When comparing two or more groups in statistics, you often need to estimate their shared variability. That’s where the pooled standard deviation comes in.
If you’ve ever searched “how to calculate the pooled standard deviation”, you’re likely analyzing test scores, experimental data, or comparing performance across multiple samples. This guide explains:
- What pooled standard deviation means
- The formula for 2+ groups
- Step-by-step calculation examples
- When it’s appropriate to use it
- How it connects to t-tests and effect size (Cohen’s d)
What is Pooled Standard Deviation?
The pooled standard deviation is a weighted average of the standard deviations from two or more independent groups.
Instead of simply averaging the standard deviations, it accounts for sample sizes, giving more weight to larger groups.
This makes it especially useful in:
- Two-sample t-tests
- ANOVA (Analysis of Variance)
- Effect size calculation (Cohen’s d)
Formula for Pooled Standard Deviation
For Two Groups
sp=(n1−1)s12+(n2−1)s22n1+n2−2s_p = \sqrt{ \frac{ (n_1 – 1)s_1^2 + (n_2 – 1)s_2^2 }{n_1 + n_2 – 2} }sp=n1+n2−2(n1−1)s12+(n2−1)s22
Where:
- sps_psp = pooled standard deviation
- s1,s2s_1, s_2s1,s2 = standard deviations of groups 1 and 2
- n1,n2n_1, n_2n1,n2 = sample sizes of groups 1 and 2
For More than Two Groups
sp=∑(ni−1)si2∑(ni−1)s_p = \sqrt{ \frac{ \sum (n_i – 1)s_i^2 }{ \sum (n_i – 1) } }sp=∑(ni−1)∑(ni−1)si2
Where:
- sis_isi = standard deviation of group iii
- nin_ini = sample size of group iii
Step-by-Step Example
Example 1: Two Groups
Group A:
- n1=10n_1 = 10n1=10, s1=4s_1 = 4s1=4
Group B:
- n2=12n_2 = 12n2=12, s2=6s_2 = 6s2=6
sp=(10−1)(42)+(12−1)(62)10+12−2s_p = \sqrt{ \frac{ (10-1)(4^2) + (12-1)(6^2) }{10+12-2} }sp=10+12−2(10−1)(42)+(12−1)(62) sp=9(16)+11(36)20s_p = \sqrt{ \frac{ 9(16) + 11(36) }{20} }sp=209(16)+11(36) sp=144+39620=27≈5.20s_p = \sqrt{ \frac{ 144 + 396 }{20} } = \sqrt{27} \approx 5.20sp=20144+396=27≈5.20
Interpretation: The pooled standard deviation across both groups is 5.20.
Example 2: Three Groups
- Group 1: n=8, s=5n=8, \; s=5n=8,s=5
- Group 2: n=10, s=4n=10, \; s=4n=10,s=4
- Group 3: n=12, s=6n=12, \; s=6n=12,s=6
sp=(7)(25)+(9)(16)+(11)(36)7+9+11s_p = \sqrt{ \frac{ (7)(25) + (9)(16) + (11)(36) }{7+9+11} }sp=7+9+11(7)(25)+(9)(16)+(11)(36) sp=175+144+39627s_p = \sqrt{ \frac{ 175 + 144 + 396 }{27} }sp=27175+144+396 sp=715/27≈5.14s_p = \sqrt{ 715/27 } \approx 5.14sp=715/27≈5.14
Interpretation: The pooled standard deviation across all three groups is 5.14.
When to Use Pooled Standard Deviation
You should use pooled standard deviation when:
- Comparing two or more independent groups
- Assuming the groups have equal (or nearly equal) population variances
- Conducting independent-sample t-tests
- Calculating effect size (Cohen’s d)
🚨 Caution: If the group variances are very different, pooled standard deviation may not be appropriate. Instead, use Welch’s t-test or report group-specific SDs.
Applications in Statistics
1. T-Tests
Pooled standard deviation is the denominator in the two-sample t-test statistic when equal variances are assumed.
2. Effect Size (Cohen’s d)
Cohen’s d formula uses pooled SD: d=xˉ1−xˉ2spd = \frac{\bar{x}_1 – \bar{x}_2}{s_p}d=spxˉ1−xˉ2
3. ANOVA
ANOVA uses pooled variance (squared pooled SD) to test whether group means differ significantly.
Practical Shortcuts
- Excel/Google Sheets: You can compute pooled SD using:
=SQRT(((n1-1)*s1^2 + (n2-1)*s2^2) / (n1+n2-2))
- Calculator: Use our online <a href=”https://gradewisecalculator.com/pooled-standard-deviation-calculator/” target=”_blank” rel=”noopener”>pooled standard deviation calculator</a> to skip manual computation.
Related Articles
- Convert Percentile to Score
- Z-Score Explained – How pooled SD relates to standardization
- Empirical vs. Parametric Percentiles – When pooled SD is not appropriate
FAQs: How to Calculate the Pooled Standard Deviation
Q1. What does pooled standard deviation mean in simple words?
It’s a single estimate of spread (variability) across multiple groups, weighted by their sample sizes.
Q2. When should I not use pooled standard deviation?
Avoid it when the group variances differ greatly — instead, use Welch’s test.
Q3. Is pooled standard deviation the same as weighted SD?
They are related, but pooled SD specifically weights variances by sample size minus one.
Q4. How do you calculate pooled variance?
It’s the square of the pooled standard deviation: sp2=∑(ni−1)si2∑(ni−1)s_p^2 = \frac{ \sum (n_i – 1)s_i^2 }{ \sum (n_i – 1) }sp2=∑(ni−1)∑(ni−1)si2
Q5. Can I calculate pooled SD for more than two groups?
Yes. Use the general formula — it works for any number of groups.
Final Thoughts
Learning how to calculate the pooled standard deviation is essential in statistics whenever you compare group means. It gives you a fair, weighted estimate of variability across multiple samples.
Whether you’re running t-tests, calculating effect size, or performing ANOVA, pooled SD helps unify variability into a single value.
➡ Want instant results? Use our pooled standard deviation calculator to save time and reduce calculation errors.