Pooled Standard Deviation Calculator

How it is calculated

You enter the size n and the sample standard deviation s of two groups. The pooled variance weights each group's variance by its degrees of freedom, n − 1: sp² = ((n1 − 1)s1² + (n2 − 1)s2²) ÷ (n1 + n2 − 2). The pooled standard deviation is its square root, sp, in the same unit as the data. Both are rounded to two decimals. Each size must be a whole number of at least 2 (a size such as 20.5, or 1, gives no result), and a negative standard deviation gives no result; a standard deviation of 0 is accepted. The inputs must be sample standard deviations (divisor n − 1), not population ones. Pooling assumes the two groups have similar spread, so a large difference between s₁ and s₂ means the pooled figure may not describe either group well.

Common mistakes

• Averaging the two standard deviations. A simple mean of 4.5 and 5.2 is 4.85, but the pooled value is 4.93 because group 2 has more observations and counts for more.
• Entering variances instead of standard deviations. The fields expect s, not s², so a variance of 20.25 typed in place of 4.5 gives a far larger pooled value.
• Entering a population standard deviation (divisor n). The formula assumes the sample version with divisor n − 1.
• Typing a group total or a fractional size as n. The size is the number of observations in each group, a whole number of at least 2; anything else shows no result.
• Pooling groups with very different spreads. If s₁ is several times s₂, the equal-variance assumption behind the pooled figure is doubtful.