Statistics

Standard Deviation

Lesson

The range and IQR describe spread roughly. Variance and standard deviation describe it precisely — they measure how far data tends to sit from the mean.

The recipe:

Compute the mean $\bar{x}$ .
For each value $x_i$ , compute the deviation $x_i - \bar{x}$ .
Square each deviation. (Squaring keeps positives and negatives from canceling.)
Average the squared deviations — this is the variance.
Take the square root — this is the standard deviation, in the original units.

\begin{aligned} \sigma^2 &= \frac{\sum (x_i - \bar{x})^2}{n} \\ \sigma &= \sqrt{\sigma^2} \end{aligned}

Note: this is the population variance and standard deviation (divide by $n$ ). A separate sample formula divides by $n - 1$ instead — used when the data is a sample drawn from a larger population. We’ll stick with the population version here.

For non-exact answers, round to two decimal places.

Worked example 1

Data: 2, 4, 6, 8, 10.

Mean:

\bar{x} = \frac{2 + 4 + 6 + 8 + 10}{5} = 6

Deviations from 6: $-4,\,-2,\,0,\,2,\,4$ . Squared: $16,\,4,\,0,\,4,\,16$ . Sum: 40.

\sigma^2 = \frac{40}{5} = 8

\sigma = \sqrt{8} \approx 2.83

Worked example 2

Data: 1, 1, 4, 4, 5, 9.

Mean: $\bar{x} = 24 / 6 = 4$ . Deviations: $-3,\,-3,\,0,\,0,\,1,\,5$ . Squared: $9,\,9,\,0,\,0,\,1,\,25$ . Sum: 44.

\begin{aligned} \sigma^2 &= \frac{44}{6} \approx 7.33 \\ \sigma &\approx 2.71 \end{aligned}

How to type your answer

Type a single number — variance or standard deviation as the question asks. Round to two decimal places when not exact. Examples: 8, 2.83, 14.14, 0.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

\text{Population variance of } 2,\ 4,\ 6,\ 8,\ 10

Problem 2

\text{Population SD of } 2,\ 4,\ 6,\ 8,\ 10

Problem 3

\text{Population variance of } 1,\ 2,\ 3,\ 4,\ 5

Problem 4

\text{Population SD of } 1,\ 2,\ 3,\ 4,\ 5

Practice

Standard problems matching the lesson.

Problem 5

\text{Population variance of } 3,\ 5,\ 7,\ 9,\ 11

Problem 6

\text{Population SD of } 3,\ 5,\ 7,\ 9,\ 11

Problem 7

\text{Population variance of } 10,\ 20,\ 30,\ 40,\ 50

Problem 8

\text{Population SD of } 10,\ 20,\ 30,\ 40,\ 50

Problem 9

\text{Population variance of } 0,\ 4,\ 4,\ 8

Problem 10

\text{Population SD of } 0,\ 4,\ 4,\ 8

Problem 11

\text{Population variance of } 5,\ 5,\ 5,\ 5,\ 5

Problem 12

\text{Population SD of } 7,\ 7,\ 7,\ 7

Problem 13

\text{Population variance of } 1,\ 1,\ 4,\ 4,\ 5,\ 9\ \text{(round to 2 dp)}

Problem 14

\text{Population SD of } 1,\ 1,\ 4,\ 4,\ 5,\ 9\ \text{(round to 2 dp)}

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 15

\text{Population variance of } 4,\ 8,\ 6,\ 5,\ 3,\ 7,\ 6,\ 9

Problem 16

\text{Population SD of } 4,\ 8,\ 6,\ 5,\ 3,\ 7,\ 6,\ 9\ \text{(round to 2 dp)}

Problem 17

\text{Population SD of } 15,\ 17,\ 19,\ 21,\ 23,\ 25\ \text{(round to 2 dp)}

Problem 18

\text{Population variance of } 1,\ 4,\ 4,\ 4,\ 5,\ 5,\ 7,\ 10

Problem 19

\text{Population SD of } -3,\ -1,\ 1,\ 3\ \text{(round to 2 dp)}

Problem 20

\text{Population SD of } 6,\ 7,\ 8,\ 9,\ 10\ \text{(round to 2 dp)}

Problem 21

\text{If variance is 49, what is the SD?}

Problem 22

\text{If SD is 6, what is the variance?}