Pooled Variance Calculator
Combines standard deviations or variances from unlimited groups into one weighted, reliable metric — with degrees of freedom handled automatically.
Pooled variance and pooled standard deviation for t-tests, ANOVA and quality control.
Format: Size, Value — e.g. 10, 2.5. You can paste directly from Excel (sample size column, then value column).
Pooled variance (s²p)
8.0893
Weighted average computed across all your groups
Pooled standard deviation
2.8442
√s²p — for t-tests and confidence intervals
Degrees of freedom
42
N − k, combined automatically
Total groups (k)
3
Total sample size
45
Step by step
- 1
Squared your SD values
2.50² = 6.25 · 3.10² = 9.61 · 2.80² = 7.84
- 2
Weighted sum
(10−1)×6.25 + (15−1)×9.61 + (20−1)×7.84 = 339.75
- 3
Divide by df
339.75 ÷ 42 = 8.0893
About this calculator
Sometimes you have data from two or more groups — Class A and Class B, or samples from several production lines — that share a similar level of spread. Instead of calculating each variance separately, pooled variance combines them into one reliable, weighted average.
Why weighted? Imagine Group A has 1,000 people and Group B has 5. You should not average their variances equally — the larger group gives more reliable data. Pooled variance gives more importance to larger groups, keeping the estimate of population variability accurate.
Formula
s²p = [(n₁−1)s₁² + (n₂−1)s₂² + … + (nₖ−1)sₖ²] ÷ [(n₁−1) + (n₂−1) + … + (nₖ−1)]
How to use it
- 1
Pick the value type: Std Dev mode squares your SD values automatically (2.5 becomes 6.25); Variance mode uses values as entered.
- 2
Type each group on a separate line as Size, Value — e.g. 10, 2.5 — or paste directly from Excel. Unlimited groups.
- 3
Read the pooled variance (for ANOVA and F-tests) and the pooled standard deviation (for t-tests and confidence intervals), with degrees of freedom computed automatically.
Example: three production lines
n=10 with SD 2.5, n=15 with SD 3.1, n=20 with SD 2.8 → variances 6.25, 9.61, 7.84 → numerator 339.75 over df 42 → pooled variance 8.0893, pooled SD 2.8442. Press Reset to reproduce every number.
How this calculator works
Many students confuse standard deviation (s) with variance (s²) — the value-type toggle prevents that mistake. In SD mode the tool squares each value before pooling; in Variance mode it uses them as entered.
Each line is one group: the first number is the sample size, the second the SD or variance. The tool weights every group's variance by its degrees of freedom (n − 1), sums, and divides by the total degrees of freedom — handling k groups at once.
df total = (n₁−1) + (n₂−1) + … = N total − k
1. The assumption of homogeneity
Pooled variance works only when groups have a similar spread — statisticians call this homoscedasticity. If one group's variance is 5 and another's is 500, pooling gives a misleading result: use Welch's t-test instead of a pooled test.
2. Degrees of freedom combine
One sample contributes n − 1 degrees of freedom; combined groups add up: df total = (n₁−1) + (n₂−1) + …, or simply N total − k. The calculator shows this value — you need it to find critical values in statistical tables.
When to use this calculator
Independent t-tests
Independent t-tests compare the means of two separate groups — two medicines, two teaching methods. The pooled variance estimates the shared standard error that the test statistic is built on.
ANOVA
In ANOVA, the Mean Square Error (MSE, also called Mean Square Within) is a pooled variance. This calculator hands you that value directly — no full ANOVA table needed.
Quality control
Factories sample products from several machines each hour. Pooling the variances of all samples gives a better estimate of the factory's overall process performance than any single machine's data.
Frequently asked questions
Can I calculate pooled variance with only sample sizes and means?
No. You also need the standard deviation or variance for each group — the mean never appears in the pooled variance formula.
Is pooled variance the same as average variance?
No — they match only when sample sizes are equal. When sizes differ, pooled variance weights larger groups more heavily, so the result stays closer to the larger group's variance.
What if one group has n = 1?
A group with a single value cannot be used. The formula relies on n − 1, which becomes zero when n = 1 — and variance needs at least two data points to exist.
When should I avoid pooled variance?
Avoid it when group variances differ too much — for example, when one is four times larger than another — or when the data is highly skewed. In those cases the pooled result does not represent the groups; Welch's t-test is the safer route.