10.2: 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:

\[
\Large
t=\frac{(M_1-M_2)-(\mu_1-\mu_2)}{s_{M_1-M_2}}
\nonumber
\]

This looks like more work to calculate, but remember that our null hypothesis states that the quantity \(\mu_1-\mu_2=0\), so we can drop that out of the equation and are left with:

\[
\Large
t=\frac{M_1-M_2}{s_{M_1-M_2}}
\nonumber
\]

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 appear. However, once we do calculate it and use it in our test statistic, everything else goes back to normal. Our decision criteria are 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.

Standard Error and Pooled Variance

Recall that the standard error is the average distance between any given sample mean and the center of its corresponding sampling distribution, and it is a function of the standard deviation of the population (either given or estimated) and the sample size. This definition and interpretation hold true for our independent samples t-test as well, but because we are working with two samples drawn from two populations, we have to first combine their estimates of standard deviation—or, more accurately, their estimates of variance—into a single value that we can then use to calculate our standard error.

The combined estimate of variance using the information from each sample is called the pooled variance and is denoted \(s_p^2\); the subscript p serves as a reminder indicating that it is the pooled 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:

\[
\Large
s_p^2=\frac{(n_1-1)s_1^2+(n_2-1)s_2^2}{n_1+n_2-2}
\nonumber
\]

This formula can look daunting at first, but it is in fact just a weighted average. Even more conveniently, some simple algebra can be employed to greatly reduce the complexity of the calculation. The simpler and more appropriate formula to use when calculating pooled variance is:

\[
\Large
s_p^2=\frac{SS_1+SS_2}{df_1+df_2}
\nonumber
\]

Using this formula, it’s very simple to see that we are just adding together the same pieces of information we have been calculating since Chapter 3. Thus, when we use this formula, the pooled variance is not nearly as intimidating as it might have originally seemed.

Once we have our pooled variance calculated, we can drop it into the equation for our standard error:

\[
\Large
s_{M_1-M_2}=\sqrt{\frac{s_p^2}{n_1}+\frac{s_p^2}{n_2}}
\nonumber
\]

Once again, although this formula may seem different than it was before, in reality, it is just a different way of writing the same thing. An alternative but mathematically equivalent way of writing our old standard error is:

\[
\Large
s_M=\frac{s}{\sqrt{n}}= \sqrt{\frac{s^2}{n}}
\nonumber
\]

Looking at that, we can now see that, once again, we are simply adding together two pieces of information—no new logic or interpretation required. Once the standard error is calculated, it goes in the denominator of our test statistic, as shown above and as was the case in all previous chapters. Thus, the only additional step to calculating an independent samples t statistic is computing the pooled variance. Let’s see an example in action in the next section.

Video: Independent Samples t-Test

Independent Samples t-Test on YouTube.

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