Wolfram Alpha:

```Pooled Variance and Standard Error
----------------------------------

The pooled variance and standard error are used in tests of
significance for differences in means or proportions between
2 independent populations.

Equal Variances
---------------

If we assume that the sample variance are the same we can use
the poooled estimate of the variance by using the following
formula:

sp2 = [(n1 - 1)s12 + (n2 - 1)s22]/[(n1 - 1) + (n2 - 1)]

df = (n1 - 1) + (n2 - 1)

= n1 + n2 - 2

This is just the weighted average of s12 + s22.  When n1 = n2
we get sp2 = (s12 + s22)/2.

Under the assumption of equal population variances, the pooled
sample variance provides a higher precision estimate of variance
than the individual sample variances. This higher precision can
lead to increased statistical power when used in statistical
tests that compare the populations, such as the t-test.

The F test can be used to determine whether the sample variances
are different or not.

Pooled Standard Error
---------------------

The pooled standard error for equal variances is:

SE = sp√(1/n1 + 1/n2)

Unequal Variances
-----------------

When the sample variances are different we canâ€™t get a pooled
estimate.  Instead, we use a standard error formula that includes
both standard deviations, but separately.  We use:

SE = √(s12/n1 + s22/n2)

SE1 = s1/√n1 = √(s12/n1) and SE2 = s2/√n2 = √(s22/n2)

In this case the df is rather complicated.  It is the integer
part of:

√(s12/n1 + s22/n2)
df = ---------------------------------------
((s12/n1)2/(n1 - 1) + ((s22/n2)2/(n2 - 1)

This is the Satterthwaite correction.  One can always use this
regardless of whether the variances are equal or not.

In the situation where we are considering proportions we make
the following substitutions:

s2 = pq = p(1 - p)

s = √(pq)

For a single proportion the SE is:

SE =  √(pq)/√n = √(pq/n)
```