Variance is the average of squared deviations from the mean; it measures data spread and is the square of standard deviation.
## Core concept
Variance (σ² or s²) quantifies how far individual data points are scattered from the arithmetic mean. Unlike standard deviation (which is in original units), variance is expressed in squared units.
Why variance matters: - Foundation for calculating standard deviation (σ = √Variance) - Used in comparing variability across datasets with different scales - Basis for coefficient of variation and combined standard deviation calculations - Essential for statistical hypothesis testing and forecasting models
Relationship with other measures: - Variance = (Standard Deviation)² - Often converted to standard deviation for practical interpretation (because SD is in original units) - Used alongside mean to fully describe a distribution's shape and spread
## Formulas and calculation methods
For ungrouped (raw) data: $$\sigma^2 = \frac{\sum(X - \bar{X})^2}{N}$$
where X = individual value, $\bar{X}$ = mean, N = number of observations
Alternative (computational) formula: $$\sigma^2 = \frac{\sum X^2}{N} - \left(\frac{\sum X}{N}\right)^2$$
For grouped data: $$\sigma^2 = \frac{\sum f(X - \bar{X})^2}{\sum f}$$
or using mid-values with frequency distribution: $$\sigma^2 = \frac{\sum fX^2}{\sum f} - \left(\frac{\sum fX}{\sum f}\right)^2$$
Sample variance (with Bessel's correction): $$s^2 = \frac{\sum(X - \bar{X})^2}{n-1}$$
(dividing by n−1 instead of n for unbiased estimate)
## Worked example
Dataset: 4, 6, 8, 10, 12
Step 1: Calculate mean = (4 + 6 + 8 + 10 + 12) ÷ 5 = 8
Step 2: Find deviations and square them: - (4 − 8)² = 16 - (6 − 8)² = 4 - (8 − 8)² = 0 - (10 − 8)² = 4 - (12 − 8)² = 16
Step 3: Sum of squared deviations = 16 + 4 + 0 + 4 + 16 = 40
Step 4: Variance = 40 ÷ 5 = 8
Standard deviation = √8 ≈ 2.83
## Common exam applications
- Comparing dispersion: Two datasets with same mean but different variances show different consistency
- Coefficient of variation: CV = (SD ÷ Mean) × 100 uses variance indirectly; helps compare datasets with different means
- Combined standard deviation: When merging two groups, variance is used as intermediate calculation before finding combined SD
- Quality control: Manufacturing processes use variance thresholds; lower variance = better consistency
- Portfolio analysis: In financial mathematics, variance of returns measures investment risk
## Common mistakes
- Forgetting to square deviations – deviations must be squared before averaging
- Mixing population and sample formulas – use N for population, (n−1) for sample variance
- Not squaring the unit – variance is always in squared units; standard deviation converts back to original units
- Confusing variance with standard deviation – they are related but distinct; variance is harder to interpret directly because of squared units
- Using raw formula when computational formula is easier – especially with large numbers, the alternative formula reduces rounding errors
## Relationship to standard deviation
Variance is the mathematical foundation for standard deviation. Examiners often ask students to calculate both; remember: Variance = SD² and SD = √Variance. This relationship is used in combined SD calculations and coefficient of variation problems.