Pooled Standard Deviation Calculator

📖 What is Pooled Standard Deviation?

Pooled standard deviation (S_p) is a single estimate of spread that combines the standard deviations of two or more groups. Rather than simply averaging the group SDs, it weights each group's variance by its degrees of freedom (n−1), giving more influence to larger samples. The result is a more precise estimate of the common population variance than any individual group SD alone.

The concept arises directly from the assumptions of the two-sample independent t-test. When you assume that both groups share the same underlying variance σ² (the equal-variance or homoscedasticity assumption), you pool the sample variances to get the best possible estimate of σ². The pooled SD is the square root of that pooled variance estimate.

Pooled SD is also the foundation of one-way ANOVA. In ANOVA with k groups, the Mean Square Within (MSW) is exactly the pooled variance, and its square root is the pooled SD. When the F-statistic tests whether group means differ, MSW is the denominator - meaning pooled SD is the baseline measure of within-group variability against which between-group differences are judged.

The key assumption when using pooled SD is that the group variances are approximately equal. You can verify this with Levene's test or Bartlett's test before pooling. If the variances differ substantially, Welch's t-test (which does not assume equal variances and does not pool the SDs) is more appropriate and more robust.

This calculator supports up to 5 groups and offers two input modes: Summary Stats (enter n and SD directly) for when you already have group statistics, and Raw Data (paste numbers) for when you want the calculator to compute each group's SD automatically from the raw observations.

📐 Formula

Two-group form:

Where:

• n_i = sample size of group i
• s_i = sample standard deviation of group i
• n_i−1 = degrees of freedom contributed by group i
• Pooled variance S_p² = the numerator divided by the denominator (before taking the square root)
• Total df = Σn_i − k (where k = number of groups)

Worked example: Two groups: Group A (n=10, s=4), Group B (n=12, s=6).
Numerator = (10−1)×16 + (12−1)×36 = 144 + 396 = 540.
Denominator (df) = (10−1) + (12−1) = 9 + 11 = 20.
Pooled variance = 540 / 20 = 27. Pooled SD = √27 ≈ 5.196.

Note that the pooled SD (5.196) lies between s₁=4 and s₂=6, closer to 6 because Group B has more observations and therefore more influence.

✍ How to Use This Calculator

1. Choose a mode — click Summary Stats if you already have each group's sample size and SD, or Raw Data to paste the actual numbers.
2. Enter group data — for Summary Stats, enter n (≥2) and SD for each group. For Raw Data, paste comma-separated numbers for each group. Use + Add Group for up to 5 groups.
3. Click Calculate Pooled SD — the pooled standard deviation, pooled variance, total degrees of freedom, and a per-group table appear instantly.
4. Interpret the output — use pooled SD in your t-test or ANOVA calculation. If the individual group SDs differ by more than a factor of 2, consider Welch's t-test instead of pooling.

💡 Example Calculations

Example 1 — Two lab groups (summary stats)

Example 2 — Unequal group sizes (clinical trial)

Example 3 — Three classroom groups

Example 4 — Raw data mode (quality control)

❓ Frequently Asked Questions

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