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Pooled Standard Deviation Calculator

Free pooled standard deviation calculator. Compute pooled SD from multiple groups. Summary stats or

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Why This Statistical Analysis Matters

Why: Statistical calculator for analysis.

How: Enter inputs and compute results.

s_p
STATISTICSDescriptive Statistics

Pooled Standard Deviation — from Multiple Groups for t-tests, ANOVA, Meta-Analysis

Combine variance from multiple groups. Summary stats or raw data. Cohen's d for 2 groups.

T-Test →ANOVA →

Real-World Scenarios — Click to Load

Group summary table
NamenMeanSD
pooled_sd_results.sh
CALCULATED
Pooled SD
2.2550
Pooled Variance
5.0850
Total N
60
Total df
58
Cohen's d
1.0200
Share:
Pooled Standard Deviation
s_p = 2.2550
Variance: 5.0850N: 60df: 58Cohen's d: 1.0200
numbervibe.com/calculators/statistics/pooled-standard-deviation-calculator

Group Comparison (Mean & SD)

Pooled SD vs Group SDs

Calculation Breakdown

COMPUTATION
Total N
60
N = Σnᵢ = 30 + 30
Total df
58
df = Σ(nᵢ−1) = 29 + 29
Σ(nᵢ−1)sᵢ²
294.9300
ext{Weighted} ext{sum} ext{of} ext{variances}
Pooled variance s_p²
5.0850
s_p² = Σ(nᵢ−1)sᵢ² / Σ(nᵢ−1) = 294.93/58
RESULT
Pooled SD s_p
2.2550
s_p = √(s_p²) = √5.0850
Cohen's d
1.0200
d = |mean₁ − mean₂| / s_p = |12.5 − 10.2| / 2.2550

For educational and informational purposes only. Verify with a qualified professional.

Key Takeaways

  • • Pooled standard deviation combines variance from multiple groups, weighted by degrees of freedom.
  • • Two groups: s_p = √(((n₁-1)s₁² + (n₂-1)s₂²) / (n₁+n₂-2)).
  • • k groups: s_p = √(Σ(nᵢ-1)sᵢ² / Σ(nᵢ-1)).
  • • Used in independent-samples t-test, ANOVA, meta-analysis.
  • • Cohen's d uses pooled SD for effect size: d = (mean₁ − mean₂) / s_p.

Did You Know?

📊Pooled SD assumes equal population variances (homoscedasticity). Use Welch t-test when variances differ.Source: NIST
🔬Clinical trials often use pooled SD for the standard error of the difference between means.Source: FDA Guidance
📏The pooled estimate has more df than any single group — more precise estimate of σ.Source: Statistical theory
📈ANOVA uses pooled variance (MS within) as the denominator of the F statistic.Source: NIST Handbook
🎯Cohen's d with pooled SD is the standard effect size for independent-samples t-test.Source: Cohen, 1988
📱A/B testing often uses pooled SD for confidence intervals on the difference.Source: Evan Miller

Expert Tips

Equal Variance Assumption

Pooled SD assumes equal population variances. Check Levene's test.

When to Use

Independent-samples t-test, ANOVA, meta-analysis combining effect sizes.

Welch Alternative

When variances differ, use Welch t-test (does not pool).

Effect Size

Cohen's d: 0.2 small, 0.5 medium, 0.8 large.

Pooled vs Separate SD

AspectPooledSeparate
AssumptionEqual population variancesVariances may differ
Use caset-test, ANOVAWelch t-test
FormulaWeighted by dfs₁, s₂ separately
dfN − kn₁−1, n₂−1

Frequently Asked Questions

What is pooled standard deviation?

A weighted average of group SDs, used when assuming equal population variances. Weight = df per group.

When is it used?

Independent-samples t-test, ANOVA (within-group variance), meta-analysis combining studies.

How does it differ from regular SD?

Regular SD is per group. Pooled SD combines all groups into one estimate.

What is Cohen's d?

Effect size: (mean₁ − mean₂) / pooled SD. 0.2=small, 0.5=medium, 0.8=large.

Can I use it with unequal variances?

Pooled SD assumes equal variances. Use Welch t-test when variances differ.

Pooled SD by the Numbers

df
Σ(nᵢ−1)
s_p²
Pooled variance
s_p
Pooled SD
Cohen's d
Effect size (2 groups)

Disclaimer: Pooled SD assumes equal population variances (homoscedasticity). Use Welch t-test when variances differ significantly. Cohen's d is for 2 groups only.

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