Standard Deviation vs. Variance

Both standard deviation and variance measure how spread out data is from the mean. Variance is the average of the squared deviations from the mean. Standard deviation is the square root of variance — it returns the measure to the original units. If test scores have a variance of 100 points², the standard deviation is 10 points.

Standard Deviation vs. Variance

	Standard Deviation	Variance
Definition	Square root of variance — average distance from the mean	Average of the squared deviations from the mean
Formula (population)	$\sigma = \sqrt{\frac{\sum(x_i - \mu)^2}{N}}$	$\sigma^2 = \frac{\sum(x_i - \mu)^2}{N}$
Formula (sample)	$s = \sqrt{\frac{\sum(x_i - \bar{x})^2}{n-1}}$	$s^2 = \frac{\sum(x_i - \bar{x})^2}{n-1}$
Units	Same units as the data (e.g. cm, dollars)	Squared units (e.g. cm², dollars²)
Interpretation	Directly interpretable — 'typical distance from mean'	Harder to interpret — squared units are abstract
Used for	Reporting spread, z-scores, confidence intervals	Mathematical calculations, ANOVA, regression theory
Relationship	$\sigma = \sqrt{\sigma^2}$	$\sigma^2 = \sigma \times \sigma$

When to Use Each

Use Standard Deviation when...

Reporting results to a non-technical audience
Describing spread in the original units
Calculating z-scores or confidence intervals
Comparing variability across datasets

Use Variance when...

Performing mathematical derivations
ANOVA (Analysis of Variance) calculations
Portfolio theory in finance (variances are additive for independent variables)
Regression and theoretical statistics

Examples

Standard deviation example

Test scores: {70, 80, 90}. Mean = 80. Deviations: -10, 0, +10. Variance = (100 + 0 + 100)/3 = 66.7. Standard deviation =

\sqrt{66.7} \approx 8.16

points. This tells you scores typically differ from the mean by about 8 points.

Why variance matters

If investment A has variance 25 and investment B has variance 16, the combined portfolio variance is 25 + 16 = 41 (assuming independence). Variances add directly — standard deviations do not:

\sqrt{25} + \sqrt{16} = 9 \neq \sqrt{41} \approx 6.4

Common Confusion Points

Students often wonder why we square the deviations in the first place. The reason: without squaring, positive and negative deviations would cancel out (average deviation from the mean is always zero). Squaring makes all deviations positive.

Don't forget the difference between population (÷N) and sample (÷(n−1)) formulas. The sample formula uses n−1 (Bessel's correction) to produce an unbiased estimate of the population variance.

Frequently Asked Questions

Is standard deviation or variance more useful?

Standard deviation is more commonly reported because it is in the same units as the data and is directly interpretable. Variance is more useful in mathematical and theoretical work because it has nice algebraic properties (like additivity for independent variables).

Can standard deviation be negative?

No. Standard deviation is defined as the square root of variance, and variance is a sum of squares — both are always non-negative. A standard deviation of 0 means all values are identical.

What is the empirical rule?

For approximately normal distributions: about 68% of data falls within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3. This is also called the 68-95-99.7 rule.

Standard Deviation vs. Variance