Variance with every step: the mean, each deviation, its square and the total.
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Variance is the average squared distance of the values from their mean. Squaring makes every deviation positive and gives more weight to values far from the mean. It is the basis of the standard deviation, ANOVA, regression and portfolio risk.
The sample variance divides the sum of squares by n − 1; the population variance divides by n. The calculator gives both, along with the standard deviation (the square root of the variance) and the full table of deviations.
| Data | 10, 12, 23, 23, 16, 23, 21, 16 |
| Data is a | Sample (divide by n − 1) |
| Sample variance s² | 27.42857143 |
| Sample standard deviation | 5.237229366 |
| Population SD σ (÷ n) | 4.898979486 (variance 24) |
| Mean | 18 |
| Count n | 8 |
| Sum | 144 |
| Sum of squared deviations | 192 |
| Standard error of the mean | 1.8516402 |
| Range | 10 to 23 (13) |
| Coefficient of variation | 29.096% |
| Within 1 SD of the mean | 6 of 8 values (75%) |
s² = 27.42857143.
| Data | 2, 4, 4, 4, 5, 5, 7, 9 |
| Data is a | Population (divide by n) |
| Population variance σ² | 4 |
| Population standard deviation | 2 |
| Sample SD s (÷ (n − 1)) | 2.138089935 (variance 4.571428571) |
| Mean | 5 |
| Count n | 8 |
| Sum | 40 |
| Sum of squared deviations | 32 |
| Standard error of the mean | 0.75592895 |
| Range | 2 to 9 (7) |
| Coefficient of variation | 40% |
| Within 1 SD of the mean | 6 of 8 values (75%) |
σ² = 4.
Squaring makes negative and positive deviations count equally and is mathematically convenient — variances of independent variables add up.
No. It is a sum of squares divided by a positive number, so it is 0 (all values equal) or more.
When the data are a sample. Dividing by n − 1 makes the sample variance an unbiased estimate of the population variance.