Variance vs Standard Deviation — Which to Use?

The Unit Problem — Why Standard Deviation Exists

Variance was the original measure of spread when Gauss and Laplace were developing the theory of errors in the early 19th century. It is mathematically elegant — the expected squared deviation from the mean. But it has one serious practical problem: its units are squared.

If you measure height in centimetres, variance is in cm². If you measure income in dollars, variance is in dollars². These units have no intuitive meaning. What does it mean to say "the variance of heights in this sample is 64 cm²"? It is technically correct but impossible to visualise. You cannot say "most people are within one unit of variance of the mean" in a way that makes sense.

Standard deviation solves this by taking the square root: √64 = 8 cm. Now you can say "most heights are within 8 cm of the mean" — a statement that is immediately interpretable. This is why standard deviation was introduced as a practical reporting tool: it translates the mathematical precision of variance back into the original units of the data.

Worked example: A dataset of daily temperatures in °C with variance = 25°C². Standard deviation = √25 = 5°C. You can say "daily temperatures typically vary by about 5°C from the mean." The variance of 25°C² conveys no intuitive information about how temperatures fluctuate day to day.

When Variance Is the Right Measure

Analysis of Variance (ANOVA): ANOVA tests whether groups have different means by partitioning total variance into between-group variance and within-group variance. The F-statistic is a ratio of variances. Standard deviation cannot be partitioned in this way — variance is the only measure that adds up correctly across independent sources.

Portfolio theory (finance): The variance of a portfolio of assets is: Var(portfolio) = Σᵢ Σⱼ wᵢwⱼ Cov(Xᵢ, Xⱼ), where Cov is the covariance. This additivity only works with variance. Standard deviation of a portfolio is not the weighted average of individual standard deviations — diversification benefits emerge precisely because variance (not standard deviation) adds linearly, allowing portfolio risk to be reduced below the weighted average of individual risks.

Linear regression: The coefficient of determination R² is the proportion of variance in the dependent variable explained by the model. Mean squared error (MSE) and its relatives (residual standard error) are all variance-based. Regression output is built on variance decomposition.

Error propagation in physics and engineering: When combining measurements with individual uncertainties, variances add (for independent errors). If position has uncertainty σ₁ and velocity has uncertainty σ₂, the uncertainty in a derived quantity is computed by summing variances, then taking the square root at the end.

When Standard Deviation Is Preferred

Standard deviation is the preferred reporting measure in almost every context where you are communicating results to humans rather than feeding numbers into further mathematical operations.

Descriptive statistics and data summaries: "The mean height is 175 cm ± 8 cm (SD)" is immediately interpretable. "The mean height is 175 cm with variance 64 cm²" requires mental calculation to be useful.

The empirical rule: the 68-95-99.7 rule uses standard deviation directly. "95% of values fall within two standard deviations of the mean" is a universally understood shorthand for the bell curve. The equivalent statement using variance ("95% of values fall within √[4σ²] of the mean") is unnecessarily awkward.

Z-scores and standardisation: z = (x − μ) / σ uses standard deviation. The z-score tells you how many standard deviations a value is from the mean — a concept that requires standard deviation in its natural form.

Error bars in charts and figures: journal figures use ± 1 SD or ± 1 SEM (standard error of the mean) as error bars. Variance is never used for error bars because the unit mismatch with the axis would be misleading.

Coefficient of Variation — When Neither Is Enough

Both variance and standard deviation have a limitation: they are absolute measures, tied to the scale of the data. You cannot compare the variability of a dataset measured in grams with one measured in kilometres just by comparing their standard deviations.

The coefficient of variation (CV = σ / μ × 100%) solves this by expressing standard deviation as a percentage of the mean. A dataset with mean 100 and SD 10 has CV = 10%. A dataset with mean 10,000 and SD 1,000 also has CV = 10%. Despite the vastly different scales and standard deviations, both have identical relative variability.

Use CV when: comparing spread across datasets with different units or scales; evaluating precision of measurements in analytical chemistry or laboratory science; comparing volatility across assets with different price levels in finance; or communicating relative risk to a non-technical audience as a percentage.