Pooled Standard Deviation Calculator

Pooled standard deviation calculator: combine two groups into one pooled standard deviation. Enter raw data lists or summary statistics (s1, n1, s2, n2) to get the pooled variance, pooled SD and degrees of freedom. For n1 = 10, s1 = 4 and n2 = 15, s2 = 6, the pooled SD is 5.308 with df = 23.

What is pooled standard deviation?

A pooled standard deviation is a single standard deviation calculated from two samples, combining their spreads into one estimate of the common population spread. It is a weighted average of the two sample variances, with each group's degrees of freedom (n − 1) as its weight: groups with more observations count more. Use it when you have two independent samples and are willing to assume both come from populations with the same variance - the equal-variance assumption behind the two-sample t-test. If you only have one sample, start with our standard deviation calculator instead.

The pooled value is the spread that a two-sample t-test and a confidence interval for the difference of two means use. Because it pools the information from both groups, it is a more stable estimate than either group's standard deviation on its own, and its degrees of freedom (n₁ + n₂ − 2) are larger than either group's alone - which is what makes the resulting test and interval more precise.

How do you calculate pooled standard deviation?

Work through four steps. First, find each group's sample variance: s₁² and s₂². Second, multiply each variance by its degrees of freedom: (n₁ − 1)·s₁² and (n₂ − 1)·s₂². Third, add those two products and divide by the total degrees of freedom, n₁ + n₂ − 2 - that is the pooled variance, sp². Fourth, take the square root: sp = √[((n₁ − 1)·s₁² + (n₂ − 1)·s₂²) / (n₁ + n₂ − 2)]. For a single sample, see our guide on how to calculate standard deviation.

Note that the formula uses sample variances (dividing by n − 1), not population variances. The group standard deviations are sample standard deviations - see population vs sample standard deviation if you are unsure which one you have. If you only know the population standard deviations, convert them first: s₁² = σ₁² · n₁ / (n₁ − 1).

When should you pool standard deviations?

Pool when the two groups are independent samples from populations you are willing to treat as having the same variance - the classic setting is a two-sample t-test or a confidence interval for a difference of two means, for example a treatment group and a control group measured on the same scale. The equal-variance assumption should be plausible from the design or the data: similar group standard deviations, and no group with a much larger spread.

Do not pool when the variances are clearly unequal, or when the data are paired rather than independent. With very different variances, the pooled value misstates both groups' spread; use Welch's t-test instead, which keeps the variances separate. As a rough guide, if the larger sample variance is more than about four times the smaller one, pooling is questionable. The pooled SD also feeds the standard error of the difference of means - and with it, the margin of error vs confidence interval trade-off for that difference.

Worked example: n1 = 10, s1 = 4 and n2 = 15, s2 = 6

Take Group A with n₁ = 10 and s₁ = 4, and Group B with n₂ = 15 and s₂ = 6. The variances are s₁² = 16 and s₂² = 36. Weight them by the degrees of freedom: (10 − 1) × 16 = 144 and (15 − 1) × 36 = 504. Add them: 144 + 504 = 648. The pooled degrees of freedom are 10 + 15 − 2 = 23, so the pooled variance is sp² = 648 ÷ 23 = 28.174, and the pooled standard deviation is sp = √28.174 = 5.308 - between the two group SDs, but closer to 6 because Group B is larger.

A raw-data check gives the same recipe. Group A [10, 12, 14] has mean 12 and sample variance (4 + 0 + 4) ÷ 2 = 4, so s₁ = 2. Group B [20, 22, 24, 26] has mean 23 and sample variance (9 + 1 + 1 + 9) ÷ 3 = 6.667, so s₂ ≈ 2.582. The pooled variance is (2 × 4 + 3 × 6.667) ÷ 5 = 28 ÷ 5 = 5.6, the pooled SD is √5.6 ≈ 2.366, and df = 3 + 4 − 2 = 5. Try both examples in the calculator above - the ledger shows every step.

Pooled standard deviation vs individual standard deviations

Each group's own standard deviation describes that group alone; the pooled standard deviation describes the common spread you assume both groups share. It is not the average of the two SDs - you average the variances, weighted by degrees of freedom, and then take the square root. That matters: averaging SDs directly understates the contribution of the more variable group.

Pooling also differs from weighting by sample size alone. The weights here are the degrees of freedom (n − 1), not n, because each group's variance is estimated with n − 1 degrees of freedom. If your data have weights rather than two groups, use our weighted standard deviation calculator instead.

Pooled standard deviation in Excel

Excel has no built-in pooled standard deviation function, so build it from the group statistics. With n₁ in A2, s₁ in B2, n₂ in A3 and s₂ in B3, the pooled variance is =((A2-1)*B2^2+(A3-1)*B3^2)/(A2+A3-2) and the pooled standard deviation is =SQRT(((A2-1)*B2^2+(A3-1)*B3^2)/(A2+A3-2)). If your data are raw lists, get each group's standard deviation first with =STDEV.S(range) - never STDEV.P, which divides by n and would understate the pooled value.

Google Sheets uses the same formulas. For a quick check, the calculator at the top of this page gives the same numbers, including a ledger you can compare with your spreadsheet cell by cell.

Frequently asked questions

What is pooled standard deviation?

A single standard deviation that combines two samples into one estimate of their common spread. It is the square root of the pooled variance: sp = √[((n₁ − 1)·s₁² + (n₂ − 1)·s₂²) / (n₁ + n₂ − 2)]. It is used in two-sample t-tests and confidence intervals for the difference of two means.

How do I calculate pooled standard deviation?

Find the sample variance of each group, multiply each by its degrees of freedom (n − 1), add the two products, divide by the total degrees of freedom (n₁ + n₂ − 2) to get the pooled variance, then take the square root.

What is the pooled SD of n1 = 10, s1 = 4 and n2 = 15, s2 = 6?

The variances are 16 and 36; (9 × 16) + (14 × 36) = 648, and 648 ÷ 23 = 28.174, so sp² = 28.174, sp = 5.308 and df = 23.

When do I use pooled standard deviation?

When you compare two independent means and assume the two populations have equal variances - the standard setting for a two-sample t-test and a confidence interval for the difference of two means. It is not for paired data or for a single sample.

What is the difference between pooled SD and ordinary SD?

An ordinary SD describes one sample. A pooled SD combines two samples: it averages their variances, weighted by degrees of freedom, and takes the square root. With only one group there is nothing to pool, so the pooled SD is just that group's SD.

Can I calculate a pooled SD from raw data?

Yes. Paste the two data lists into the calculator and choose "Raw data lists": it finds each group's mean, sample variance and size, then pools them exactly as the summary formula does.

What are the degrees of freedom for a pooled SD?

df = n₁ + n₂ − 2: each group contributes n − 1 degrees of freedom (one is used up by estimating its mean), and pooling adds them. The two-sample t-test uses the same df.

What if the two variances are very different?

Do not pool. The pooled value assumes equal population variances, and with clearly unequal variances it misstates the spread of both groups. Use Welch's t-test instead, which does not pool, or investigate why the spreads differ.

Keep exploring: once you have the pooled SD and its degrees of freedom, use them to compare two means - or turn individual values into standard scores with our z-score calculator. For proportions rather than means, use our one-proportion z-test calculator, and browse all our free statistics tools. For the formal definitions behind these formulas, see the NIST/SEMATECH e-Handbook of Statistical Methods.