Direct comparison
Standard deviation vs variance
Variance is the average of the squared deviations from the mean; the standard deviation is the square root of the variance, returning the measure to the original units of the data.
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Side-by-side comparison
| Dimension | Standard deviation | Variance |
|---|---|---|
| What it measures | The typical distance of values from the mean. | The average squared distance of values from the mean. |
| Relationship | The square root of the variance. | The square of the standard deviation. |
| Units | The same units as the original data (for example, kilograms). | The squared units of the data (for example, kilograms squared). |
| Symbol (population) | σ (sigma). | σ² (sigma squared). |
| How it is found | Take the square root of the variance. | Average the squared deviations from the mean. |
| Interpretability | Directly interpretable on the data’s own scale. | Harder to interpret because units are squared. |
| Sensitivity to outliers | Sensitive — uses squared deviations, so extremes count heavily. | Even more sensitive, as deviations are squared and not rooted back. |
| Sample denominator | Square root of the sample variance. | Sum of squared deviations divided by n − 1 (sample). |
| Typical use | Reporting spread, z-scores, error bars and confidence intervals. | Algebra and theory, ANOVA, and additive variance components. |
Two views of the same spread
Variance and standard deviation are two expressions of the same idea: how far values typically fall from the mean. You compute variance first — averaging the squared deviations from the mean — and then take its square root to get the standard deviation. Squaring in the variance step ensures positive and negative deviations do not cancel, but it leaves the result in awkward squared units; the square root undoes this, which is why the standard deviation is usually preferred for reporting. Variance keeps its place in theory because it has neat additive properties (variances of independent quantities add), which the standard deviation does not.
Common questions
FAQ
Why use variance at all if the standard deviation is easier to read?+
Variance has mathematical properties the standard deviation lacks. Most importantly, the variances of independent quantities add together, which makes variance the natural currency of techniques such as analysis of variance and of theoretical derivations. The standard deviation is preferred for reporting because it shares the data’s units, but variance underpins much of the underlying theory.
How do I convert between variance and standard deviation?+
They are linked by a square. The standard deviation is the square root of the variance, and the variance is the standard deviation squared. So if the variance is 16, the standard deviation is 4; if the standard deviation is 5, the variance is 25. The conversion is exact and works in both directions.
Why is the standard deviation in the same units as the data but variance is not?+
Variance averages squared deviations, so its units are the data’s units squared — for weights in kilograms, variance is in kilograms squared, which has no intuitive meaning. Taking the square root to obtain the standard deviation returns the measure to the original units, letting you state spread on the same scale as the data themselves.
