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The chi-square test can be used to test independence as well as goodness of fit.
An example of a test of independence would be if sex and political affiliation are connected. So you would gather your sample, your expected value, find your critical value, and if the chi-square test is greater than the critical value, you can reject the null, otherwise, you fail to reject the null. (you never accept the null)
The chi-square probability density function is
and pk(x) = 0 for . Here Γ denotes the gamma function. Tables of this distribution — usually in its cumulative form — are widely available (see the External links below for online versions), and the function is included in many spreadsheets (for example OpenOffice.org calc or Microsoft Excel) and all statistical packages.
If p independent linear homogeneous constraints are imposed on these variables, the distribution of X conditional on these constraints is , justifying the term "degrees of freedom". The characteristic function of the Chi-square distribution is
- φ(t) = (1 - 2it) - k / 2
The chi-square distribution has numerous applications in inferential statistics, for instance in chi-square tests and in estimating variances. It enters the problem of estimating the mean of a normally distributed population and the problem of estimating the slope of a regression line via its role in Student's t-distribution. It enters all analysis of variance problems via its role in the F-distribution, which is the distribution of the ratio of two independent chi-squared random variables.
The normal approximation
If , then as k tends to infinity, the distribution of X tends to normality. However, the tendency is slow (the skewness is and the kurtosis is 12 / k) and two transformations are commonly considered, each of which approaches normality faster than X itself:
Fisher showed that is approximately normally distributed with mean and unit variance.
Wilson and Hilferty showed in 1931 that is approximately normally distributed with mean 1 - 2 / (9k) and variance 2 / (9k).
Note that 2 degrees of freedom leads to an exponential distribution.
The chi-square distribution is a special case of the gamma distribution.
The information entropy is given by:
where ψ(x) is the Digamma function.
- On-line calculator for the significance of chi-square, in Richard Lowry's statistical website at Vassar College.
- Distribution Calculator Calculates probabilities and critical values for normal, t-, chi2- and F-distribution
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