This pooled standard deviation calculator combines within-group sample spread from raw observations, sample SDs or sample variances. It shows pooled variance, degrees of freedom and each group's contribution.
Pool within-group spread with degrees of freedom
Pooled standard deviation combines spread within separate samples. It weights each sample variance by that sample's degrees of freedom, then takes the square root. The variance weights use sample size minus one; averaging the SDs directly does not give the same quantity. All groups must use the same measurement unit. SD keeps that unit, while variance and the within-group squared-deviation sum use its square. Keep raw groups in separate input boxes, or switch to known sample sizes and sample SDs. A separate sample-variance mode accepts squared-unit summaries directly. Additional groups are optional. Every entered group contributes, even when the calculation details show only the first group rows. The arithmetic result describes within-group pooling. Interpreting it as an estimate of a shared population spread requires an appropriate common-variance model; the tool does not test that assumption. The NYU Stern pooling note states the usual normal/common-SD model and allows the group means to differ. This calculator supplies the descriptive quantity, not a hypothesis test or confidence interval.
Enter sample SDs or sample variances. Population-denominator summaries are not interchangeable.
Use the same measurement unit for all groups. Sample variances use the squared unit.
Pool within-group spread. A difference between group means does not enter this numerator.
Check the pooled variance and its square root
The raw preset has group observations 2 4 6 and 10 12 14 16. Their counts are 3 and 4, and their centered squared-deviation sums are 8 and 20. The sum is 28. Each group contributes its count minus one, so total degrees of freedom are 5.
Divide the within-group sum by total degrees of freedom to obtain pooled variance 5.6. Its square root gives pooled SD approximately 2.366431913. The engine keeps unrounded quantities throughout calculation; displayed working is rounded afterward.
The known-SD preset uses sample sizes 40 and 38 with SDs 18.5 and 16.7. The engine reports total degrees of freedom 76, pooled variance 311.4036842 and pooled SD approximately 17.64663379. Squaring each SD and weighting the resulting variance by its sample degrees of freedom reproduces the result. It is not the arithmetic average of those SDs.
Enter raw groups, sample SDs or sample variances
With raw observations, the tool computes each group's arithmetic mean and centered sum of squared deviations. Dividing that group sum by its sample degrees of freedom gives the group sample variance. Pooling uses the raw centered sums directly, so a rounded displayed group SD is not squared back into the calculation. The OpenStax spread definitions explain the sample denominator.
With known SDs, enter each sample size and sample SD. With known variances, enter each sample size and sample variance instead. The tool does not infer group means or reconstruct observations from those summaries. Means are unnecessary for this within-group quantity. Every group needs at least two observations; nonnegative SDs and variances are allowed, including zero.
The additional-group fields use one group per line. For additional raw groups, put the observations for each extra group on its own line; newlines in either main group box still separate observations within that box. For summary groups, enter one sample-size/spread pair per line, matching the selected SD or variance mode.
The three-variance preset has 3 groups and 15 total observations. Its total degrees of freedom are 12, within-group sum is 72, pooled variance is 6 and pooled SD is approximately 2.449489743. NIST's degrees-of-freedom pooling formula supplies the general weighted-variance identity in a calibration setting. Here each ordinary sample contributes its size minus one; the tool does not import calibration-specific degrees of freedom.
Keep pooled spread separate from combined and paired spread
The standard deviation of all observations concatenated into one dataset includes spread between group means. Within-group pooling does not. Changing group 2 the raw preset to 1010 1012 1014 1016 changes its mean from 13 to 1,013 while pooled SD stays approximately 2.366431913. This is a property of the computed quantity, not evidence that the samples have equal population variances.
Paired-difference SD is another quantity. Separate group sizes and SDs do not determine the correlation or paired differences. This tool does not estimate paired-difference spread, choose a statistical test, report a p-value or confidence interval, or establish a common variance from the entered data.
For the spread of one actual dataset, the Standard Deviation Calculator supports raw observations, exact frequencies and grouped midpoint estimates. The Coefficient of Variation Calculator compares SD with a nonzero mean and explains measurement-scale limits. The Z-Score Calculator measures an observation's distance from a reference mean in SD units. The Mean Median Mode Calculator examines centers without pooling sample spreads.
Local limits are 5,000 raw observations across all groups, 5,000 summary groups, 1,000,000,000 total summarized observations and 100,000 characters across active inputs. Individual number tokens support 128 characters, and nonzero entered magnitudes range from 1e-100 to 1e+100. The detail display lists the first 100 groups; every accepted group contributes to the result.
The engine uses IEEE754 numbers with compensated sums and centered raw-group spread. Displays use 10 significant digits after calculation. Differences lost when an input is represented cannot be recovered. Negative SDs/variances, invalid tokens and undersized samples are rejected rather than silently omitted.