Enter up to 10 values to calculate both population variance and sample variance. Variance measures the average squared deviation from the mean.
Variance measures how spread out a dataset is from its mean — specifically, the average squared deviation from the mean. While standard deviation (its square root) is more intuitive because it shares units with the original data, variance has mathematical advantages that make it the workhorse of statistical theory: variances add when independent random variables combine, while standard deviations don't.
This calculator returns both population variance (σ², divides by N) and sample variance (s², divides by N-1, Bessel's correction). Almost all real-world datasets are samples of larger populations, so sample variance is the appropriate choice in practice. Population variance only applies when you've measured every member of your population.
Variance is the heart of analysis of variance (ANOVA), regression diagnostics, portfolio theory, and statistical process control. While most communication uses standard deviation, calculations in advanced statistics typically use variance for cleaner algebra.
**Scenario:** Product A weights: 100, 102, 98, 100, 101. Product B weights: 100, 80, 120, 100, 90. **Calculation:** A mean: 100.2, deviations: -0.2, 1.8, -2.2, -0.2, 0.8. Squared: 0.04, 3.24, 4.84, 0.04, 0.64 = 8.8. Sample variance: 8.8/4 = 2.2. B mean: 98, deviations: 2, -18, 22, 2, -8. Squared: 4, 324, 484, 4, 64 = 880. Sample variance: 880/4 = 220. **Result:** Product A variance = 2.2; Product B variance = 220. Product A is 100× more consistent than B. Variance reveals quality control differences not apparent from means alone.
**Scenario:** Stock returns over 5 years: 8%, 12%, -5%, 15%, 20%. **Calculation:** Mean: 10%. Deviations: -2, 2, -15, 5, 10. Squared: 4, 4, 225, 25, 100 = 358. Sample variance: 358/4 = 89.5. Sample SD: √89.5 = 9.46%. **Result:** Portfolio variance 89.5; SD 9.46% (volatility). Compare against benchmark like S&P 500 (~15% annual SD) to assess relative risk. Lower variance = more predictable returns; higher = riskier.
**Scenario:** Tool dimension specifications: 50.00 ± 0.05mm. Measured: 49.97, 50.02, 50.01, 49.99, 50.00, 50.03, 49.98, 50.02. **Calculation:** Mean: 50.00. Deviations summed (squared): 0.0009, 0.0004, 0.0001, 0.0001, 0, 0.0009, 0.0004, 0.0004 = 0.0032. Sample variance: 0.0032/7 = 0.000457. Sample SD: √0.000457 = 0.0214mm. **Result:** SD of 0.0214mm is within tolerance (0.05mm). Variance shows the manufacturing process is producing consistent parts. Quality control charts track variance over time.
**Calculate variance for:**
- **Statistical theory work**: variance is mathematically convenient. - **Multi-variable analysis**: variances add for independent variables. - **Regression and ANOVA**: variance partitioning is fundamental. - **Portfolio theory**: portfolio variance combines individual variances. - **Process control**: variance tracks manufacturing consistency. - **Comparing data quality**: low variance = consistent.
**Variance interpretation:**
- **Zero variance**: all values identical (rare in real data). - **Small variance**: high consistency, tight clustering. - **Large variance**: spread out, less predictable. - **Compare to mean**: CV = SD/mean × 100%.
**Population vs sample variance:**
- **Population (σ²)**: when you have all data; rare. - **Sample (s²)**: most cases; standard practice. - **Bessel correction**: N-1 instead of N; corrects for sample mean usage. - **Effect**: sample variance is slightly larger than population variance.
**Variance in portfolio theory:**
- **Single stock variance**: stock's risk. - **Portfolio variance**: combines individual variances + covariances. - **Diversification**: combining uncorrelated assets reduces overall variance. - **Modern Portfolio Theory**: maximizing return per unit of variance.
**ANOVA basics:**
- **Total variance**: sum of all squared deviations from grand mean. - **Between-group variance**: variation explained by group differences. - **Within-group variance**: variation within groups. - **F-statistic**: ratio of between to within variance.
**Common variance applications:**
| Field | Use | |---|---| | Engineering | Quality control, reliability | | Finance | Risk measurement, volatility | | Biology | Trait variation, ANOVA | | Psychology | Test scores, group comparisons | | Education | Performance variation | | Manufacturing | Process consistency | | Healthcare | Treatment effect variability |
**Pitfalls:**
- **Outliers affect strongly**: one extreme value can dominate. - **Comparison across scales**: use CV not variance directly. - **Small samples**: variance estimates have high uncertainty. - **Non-normal data**: median and IQR may be more appropriate.
**Related measures of spread:**
- **Range**: max - min, simplest. - **IQR**: Q3 - Q1, robust to outliers. - **MAD**: median absolute deviation. - **Standard deviation**: √variance, more interpretable. - **Coefficient of variation**: SD/mean, dimensionless.
**Software:**
- **Excel**: VAR.S (sample), VAR.P (population). - **R**: var() defaults to sample. - **Python (pandas)**: .var(ddof=1) for sample. - **SPSS**: Descriptive Statistics → Variance.
**When NOT to use variance:**
- For skewed distributions, use IQR. - With outliers, use median absolute deviation. - For ratio comparisons, use coefficient of variation. - For ordinal data, use range or specialized measures.
**Practical reporting:**
- Always report sample size with variance. - Note whether population or sample. - Include mean and standard deviation alongside. - Verify normality assumption for tests using variance.
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Population Variance
2.96
Sample Variance
3.7
Population Std Dev
1.720465
Sample Std Dev
1.923538
Mean
5.2
Sum of Squares
14.8
Count
5