Formula & Calculator
Population Variance
Measures the average squared deviation of data points from the population mean, quantifying overall data spread.
Interpretation
σ² = Σ(xᵢ−μ)²/N. Average of squared deviations from the mean. Measures spread for a population. Base for standard deviation. Used in statistical inference.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| σ² | Population variance | |
| xᵢ | Each data value | |
| μ | Population mean | |
| N | Population size |
What it means
Population variance (σ²) is the average of the squared differences between each data point and the population mean. It is a measure of dispersion that gives more weight to outliers due to the squaring. The variance is in squared units, making it less interpretable than the standard deviation, but it is fundamental for many statistical calculations (ANOVA, regression). It is used in probability theory and inferential statistics. The square root gives the standard deviation, which is on the original scale. Understanding variance is essential for understanding variability in data and for comparing distributions. It is also used in the calculation of the coefficient of determination (R²).
Worked example
Population Variance – Two Examples
Real‑World| Parameter | Value |
|---|---|
| Data | 2,4,6,8,10 |
| μ | 6 |
| N | 5 |
| Parameter | Value |
|---|---|
| Data | 1,2,3,4,100 |
| μ | 22 |
| N | 5 |
Common mistakes
- Population variance: Uses N (population size) in the denominator.
- Mean μ: The population mean – not the sample mean.
- Units: Variance is in squared units of the data.
- Non‑negative: Variance is always ≥ 0.
- Zero variance: Occurs when all values are equal.
Applications
Population variance is the average of the squared deviations from the mean, measuring the spread of a population. It is the square of the standard deviation and is used in many statistical formulas, such as hypothesis tests and ANOVA. Engineers use variance to assess process variability and to design experiments. In finance, it quantifies risk. In quality control, it helps determine process capability. By calculating variance, professionals can compare variability between populations, identify sources of variation, and make informed decisions about process improvement. Understanding variance is essential for statistical inference, as it appears in confidence intervals, regression, and design of experiments. It is a foundational concept in statistics.
- Process capability and variability analysis
- Design of experiments (ANOVA, factorial designs)
- Financial risk and volatility assessment
- Quality improvement – identifying sources of variation
- Statistical inference and hypothesis testing
Frequently Asked Questions
σ² = Σ(xᵢ−μ)² / N. It measures the average squared deviation of each data point from the population mean (μ). The squared units make it less interpretable than standard deviation.
Population variance uses N in the denominator; sample variance uses n−1 (Bessel's correction) to provide an unbiased estimate of the population variance.
Because the deviations are squared, the units are squared units (e.g., if data is in meters, variance is in m²). This is why standard deviation (the square root) is often preferred for interpretation.
A large variance indicates that the data points are widely spread from the mean, showing high variability or dispersion.
Variance is the foundation for analysis of variance (ANOVA), where it partitions total variability into between‑group and within‑group components. In regression, it is used to assess the goodness‑of‑fit via R².
Variance of a single variable is the covariance of the variable with itself (Cov(X,X) = Var(X)). Covariance extends the concept to two variables.
Use the weighted formula: σ² = Σ fᵢ (xᵢ − μ)² / N, where fᵢ are frequencies and xᵢ are class midpoints.
Adding a constant does not change the variance (σ² remains the same) because the deviations from the mean are unchanged.
Multiplying by a multiplies the variance by a². This reflects the scale change in squared units.
Bessel's correction ensures that the sample variance is an unbiased estimator of the population variance. Without it, the sample variance would systematically underestimate the true variance.