Variance & Standard Deviation Calculator

Variance and standard deviation measure how spread out a dataset is around its mean — the core question in statistics beyond just knowing the average. Two datasets can have identical means but completely different distributions: {50, 50, 50, 50} and {0, 0, 100, 100} both have means of 50, but the first has zero variance and the second has a variance of 2,500. Understanding this spread is essential for evaluating investment risk, quality control in manufacturing, research significance testing, and any situation where the average alone is insufficient information about a distribution.

The Empirical Rule: What Standard Deviation Tells You

For normally distributed data, the empirical rule (68-95-99.7 rule) states: approximately 68% of values fall within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations. These percentages are specific to normal distributions but serve as useful approximations for many real-world distributions that are roughly bell-shaped.

Investment application: Jason, 43, in Seattle, Washington analyzes monthly stock returns with a mean of 1.2% and standard deviation of 4.3%. Under the empirical rule: 68% of months fall between 1.2% - 4.3% = -3.1% and 1.2% + 4.3% = 5.5%. 95% of months fall between 1.2% - 8.6% = -7.4% and 1.2% + 8.6% = 9.8%. In any given month, a loss worse than -7.4% should occur roughly 2.5% of the time — about once every 40 months, or once every 3.3 years. This is probabilistic risk assessment from standard deviation.

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Variance: The Core Calculation

Variance is the average of squared deviations from the mean. Population variance (σ²) is used when you have data on every member of a group you care about. Sample variance (s²) is used when your data is a sample from a larger population — the calculation divides by (n-1) instead of n to produce an unbiased estimate of the population variance.

Population variance formula: σ² = Σ(xᵢ - μ)² / N, where μ is the population mean and N is the population size. Sample variance formula: s² = Σ(xᵢ - x̄)² / (n-1), where x̄ is the sample mean and n is the sample size. The (n-1) denominator correction (called Bessel's correction) accounts for the fact that a sample mean, computed from the same data, is already the "best fitting" center — meaning deviations from the sample mean are systematically smaller than deviations from the true population mean, and dividing by (n-1) corrects for this underestimation.

Dataset: {4, 7, 13, 16}. Mean: (4+7+13+16)/4 = 40/4 = 10. Deviations from mean: -6, -3, 3, 6. Squared deviations: 36, 9, 9, 36. Sum: 90. Population variance: 90/4 = 22.5. Sample variance: 90/3 = 30. Standard deviation (population): √22.5 = 4.74. Standard deviation (sample): √30 = 5.48.

Variance in Probability Distributions

For a discrete probability distribution, variance is: σ² = Σ(xᵢ - μ)² × P(xᵢ), where each outcome xᵢ has probability P(xᵢ). Mean first: μ = Σxᵢ × P(xᵢ). Variance: weight each squared deviation by its probability.

A game pays $10 with probability 0.3, $0 with probability 0.5, and -$5 with probability 0.2. Mean: 10(0.3) + 0(0.5) + (-5)(0.2) = 3 + 0 - 1 = $2. Variance: (10-2)²(0.3) + (0-2)²(0.5) + (-5-2)²(0.2) = 64(0.3) + 4(0.5) + 49(0.2) = 19.2 + 2.0 + 9.8 = 31.0. Standard deviation: √31 = $5.57. The game has positive expected value ($2) but considerable variability ($5.57 standard deviation) — quantifying the risk of participating.

Standard Deviation: Variance in Original Units

Variance is in squared units (squared dollars, squared inches) — useful mathematically but unintuitive for interpretation. Standard deviation is the square root of variance, returning to the original units of measurement and making it directly comparable to the data values.

Standard deviation tells you approximately how far a typical data point is from the mean. For the dataset {4, 7, 13, 16} with sample standard deviation 5.48: a "typical" data point is about 5.48 units from the mean of 10. Look at the data: 4 is 6 away, 7 is 3 away, 13 is 3 away, 16 is 6 away. Average absolute deviation is (6+3+3+6)/4 = 4.5. Standard deviation of 5.48 is larger — it's not the average absolute deviation, which some find more intuitive but which has less desirable mathematical properties.

Coefficient of Variation: Relative Spread

Standard deviation measures spread in absolute terms — useful for comparing datasets in the same units and similar scale. The coefficient of variation (CV) measures spread relative to the mean: CV = (Standard deviation / Mean) × 100%. CV allows comparing variability across datasets with different means or different units.

Investment A: mean return 12%, standard deviation 18%. CV = 18/12 × 100% = 150%. Investment B: mean return 5%, standard deviation 6%. CV = 6/5 × 100% = 120%. Investment B is more variable in absolute standard deviation terms (6% vs 18% — wait, actually A has higher absolute SD). Let me redo: Investment B has lower absolute SD (6%) than Investment A (18%), so A is absolutely more volatile. But the CV of 150% for A versus 120% for B shows that A is also relatively more volatile (more risk per unit of expected return). CV is particularly useful in quality control, biological research, and financial analysis where comparing relative variability is more meaningful than absolute variability.

Variance in Multiple Variables: Covariance

When dealing with two variables rather than one, covariance measures how they vary together. Positive covariance: when one is above its mean, the other tends to be above its mean too (they move together). Negative covariance: they move in opposite directions. Zero covariance: no linear relationship.

Covariance formula: Cov(X,Y) = Σ(xᵢ - x̄)(yᵢ - ȳ) / (n-1). Correlation normalizes covariance to a [-1, +1] range: r = Cov(X,Y) / (σₓ × σᵧ). Correlation of +1 means perfect positive linear relationship; -1 means perfect negative; 0 means no linear relationship. In portfolio finance, variance of a two-asset portfolio: σ²ₚ = w₁²σ₁² + w₂²σ₂² + 2w₁w₂Cov(X₁,X₂), where w₁ and w₂ are portfolio weights. Negative covariance between assets reduces portfolio variance below the weighted average of individual variances — the mathematical basis for diversification benefit. Holding assets that move in different directions reduces the portfolio's overall variability even if individual asset volatility is unchanged.