Independent samples with pooled variance

Independent samples t-statistic

The test statistic for our independent samples 𝑡-test takes on the same logical structure and format as our other 𝑡-tests: our observed effect minus our null hypothesis value, all divided by the standard error:

This looks like more work to calculate, but remember that our null hypothesis states that the quantity μ1−μ2=0, so we can drop that out of the equation and are left with:

Our standard error in the denomination is still standard deviation (𝑠) with a subscript denoting what it is the standard error of. Because we are dealing with the difference between two separate means, rather than a single mean or single mean of difference scores, we put both means in the subscript. Calculating our standard error, as we will see next, is where the biggest differences between this 𝑡-test and other 𝑡-tests appears. However, once we do calculate it and use it in our test statistic, everything else goes back to normal. Our decision criteria is still comparing our obtained test statistic to our critical value, and our interpretation based on whether or not we reject the null hypothesis is unchanged as well.

the subscript, p or w, serves as a reminder indicating that it is the pooled variance, which is also called the weighted variance.. The term “pooled variance” is a literal name because we are simply pooling or combining the information on variance – the Sum of Squares and Degrees of Freedom – from both of our samples into a single number. The result is a weighted average of the observed sample variances, the weight for each being determined by the sample size, and will always fall between the two observed variances. The computational formula for the pooled variance is:

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