Chi-squared distribution

A continuous probability distribution equal to the sum of squares of independent standard normal random variables. Its degrees of freedom determine its shape.

Standard normal distribution: The normal distribution with mean zero and variance one. Independent draws from this distribution are squared and summed to construct the chi-squared distribution.
Pearson's chi-squared test: A statistical test that compares observed categorical counts with counts expected under a null hypothesis. Its test statistic is approximated by a chi-squared distribution under suitable conditions.
Normal distribution: A symmetric continuous probability distribution characterized by its mean and variance. The standard normal variables being squared are a special case of this broader distribution.
Student's t-distribution: A symmetric distribution arising from a standard normal variable divided by the square root of an independent scaled chi-squared variable. It uses a chi-squared variable in its construction but describes a signed statistic with heavier tails.
Degrees of freedom: The number of independent quantities that can vary in a statistical calculation. Here, the degrees of freedom count the standard normal squares in the defining sum.
Chi-squared goodness-of-fit test: A test of whether observed categorical frequencies agree with specified expected frequencies. The statistic’s reference distribution is chi-squared when expected counts and model constraints justify the approximation.
Variance: A measure of the average squared deviation of values from their mean. Sums of squared normal deviations connect the distribution to inference about variance.
F-distribution: A distribution of the ratio of two independent chi-squared variables, each scaled by its degrees of freedom. It compares variance estimates through a ratio, unlike the single sum of squares represented here.
Gamma distribution: A family of continuous probability distributions defined on positive values and parameterized by shape and scale. The chi-squared distribution is a gamma distribution with shape equal to half its degrees of freedom and scale two.
Likelihood-ratio test: A statistical procedure that compares the fit of nested models using their likelihoods. Under regularity conditions, twice the log-likelihood ratio is asymptotically chi-squared.

Linked from 6 pages