# Chi-squared test

chi-square testchi-squared statisticChi-squaredchi square testchi squarechi squaredchi squared testchi-squareChi-square tests&chi;'' 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
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.

### 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