Formula
Mean = sum of values ÷ n. Population variance σ² = Σ(x − mean)² ÷ n. Sample variance s² = Σ(x − mean)² ÷ (n − 1).
Math & Statistics
Variance Calculator
Calculate population variance and sample variance for a small data set, with the mean, squared differences and denominator shown clearly.
Math & Statistics
Variance Calculator
Live answer16.5 sample variancepopulation variance 13.2 · mean 9 · squared-difference total 66 across 5 values
Live result16.5 sample variancepopulation variance 13.2 · mean 9 · squared-difference total 66 across 5 values
Formula used
Mean = sum of values ÷ n. Population variance σ² = Σ(x − mean)² ÷ n. Sample variance s² = Σ(x − mean)² ÷ (n − 1).
This is the method behind the answer, so the result can be checked rather than simply trusted.Visual grid
This number is one point on a larger pattern
Variance is not just a final answer. It is a step on a line: before and after, input and output, assumption and result.
Micro-timehours, minutes, shiftsHuman scaledays, weeks, projectsMacro-timemonths, years, calendars
InputFormulaResult
16.5 sample varianceCalculationTime keeps the path visible: the input, the method and the final number belong together.
CalculationTime
Variance Calculation Report
16.5 sample variancepopulation variance 13.2 · mean 9 · squared-difference total 66 across 5 values
Inputs
- Value 1
- 4
- Value 2
- 7
- Value 3
- 9
- Value 4
- 10
- Value 5
- 15
- Value 6
- 0
- Value 7
- 0
- How many values to include
- 5 2 to 7
Method
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.
Assumptions
- The active count is rounded to a whole number from 2 to 7.
- Only Value 1 through the active count are included; later boxes are ignored.
- Population variance divides by n when the entered values are the whole group being described.
- Sample variance divides by n − 1 when the entered values are a sample used to estimate a wider population.
Notes
Use this space on the printed report for client, supplier, classroom, job-location, measurement, quote or approval notes.
Worked example
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.
Professional note
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.
Regional and unit assumptions
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.
Assumptions and limitations
- The active count is rounded to a whole number from 2 to 7.
- Only Value 1 through the active count are included; later boxes are ignored.
- Population variance divides by n when the entered values are the whole group being described.
- Sample variance divides by n − 1 when the entered values are a sample used to estimate a wider population.
- This page uses unweighted numeric values; grouped frequency tables and weighted variance need a separate method.
Methodology & Accuracy
How this calculator is checked
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.
Formula used
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
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
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.
Related calculators
Questions
How do you calculate variance?
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.
What is the difference between population variance and sample variance?
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.
Is variance the same as standard deviation?
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.
Can variance be zero?
Yes. If every included value is the same, every difference from the mean is zero, so both population and sample variance are zero.
Why are the units squared?
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.
Calculation note
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.
Variance starts with distance from the mean
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 and sample variance answer different questions
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.
The printable report is the audit trail
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.