statistics Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...

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

degrees of freedom In many scientific fields, the degrees of freedom of a system is the number of parameters of the system that may vary independently. For example, a point in the plane has two degrees of freedom for translation: its two coordinates; a non-infinite ...

of a

linear combination In mathematics, a linear combination or superposition is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' a ...

of independent

sample variance In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion, ...

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 differen ...

. 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 a Function (mathematics), function that gives the probabilities of occurrence of possible events for an Experiment (probability theory), experiment. It is a mathematical descri ...

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

chi-squared distribution In probability theory and statistics, the \chi^2-distribution with k Degrees of freedom (statistics), degrees of freedom is the distribution of a sum of the squares of k Independence (probability theory), independent standard normal random vari ...

, 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. 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.Upton, G., Cook, I. (2008) ''Oxford Dictionary of Statistics'', OUP. . Inferential statistical analysis infers properties of ...

tests. The simplest application of this equation is in performing Welch's ''t''-test. An improved equation was derived to reduce underestimating the effective degrees of freedom if the pooled sample variances have small degrees of freedom. Examples are jackknife and imputation-based variance estimates.

References

Further reading