Standard Deviation

The square root of variance, expressing spread in the same units as the data.

Standard deviation is the typical distance from the mean, in the original units

Definition

The standard deviation is the square root of the variance. Where variance is measured in squared units, standard deviation is back in the same units as the original data — making it much easier to interpret.

\sigma = \sqrt{\frac{1}{n}\sum_{i=1}^{n}(x_i - \bar{x})^2}

A small standard deviation means the data clusters tightly around the mean. A large one means it is spread out.

Interpreting standard deviation

Two students both average 70 on their tests:

Student A: 68, 70, 71, 70, 71 → $\sigma \approx 1$
Student B: 40, 95, 55, 90, 50 → $\sigma \approx 22$

Same mean, very different consistency. Standard deviation captures that difference.

Computing standard deviation

Dataset: 2, 4, 4, 4, 5, 5, 7, 9. Mean $= 5$ , variance $= 4$ .

\sigma = \sqrt{4} = 2

Most values fall within 2 of the mean — which matches a quick look at the data.

Try it

A dataset has values 10, 20, 30, 40, 50. Find the mean, variance, and standard deviation.

Solution

Mean $= 30$ . Deviations: $-20, -10, 0, 10, 20$ . Squared: $400, 100, 0, 100, 400$ .

Variance $= \frac{1000}{5} = 200$ . Standard deviation $= \sqrt{200} = 10\sqrt{2} \approx 14.1$ .

Related concepts

Needs first

Variance

Linear Regression Normal Distribution Quantization Sampling

Uses this

T-Test

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