**Instructions**: This calculator computes the pooled variance and standard deviation for two given sample standard deviations \(s_1\) and \(s_2\), with corresponding sample sizes \(n_1\) and \(n_2\).

## How to Compute Pooled Variances

A pooled variance is an estimate of population variance obtained from two sample variances when it is assumed that the two samples come from population with the same population standard deviation. In that situation, none of the sample variances is a better estimate than the other, and the two sample variances provided are "pooled" together, in a sort of weighted average manner, to compute the pooled variance

The formula for a pooled-variance given two sample variances is:

\[ s_p^2 = \frac{(n_1-1)s_1^2 + (n_2-1)s_2^2}{n_1+n_2-2} \]On the other hand, the pooled standard deviation is:

\[s_p = \sqrt{ \frac{(n_1-1)s_1^2 + (n_2-1)s_2^2}{n_1+n_2-2}}\]On cool way of expressing the above formulas is based on the idea of the *Sum of Squares* (\(SS\)). In Social Sciences the sum of squares of a sample is defined as

But using the definition of sample variance, it is direct to see that

\[SS = \sum_{i=1}^n \left( X - \bar X \right)^2 = (n-1) s^2\]So then, we multiply the sample variance by \(n-1\) and we get the sum of squares \(SS\). Also, we know that for the one-sample case, we have that \(df = n-1\). Therefore, the pooled variance can be written very simply as:

\[ s_p^2 = \frac{(n_1-1)s_1^2 + (n_2-1)s_2^2}{n_1+n_2-2} = \frac{ SS_1 + SS_2}{df_1+df_2}\]The idea of pooled variances requires the assumption that the population variances are equal. For the case of unequal population variances, you should use this unpooled variances calculator.

One context in which the idea of pooled variances is used is for t-test for two independent variances. For a t-test calculator (where the idea of pooled variances is used), check it here.

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