Variance vs. standard deviation
Variance and standard deviation both measure how spread out data or a random variable is around the mean. The variance is the mean squared distance from the mean, in squared units. The standard deviation is the square root of the variance, in the same units as the data.
Definitions
Level:
Missing definition: variance
Missing definition: standard_deviation
Examples
| Example | variance | standard_deviation | Why |
|---|---|---|---|
| The heights $1.6$ m, $1.7$ m, and $1.8$ m | $\sigma^2 = 1/150 \approx 0.0067$ m$^2$ | $\sigma = \sqrt{6}/30 \approx 0.082$ m | The heights have variance about $0.0067$ square meters but standard deviation about $0.082$ m, which is directly comparable with the heights themselves. |
| The data $2, 5, 9$ | $\frac{74}{9} \approx 8.22$ | $\sqrt{74 / 9} \approx 2.87$ | |
| The roll of a fair die | $\operatorname{Var}(X) = \mathbb{E}[X^2] - 3.5^2 = \frac{91}{6} - \frac{49}{4} = \frac{35}{12} \approx 2.92$ | $\sqrt{35 / 12} \approx 1.71$ |