Calculate population variance and sample variance for a small data set, with the mean, squared differences and denominator shown clearly.
Live math canvas
Variance Calculator: 16.5 sample variance. population variance 13.2 · mean 9 · squared-difference total 66 across 5 values
Formula applied
CalculationTime keeps the method visible so the number can be checked instead of blindly trusted.
Mean = sum of values ÷ n. Population variance σ² = Σ(x − mean)² ÷ n. Sample variance s² = Σ(x − mean)² ÷ (n − 1).Your live breakdown
These values come from the controls above and update when the calculator changes.
Resulting answer
population variance 13.2 · mean 9 · squared-difference total 66 across 5 values
Assumptions used
Use population variance for a complete group; use sample variance for an estimate from a sample.
Master’s Tip
Master’s Tip: print the mean, squared-difference total and denominator choice beside the answer. Most variance mistakes are denominator mistakes or value-inclusion mistakes, not arithmetic mysteries.
Printable record
Save the inputs, result, formula, assumptions, page URL and date together so the calculation can be reviewed later.
Mean = sum of values ÷ n. Population variance σ² = Σ(x − mean)² ÷ n. Sample variance s² = Σ(x − mean)² ÷ (n − 1).
For 4, 7, 9, 10 and 15, the mean is 45 ÷ 5 = 9. Squared differences are 25, 4, 0, 1 and 36, for a total of 66. Population variance is 66 ÷ 5 = 13.2. Sample variance is 66 ÷ 4 = 16.5.
Master’s Tip: print the mean, squared-difference total and denominator choice beside the answer. Most variance mistakes are denominator mistakes or value-inclusion mistakes, not arithmetic mysteries.
Standard or basis: descriptive-statistics arithmetic for population and sample variance. The page follows the common n and n − 1 denominator distinction used in statistics teaching and data summaries; no curriculum, scientific or regulatory standard is claimed.
Methodology & Accuracy
CalculationTime pages are built around visible arithmetic: the formula, assumptions, worked example and practical limitations are shown so the result can be checked rather than simply trusted.
Mean = sum of values ÷ n. Population variance σ² = Σ(x − mean)² ÷ n. Sample variance s² = Σ(x − mean)² ÷ (n − 1).
Standard or basis: descriptive-statistics arithmetic for population and sample variance. The page follows the common n and n − 1 denominator distinction used in statistics teaching and data summaries; no curriculum, scientific or regulatory standard is claimed.
Where a calculator follows a named legal, trade or industry standard, that standard is cited visibly. Otherwise the page uses transparent general arithmetic and states its limits.Master’s Tip: print the mean, squared-difference total and denominator choice beside the answer. Most variance mistakes are denominator mistakes or value-inclusion mistakes, not arithmetic mysteries.
Find the mean, subtract the mean from each value, square those differences, add them, divide by the correct denominator, and keep the result in squared units.
Population variance divides by n when the values are the whole group. Sample variance divides by n − 1 when the values are a sample used to estimate a larger population.
No. Variance is the average squared difference from the mean. Standard deviation is the square root of variance, which returns the spread to the original unit scale.
Yes. If every included value is the same, every difference from the mean is zero, so both population and sample variance are zero.
Variance squares each difference from the mean, so its unit is squared. If the data is in metres, variance is in square metres; standard deviation returns to metres.
Variance is a measure of spread around the mean. It is useful when a data record needs the squared-difference audit trail before taking the square root for standard deviation.
Each included value is compared with the mean. The differences are squared so negative and positive deviations do not cancel each other out, and larger deviations carry more weight.
Population variance describes the exact group entered. Sample variance estimates a wider population from a smaller set, so the denominator changes from n to n − 1.
A variance answer is easiest to trust when the report keeps the included values, mean, squared-difference total, denominator and result together. That is useful for homework, lab notes, quality checks and spreadsheet verification.