when the null hypothesis is true.

1900: Karl Pearson develops the chi squared test to determine "whether a given form of frequency curve will effectively describe the samples drawn from a given population."

### Karl Pearson

**PearsonPearson, KarlCarl Pearson**

In the 19th century, statistical analytical 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.

These techniques, which are widely used today for statistical analysis, include the chi-squared test, standard deviation, and correlation and regression coefficients.

### Fisher's exact test

**Fisher exact testexact testFisher test**

For an exact test used in place of the 2 x 2 chi-squared test for independence, see Fisher's exact test.

With large samples, a chi-squared test (or better yet, a G-test) can be used in this situation.

### Variance

**sample variancepopulation variancevariability**

Chi-squared tests are often constructed from a sum of squared errors, or through the sample variance. One test statistic that follows a chi-squared distribution exactly is the test that the variance of a normally distributed population has a given value based on a sample variance.

The F test and chi square tests are both adversely affected by non-normality and are not recommended for this purpose.

### Test statistic

**Common test statisticst''-test of test statistics**

One test statistic that follows a chi-squared distribution exactly is the test that the variance of a normally distributed population has a given value based on a sample variance.

### McNemar's test

**McNemar testMcNemarMcNemar’s chi-square statistic**

::They calculated a chi-squared statistic [...] [they] had made an error in their analysis by ignoring the pairings.[...] [their] samples were not independent, because the siblings were paired [...] we set up a table that exhibits the pairings:

### G-test

**G''-testFisher's G testG''-tests**

In statistics, G-tests are likelihood-ratio or maximum likelihood statistical significance tests that are increasingly being used in situations where chi-squared tests were previously recommended.

### Probability distribution

**distributioncontinuous probability distributiondiscrete probability distribution**

Using the chi-squared distribution to interpret Pearson's chi-squared statistic requires one to assume that the discrete probability of observed binomial frequencies in the table can be approximated by the continuous chi-squared distribution.

### Sampling distribution

**finite sample distributiondistributionsampling**

test''', is any statistical hypothesis test where the sampling distribution of the test statistic is a chi-squared distribution when the null hypothesis is true.

### Null hypothesis

**nullnull hypotheseshypothesis**

test''', is any statistical hypothesis test where the sampling distribution of the test statistic is a chi-squared distribution when the null hypothesis is true.

### Lack-of-fit sum of squares

**error sum of squaressum of squared errors**

Chi-squared tests are often constructed from a sum of squared errors, or through the sample variance.

### Central limit theorem

**Lyapunov's central limit theoremlimit theoremscentral limit**

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.

### Normal distribution

**normally distributedGaussian distributionnormal**

In the 19th century, statistical analytical 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.

### George Biddell Airy

**George AiryAirySir George Biddell Airy**

In the 19th century, statistical analytical 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.

### Mansfield Merriman

**Professor Merriman**

### Skewness

**skewedskewskewed distribution**

Until the end of 19th century, Pearson noticed the existence of significant skewness within some biological observations.

### Pearson distribution

**Pearson Type III distributionPearson's system of continuous curvesPearson**

In order to model the observations regardless of being normal or skewed, Pearson, in a series of articles published from 1893 to 1916, devised the Pearson distribution, a family of continuous probability distributions, which includes the normal distribution and many skewed distributions, and proposed a method of statistical analysis consisting of using the Pearson distribution to model the observation and performing the test of goodness of fit to determine how well the model and the observation really fit.

### Binomial test

**binomial**

For an exact test used in place of the 2 x 1 chi-squared test for goodness of fit, see Binomial test.

### Cochran–Mantel–Haenszel statistics

**Cochran–Mantel–Haenszel testCochran-Mantel-Haenszel testCochran-Mantel-Haenzel Test**

### Tukey's test of additivity

**analytical trickeryTukey test of additivityTukey's F-test for interaction (non-additivity)**

### Portmanteau test

**portmanteau test in time series**

### Time series

**time series analysistime-seriestime-series analysis**

### Autocorrelation

**autocorrelation functionserial correlationautocorrelated**