Statistical Analysis for Education and Psychology Researchers

This pooled variance estimate is the average of the two separate sample variances
weighted by their respective sample sizes. The degrees of freedom associated with the
pooled variance estimate of the t-statistic are n 1 +n 2 −2; one degree of freedom is
associated with each sample variance (sample variance has n−1 df). The homogeneity of
variance assumption is more reasonable with experimental designs because such designs
generally assume that groups are equivalent prior to treatment.

Separate Variance Estimate of the Approximate t′-ratio (unequal

variance)

When the assumption of homogeneity of variance is unreasonable (procedures for
checking are described under the heading Test assumptions) the approximate t-statistic
denoted t′ is calculated using the separate variance estimates from each sample. The
sampling distribution of t′ does not have a t-distribution and does not have n 1 +n 2 − 2
degrees of freedom. The exact sampling distribution has been evaluated but is not
generally used. Instead, approximate procedures have been developed for determining the
critical values to use with the approximate t′ distribution.
One procedure is the Satterthwaite (1946) approximation which is used in many
propriety statistical packages. This procedure uses a modified degrees of freedom,
estimated from the sample sizes, to approximate the t-distribution.
The df for the Satterthwaite (1946) approximation are,

df

for

sepa

rate

vari

ance

esti

mat

e—

8.16

This approximation often results in degrees of freedom which are not whole numbers.
Associated probabilities cannot therefore be found directly from statistical tables. The p-
value has to be interpolated based on tabled entries or a function can be used in computer
packages (TINV function in SAS) to produce exact p-values. Standard computational
procedures such as PROC TTEST in SAS automatically print equal and unequal
variance estimates and exact p-values.

Test Assumptions

The assumptions for the independent t-test are:

• The populations from which the samples are selected should have an approximate
  normal distribution. What is meant here is that the sampling distributions of the

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