The square of a variable with a standard normal distribution has a chi-squared distribution with one degree of freedom. It is a special case of a gamma distribution.
This chi-squared distribution is extremely skew. Its mean and variance are found from those of the gamma distribution.
The sum of n independent chi-squared variables (1 df) has a chi-squared (n df) distribution. It is also a special case of a gamma distribution.
The general chi-squared distribution with n degrees of freedom is also skew but becomes closer to symmetric when n increases. Its mean and variance are also given.
The variance of a random sample from a normal distribution has a distribution that is proportional to a chi-squared distribution.
A pivot can be based on a normal sample's variance. A confidence interval for the underlying normal distribution's variance σ² can be found from it.
The sample variances from several normal random samples can be combined. If the underlying normal variances are equal, this pooled variance has a chi-squared distribution, allowing a confidence interval for σ² to be found.
The ratio of a standard normal variable and the square root of an independent chi-squared variable (divided by its degrees of freedom) has a t distribution. Its pdf is given.
Formulae for the mean and variance of the t distribution are given. Its tails are longer than those of a standard normal distribution, but the two distributions' shapes are close when the degrees of freedom are large.
A function of the sample mean, sample variance and μ is shown to have a t distribution if the sample comes from a normal distribution. It is a pivot for μ.
The pivot on the previous page can be used to find a confidence interval for a normal distribution's mean, μ.
The difference between two sample means from normal distributions is normally distributed. The "pooled" estimator of the common variance has a chi-squared distribution.
A pivot can be found from the difference between the sample means and pooled estimate of the variance. This can be used to find a confidence interval for the difference between the means.
The ratio of two independent chi-squared variables (divided by their degrees of freedom) has an F distribution. Its probability density function is shown.
Formulae are given for the F distribution's mean and variance. It is very skew unless both of its degrees of freedom are large.
The ratio of two sample variances from normal distributions has a distribution proportional to an F distribution. A confidence interval for the ratio of the normal variances can be found from it.
The square of a random variable with a t distribution has an F distribution.