Descriptive Statistics
Variance Calculator
Sample variance divides the corrected sum of squares by n minus 1, and the population variance of the same list is shown beside it.
Worksheet
Press Calculate to see the result.
Sample variance divides the corrected sum of squares by n minus 1, and the population variance of the same list is shown beside it.
What this calculator answers
The values 6, 10, 12, and 16 have a mean of 11. Their squared gaps from 11 add to 52. Dividing 52 by 3, which is n minus 1, gives the sample variance 17.3333, and that is the headline. Dividing the same 52 by 4 gives the population variance 13. Both numbers are in squared units. Showing them side by side makes the divisor visible instead of hiding it in a default. The square roots are printed too, so a variance is not quietly read as a standard deviation.
The headline is the sample variance. The population variance sits beside it. Neither figure is selected by a checkbox on this page. 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
Sample and population variance uses this relationship.
sample = (sum of squared deviations) / (n - 1). population = (sum of squared deviations) / n.
- SS. Corrected sum of squares, the shared numerator.
- n - 1. Sample divisor, used by the headline.
- n. Population divisor, used by the second variance.
- unit. Both variances are in the square of the data's unit.
The numerator is fixed once the mean is fixed. The only choice left is the divisor. A sample divisor pays one degree of freedom for estimating the mean. A population divisor is right only when the list is the entire group. 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 values are 6, 10, 12, 16. The count is 4 and the mean is 11. The corrected sum of squares is 52.
The sample variance divides 52 by 3, giving 17.3333. That sample variance is the headline. The population variance divides the same 52 by 4, giving 13.
The square roots are 4.1633 for the sample and 3.6056 for the population. Both variances are in squared units. The two divisors are shown together so neither one is hidden.
| Sample variance | 17.3333 |
|---|---|
| Population variance | 13 |
| Sample standard deviation | 4.1633 |
| Population standard deviation | 3.6056 |
| Corrected sum of squares | 52 |
A second situation
16 becomes 30, which inflates both variances
The values are 6, 10, 12, 30. The count is 4 and the mean is 14.5. The corrected sum of squares is 339.
The sample variance divides 339 by 3, giving 113. That sample variance is the headline. The population variance divides the same 339 by 4, giving 84.75.
The square roots are 10.6301 for the sample and 9.206 for the population. Both variances are in squared units. The two divisors are shown together so neither one is hidden.
| Sample variance | 113 |
|---|---|
| Population variance | 84.75 |
| Sample standard deviation | 10.6301 |
| Population standard deviation | 9.206 |
| Corrected sum of squares | 339 |
Replacing 16 with 30 raises the corrected sum of squares from 52 to 339. The sample variance rises from 17.3333 to 113, and the population variance rises from 13 to 84.75. 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
16 was replaced by 30. The headline result went from 17.3333 to 113.
The nudged inputs
The values are 6, 10, 12, 30. The count is 4 and the mean is 14.5. The corrected sum of squares is 339.
The sample variance divides 339 by 3, giving 113. That sample variance is the headline. The population variance divides the same 339 by 4, giving 84.75.
The square roots are 10.6301 for the sample and 9.206 for the population. Both variances are in squared units. The two divisors are shown together so neither one is hidden.
| Sample variance | 113 |
|---|---|
| Population variance | 84.75 |
| Sample standard deviation | 10.6301 |
| Population standard deviation | 9.206 |
| Corrected sum of squares | 339 |
How to read the result
Read 17.3333 as a sample variance in squared units. Its square root is the sample standard deviation. The population variance 13 is smaller only because the divisor is larger, not because the data became tighter. 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 mean is estimated from this same list.
- The headline should use n minus 1 even if you also want the population figure.
- All values share one unit.
- You want both divisors visible.
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
- Reporting the sample variance in the original unit.
- Using the population divisor for a sample and calling the result unbiased.
- Taking the square root and still labeling it a variance.
- Averaging the variances of groups with different sizes without weighting by degrees of freedom.
Where a different calculator is the honest choice
Use the standard deviation calculator when the square root should be the headline and a checkbox should pick the divisor. Use the sum of squares calculator when you need the numerator before any division. Use the coefficient of variation when the spread should be divided by the mean. 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: Standard Deviation Calculator, Sum of Squares Calculator, Coefficient of Variation Calculator.
Check yourself
Values are 6, 10, 12, and 16. What is the sample variance, and what is the population variance?
The sample variance is 17.3333 and the population variance is 13.
Using the result in a sentence
A variance is the input to a pooled estimate, an ANOVA piece, and any later step that adds squared units rather than raw units. Keep the count, the unit, and the assumption list beside any copy of the number that leaves this browser tab.
With four points, the two divisors already differ by 25 percent. The gap shrinks as n grows, but a tiny sample never makes the choice unimportant. One distant point still owns the squared numerator. 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 Variance Calculator commit to?
This page uses Sample and population variance: sample = (sum of squared deviations) / (n - 1). population = (sum of squared deviations) / n.. The numerator is fixed once the mean is fixed. The only choice left is the divisor. A sample divisor pays one degree of freedom for estimating the mean. A population divisor is right only when the list is the entire group. 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 Numbers, 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 sample variance is 17.3333. Read 17.3333 as a sample variance in squared units. Its square root is the sample standard deviation. The population variance 13 is smaller only because the divisor is larger, not because the data became tighter. 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?
Reporting the sample variance in the original unit. With four points, the two divisors already differ by 25 percent. The gap shrinks as n grows, but a tiny sample never makes the choice unimportant. One distant point still owns the squared numerator. 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 standard deviation calculator when the square root should be the headline and a checkbox should pick the divisor. Use the sum of squares calculator when you need the numerator before any division. Use the coefficient of variation when the spread should be divided by the mean. 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 sample variance is 17.3333 and the population variance is 13. 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.