Descriptive Statistics
Pooled Variance Calculator
Pooled variance combines two sample variances with weights n minus 1, and the headline is that combined variance, not its square root.
Worksheet
Press Calculate to see the result.
Pooled variance combines two sample variances with weights n minus 1, and the headline is that combined variance, not its square root.
What this calculator answers
A first sample of 10 has standard deviation 4, and a second sample of 12 has standard deviation 5. The quantities that add across groups are the sums of squares, which means the standard deviations must be squared before they are weighted. Each weight is one less than the sample size. The pooled variance divides the combined sum of squares by n1 + n2 - 2. Leaving the result as a variance, rather than a standard deviation, is the point of this page: later steps that add variances want the square unit.
The headline is the pooled variance. The square root is shown as a supporting figure so it is not mistaken for the headline. Supporting counts and intermediate terms sit under the headline so a reader can see what was divided by what. When a checkbox or a tail setting changes the divisor or the rejection region, that choice appears in the steps rather than vanishing into a silent default.
The formula, written out
Pooled variance uses this relationship.
sp^2 = ((n1 - 1) s1^2 + (n2 - 1) s2^2) / (n1 + n2 - 2)
- s1^2. Square of the first sample standard deviation.
- n1 - 1. Degrees of freedom, and the weight, for group 1.
- sp^2. Pooled variance, the headline.
- n1 + n2 - 2. Denominator after the two sums of squares are added.
Multiplying a sample variance by n minus 1 recovers that sample's corrected sum of squares. Adding the two sums and dividing by the total degrees of freedom is the weighted average of the two variances. The square root is a different page. The algebra is the named textbook identity for this worksheet. Intermediate products and quotients appear in order so a calculator and a pencil can agree to the displayed rounding. If a neighbouring page uses a different divisor or a continuity correction, that page says so on its own ticket.
A full worked example
Example loaded in the worksheet
The form is prefilled with a small example so every arithmetic step can be checked by hand. Change one field after you finish the walkthrough and confirm the ticket moves the way the formula predicts.
The inputs are sample standard deviations, so each one is squared before it is pooled. Group 1: n = 10, s = 4, sum of squares = 9 * 16 = 144. Group 2: n = 12, s = 5, sum of squares = 11 * 25 = 275.
Add the sums of squares: 144 + 275 = 419. The denominator is 10 + 12 - 2 = 20. The pooled variance is 419 / 20 = 20.95.
That variance is the headline. Its square root is 4.5771, shown only as a supporting figure. This page does not convert the variance into a test statistic.
| Pooled variance | 20.95 |
|---|---|
| Square root of pooled variance | 4.5771 |
| Degrees of freedom | 20 |
A second situation
The second standard deviation rises from 5 to 8
The inputs are sample standard deviations, so each one is squared before it is pooled. Group 1: n = 10, s = 4, sum of squares = 9 * 16 = 144. Group 2: n = 12, s = 8, sum of squares = 11 * 64 = 704.
Add the sums of squares: 144 + 704 = 848. The denominator is 10 + 12 - 2 = 20. The pooled variance is 848 / 20 = 42.4.
That variance is the headline. Its square root is 6.5115, shown only as a supporting figure. This page does not convert the variance into a test statistic.
| Pooled variance | 42.4 |
|---|---|
| Square root of pooled variance | 6.5115 |
| Degrees of freedom | 20 |
Raising s2 from 5 to 8 changes the second sum of squares from 275 to 704. The pooled variance rises from 20.95 to 42.4, which is a change in the squared unit, not a change you should describe as if it were already a standard deviation. The second example changes the feature that this formula is sensitive to, and the ticket shows the new figure beside the same definition. That contrast is the fastest way to see which input owns the movement.
What moves when the input moves
s2 changed from 5 to 8. The sample sizes and s1 stayed put. The headline result went from 20.95 to 42.4.
The nudged inputs
The inputs are sample standard deviations, so each one is squared before it is pooled. Group 1: n = 10, s = 4, sum of squares = 9 * 16 = 144. Group 2: n = 12, s = 8, sum of squares = 11 * 64 = 704.
Add the sums of squares: 144 + 704 = 848. The denominator is 10 + 12 - 2 = 20. The pooled variance is 848 / 20 = 42.4.
That variance is the headline. Its square root is 6.5115, shown only as a supporting figure. This page does not convert the variance into a test statistic.
| Pooled variance | 42.4 |
|---|---|
| Square root of pooled variance | 6.5115 |
| Degrees of freedom | 20 |
How to read the result
Read 20.95 as a variance. Its square root is about 4.5771, and that root is not the headline. If you needed the root, the pooled standard deviation page is the one that leads with it. Say the unit, the sample size or contract terms, and the formula name in the same sentence as the figure. A number without those three pieces invites a later reader to invent a definition that this page never used.
Write the result with its unit, the inputs that produced it, and the name of the formula. Write the result with its unit, the inputs that produced it, and the name of the formula. If the ticket shows a decision sentence, quote the alpha and the tail with it. If the ticket shows a money or dose figure, quote the rate and the schedule that produced it.
Assumptions that have to stay true
- The typed spreads are sample standard deviations, not variances and not ranges.
- Both groups are treated as sharing one variance.
- Both sizes are integers of at least 2.
- The two standard deviations use the same unit.
If an assumption above is false, do not average the discomfort away. Change the worksheet or change the question. A correct arithmetic step on the wrong model is still the wrong answer for the decision you are trying to make.
Mistakes that change the answer
- Pooling the standard deviations without squaring.
- Entering a variance and squaring it again.
- Dividing by n1 + n2 instead of by n1 + n2 - 2.
- Reporting the variance in the original unit of the data.
Where a different calculator is the honest choice
Use the pooled standard deviation calculator when the headline should be the square root and the standard error of a difference. Use the variance calculator for one list. Use the standard deviation calculator when you still have the raw numbers. Related links under the essay open pages that keep their own formulas instead of hiding a mode switch inside this ticket.
When the job changes, move to a page built for that job: Pooled Standard Deviation Calculator, Variance Calculator, Standard Deviation Calculator.
Check yourself
n1 is 10, s1 is 4, n2 is 12, and s2 is 5. What is the pooled variance?
The pooled variance is 20.95.
Using the result in a sentence
Textbook two-sample problems that hand you n and s, rather than the raw rows, are the place this variance is assembled. Keep the count, the unit, and the assumption list beside any copy of the number that leaves this browser tab.
The page cannot tell whether the two groups really share a variance. It also cannot repair standard deviations that were computed with the wrong divisor before you typed them. Pair the headline with the steps before you staff a route, set a grade boundary, or compare two contracts from a single number.
Questions people ask next
01What definition does the Pooled Variance Calculator commit to?
This page uses Pooled variance: sp^2 = ((n1 - 1) s1^2 + (n2 - 1) s2^2) / (n1 + n2 - 2). Multiplying a sample variance by n minus 1 recovers that sample's corrected sum of squares. Adding the two sums and dividing by the total degrees of freedom is the weighted average of the two variances. The square root is a different page. If a textbook uses a different divisor, a different continuity correction, or a different rounding rule, the last digits can move, so the definition is printed above the result. The worksheet asks for Sample size n1, Standard deviation s1, Sample size n2, Standard deviation s2, and it refuses a blank that the formula cannot interpret. Keep that printed definition in the sentence you write next to the figure so a later reader can rebuild the arithmetic without guessing which convention you chose.
02How should the worked example be read out loud?
In the loaded example the pooled variance is 20.95. Read 20.95 as a variance. Its square root is about 4.5771, and that root is not the headline. If you needed the root, the pooled standard deviation page is the one that leads with it. Say the unit and the count before you say whether the number is large or small. Read the steps in order rather than jumping to the headline, because the intermediate terms are where a swapped percent and proportion usually hide. If the ticket shows both a primary figure and a comparison figure, name which one you are quoting.
03What is the fastest way to get a misleading result?
Pooling the standard deviations without squaring. The page cannot tell whether the two groups really share a variance. It also cannot repair standard deviations that were computed with the wrong divisor before you typed them. A second common failure is leaving a checkbox or tail setting in the wrong state and then treating the number as if the default had matched your plan. Reset the example, change one input, and confirm the ticket moves in the direction the formula predicts before you trust a live run.
04When should I leave this page for a different calculator?
Use the pooled standard deviation calculator when the headline should be the square root and the standard error of a difference. Use the variance calculator for one list. Use the standard deviation calculator when you still have the raw numbers. The related links under the essay go to those worksheets, which keep their own formulas instead of hiding them in a mode switch. If the question changed from a summary to a test, from a payment to a refinance, or from a raw score to a percentile, switch pages rather than forcing this ticket to answer a job it was not built for.
05What belongs in the sentence next to the number?
The pooled variance is 20.95. Name the formula, the sample size or the inputs, and the unit before you compare the figure with someone else's. A bare number without those three pieces cannot be audited when a score, a rate, or a lab value is challenged later. Store the inputs beside the result whenever the figure will travel into a report.