Calculate Pooled Standard Deviation
Use this tool to compute pooled standard deviation for two groups (most common) or for multiple groups.
Two-Group Calculator
Multiple-Group Calculator (Optional)
Enter matching lists. Example: sample sizes = 20, 22, 18 and SDs = 5.1, 4.8, 6.0.
What is pooled standard deviation?
Pooled standard deviation is a weighted estimate of spread across two or more groups. Instead of averaging standard deviations directly, it combines variances while accounting for each group’s degrees of freedom. This is why pooled SD is a core part of many statistical procedures, including the independent-samples t-test and effect size calculations like Cohen’s d.
Why use pooled SD instead of a simple average?
A plain average of SD values can mislead when sample sizes are different. Pooled SD gives more influence to groups with more information (larger n), and it uses the mathematically correct weighting scheme: n - 1 for each group.
- It respects sample size differences.
- It pools variance, not raw SDs.
- It aligns with standard inferential methods.
Two-group pooled standard deviation formula
Core equation
sp = √[ ((n1 - 1)s12 + (n2 - 1)s22) / (n1 + n2 - 2) ]
Where:
- n₁, n₂ are sample sizes
- s₁, s₂ are sample standard deviations
- sp is the pooled standard deviation
Quick worked example
Suppose Group A has n = 25 and SD = 10, while Group B has n = 30 and SD = 12.
- Numerator = (24)(10²) + (29)(12²) = 2400 + 4176 = 6576
- Denominator = 25 + 30 - 2 = 53
- Pooled variance = 6576 / 53 = 124.0755
- Pooled SD = √124.0755 = 11.139
This means the combined within-group spread is about 11.14.
Multiple-group pooled standard deviation
If you have more than two groups, use the generalized pooled formula shown in the calculator. The same principle applies: sum weighted variances using each group’s n - 1, divide by total degrees of freedom, then take the square root.
When pooled SD is appropriate
- Comparing group means when variance is assumed similar across groups.
- Computing Cohen’s d for two independent groups.
- Creating a single within-group spread estimate in reports.
If group variances are dramatically different, consider methods that do not assume homogeneity of variance (for example, Welch’s t-test).
Common mistakes to avoid
- Averaging SDs directly: this is not statistically correct.
- Using n instead of n - 1: pooled variance uses degrees of freedom.
- Using negative SD values: SD cannot be negative.
- Ignoring variance inequality: pooled methods assume similar variance.
How this connects to effect size (Cohen’s d)
Once you have pooled SD, Cohen’s d is straightforward:
d = (M₁ - M₂) / sp
A larger absolute d indicates a larger standardized difference between group means.
Final thoughts
A pooled standard deviation calculator saves time and reduces arithmetic errors, especially when working with multiple studies, classroom data, A/B tests, or research projects. Use the calculator above for both two-group and multi-group scenarios, and always check whether equal-variance assumptions are reasonable before relying on pooled metrics.