Statistics · Descriptive Statistics
What is Variance?
The average squared deviation from the mean — why it matters, how to calculate it, and how it relates to standard deviation and risk.
Two investment portfolios both return an average of 8% per year. But Portfolio A returns between 6% and 10% year after year, while Portfolio B swings between −20% and +36%. The mean is the same — but the risk is entirely different. The statistic that captures this difference in spread is variance — and it underpins not just statistics, but modern finance, insurance, and risk management.
📘 Key Term
Variance (σ² or s²) is the average of the squared deviations from the mean. It measures how spread out data values are around the mean. A high variance means values are widely dispersed; a low variance means they cluster tightly around the mean. Variance is always non-negative.
Variance Formulas
| Population Variance (σ²) | Sample Variance (s²) | |
|---|---|---|
| Formula | σ² = Σ(xᵢ − μ)² / N | s² = Σ(xᵢ − x̄)² / (n−1) |
| Denominator | N (entire population) | n−1 (Bessel’s correction) |
| Units | Squared original units | Squared original units |
Step-by-Step Calculation
Annual returns (%): 5, 8, 3, 10, 4 (n = 5)
Step 1: Mean = (5+8+3+10+4)/5 = 30/5 = 6%
Step 2: Deviations: −1, +2, −3, +4, −2
Step 3: Squared deviations: 1, 4, 9, 16, 4
Step 4: Sum of squared deviations = 34
Step 5: Sample variance = 34/(5−1) = 34/4 = 8.5 (% squared) | Standard deviation = √8.5 = 2.92%
Variance vs Standard Deviation
Variance and standard deviation both measure spread, but serve different purposes:
📊 Variance is used in mathematical derivations and statistical theory (e.g. ANOVA, regression) because squaring removes sign and enables algebraic manipulation.
📊 Standard deviation (= √variance) is reported in practice because it’s in the same units as the original data — far more interpretable than squared units.
💡 Key Insight
In finance, variance of returns is the mathematical definition of risk in Modern Portfolio Theory (Markowitz, 1952). Investors choosing between portfolios with the same expected return rationally prefer the one with lower variance. The variance of a portfolio is not simply the average of individual asset variances — it depends critically on the correlations between assets, which is why diversification reduces portfolio variance.
⚠️ Common Error
Students sometimes try to interpret variance directly as a meaningful quantity — e.g. ‘the variance of 8.5% squared means values deviate by 8.5%.’ This is wrong. Variance is in squared units and has no direct intuitive interpretation as a spread. Use standard deviation for meaningful spread statements. Variance is valuable for mathematical operations, not for direct communication of spread.
Q1. Calculate the sample variance for the following house prices (£000s): 200, 220, 180, 250, 190. State which measure — variance or standard deviation — is more useful for reporting to homebuyers and explain why. [6 marks]
Answer: Mean = (200+220+180+250+190)/5 = 1040/5 = 208. Deviations: −8, +12, −28, +42, −18. Squared deviations: 64, 144, 784, 1764, 324. Sum = 3080. Sample variance = 3080/(5−1) = 3080/4 = 770 (£000 squared). Standard deviation = √770 = £27.75k. Standard deviation is more useful for reporting to homebuyers because it is expressed in the same units as house prices (£000s) — telling buyers prices typically deviate from the mean by about £27,750. Variance of 770 (£000 squared) has no intuitive meaning for non-statisticians.
References
1. Moore, D.S., McCabe, G.P. and Craig, B.A. (2021) Introduction to the Practice of Statistics. W.H. Freeman.
2. Markowitz, H. (1952) ‘Portfolio Selection’, Journal of Finance, 7(1), pp. 77–91.
3. Triola, M.F. (2022) Elementary Statistics. Pearson.
4. Field, A. (2018) Discovering Statistics Using IBM SPSS Statistics. SAGE.
5. Freedman, D., Pisani, R. and Purves, R. (2007) Statistics. W.W. Norton.