"Pooled variance" is more of an advanced topic. If you don't understand it, you can
still use the t-test, but understanding pooled variance will
help make sense of why we do certain things with our T tests. The pooled
variance is really a weighted mean of the two samples' variances. You have two
samples. Each sample has its own variance. When you combine the two samples in the
t tests, you need to account for the relative amount of variance. So think of
it like this: imagine that you have a glass of 2% milk and a glass of skim
milk (which would be 0%). If you poured them together ,wouldn't you expect to
have something like 1% milk? Split the difference between 0% & 2%?
They have different amounts of milk fat, but because the glasses are the same
size, you could just average them. But what if you had half a gallon of 2% milk
and you mix it with a quart of skim milk? If you poured them together, you would
not expect 1%. There Is more 2% milk than skim milk, so the combination will be
closer to 2%. And that is the logic behind pooled variance. If you are
comparing two samples of exactly the same size, then you could just average
their variability. But if one sample was larger than the other, then you would
need to WEIGHT the variability of each sample by its sample size. wWhen we do an
independent samples t-test, we need to calculate the pooled variance. The pooled
variance is really the weighted mean of the variance of the two samples.
To calculate pooled variance, you multiply each groups' variance by its sample size.
You pool them, and then you divide by the overall sample size. Here is what the
equation looks like: notice each sample size or "degrees of freedom" (n - 1)
is multiplied by its variance, and then the two are added together. Then both
of them are divided by the overall sample size. Of course, if you're using software,
this will be done for you, without you having to ask. But still it is important
to know what's going on "behind the scenes". You will use the pooled variance,
with the standard error of the difference between sample means, usually
called the "standard error of the difference". The standard error of the
difference is the error term for the denominator of the t-test. Let's look at
how these formulas work together. You remember the formula for variance?
We divide the sum of squares by the degrees of freedom to get the variance. We use
n - 1 as a corrective factor, since we are estimating a population parameter,
from a sample statistic. You would calculate the variance of each sample
separately, so you would use this formula twice. We create the pooled variance by
multiplying that variance of each sample (that we just calculated), by each sample
size; adding them together and then dividing by the total sample size.
The pooled variance is then divided by each sample size. Those are added, and then
take the square root of the sum. That is the standard error of the difference.
And finally, the t-test is calculated by subtracting the mean of the second
sample from the mean of the first sample, (that's the mean difference) and then
dividing the mean difference by the standard error of the difference. I know
that this might be challenging to understand, at first, but I think that if you
spend a little time with it, it will become more intuitive. Yes, you
can still do a t-test even if you don't fully understand pooled variance, but I
think that it is worth the effort to understand the logic behind how a t-test
is calculated.