statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...

and uncertainty analysis, the Welch–Satterthwaite equation is used to calculate an approximation to the effective

degrees of freedom Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...

of a linear combination of independent

sample variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...

s, also known as the pooled degrees of freedom, corresponding to the

pooled variance In statistics, pooled variance (also known as combined variance, composite variance, or overall variance, and written \sigma^2) is a method for estimating variance of several different populations when the mean of each population may be different ...

. For sample variances , each respectively having degrees of freedom, often one computes the linear combination. :

\chi' = \sum_^n k_i s_i^2.

where

k_i

is a real positive number, typically

k_i=\frac

. In general, the

probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...

of cannot be expressed analytically. However, its distribution can be approximated by another

chi-squared distribution In probability theory and statistics, the chi-squared distribution (also chi-square or \chi^2-distribution) with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables. The chi-squa ...

, whose effective degrees of freedom are given by the Welch–Satterthwaite equation :

\nu_ \approx \frac

There is ''no'' assumption that the underlying population variances are equal. This is known as the

Behrens–Fisher problem In statistics, the Behrens–Fisher problem, named after Walter Behrens and Ronald Fisher, is the problem of interval estimation and hypothesis testing concerning the difference between the means of two normally distributed populations when t ...

. The result can be used to perform approximate

statistical inference Statistical inference is the process of using data analysis to infer properties of an underlying probability distribution, distribution of probability.Upton, G., Cook, I. (2008) ''Oxford Dictionary of Statistics'', OUP. . Inferential statistical ...

tests. The simplest application of this equation is in performing Welch's ''t''-test.

See also

References

Further reading