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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:

  1. Compute the mean xˉ\bar{x}.
  2. For each value xix_i, compute the deviation xi−xˉx_i - \bar{x}.
  3. Square each deviation. (Squaring keeps positives and negatives from canceling.)
  4. Average the squared deviations — this is the variance.
  5. Take the square root — this is the standard deviation, in the original units.
σ2=∑(xi−xˉ)2nσ=σ2\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 nn). A separate sample formula divides by n−1n - 1instead — 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:

xˉ=2+4+6+8+105=6\bar{x} = \frac{2 + 4 + 6 + 8 + 10}{5} = 6

Deviations from 6: −4, −2, 0, 2, 4-4,\,-2,\,0,\,2,\,4. Squared: 16, 4, 0, 4, 1616,\,4,\,0,\,4,\,16. Sum: 40.

σ2=405=8\sigma^2 = \frac{40}{5} = 8
σ=8≈2.83\sigma = \sqrt{8} \approx 2.83

Worked example 2

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

Mean: xˉ=24/6=4\bar{x} = 24 / 6 = 4. Deviations: −3, −3, 0, 0, 1, 5-3,\,-3,\,0,\,0,\,1,\,5. Squared: 9, 9, 0, 0, 1, 259,\,9,\,0,\,0,\,1,\,25. Sum: 44.

σ2=446≈7.33σ≈2.71\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

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

Problem 2

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

Problem 3

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

Problem 4

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

Practice

Standard problems matching the lesson.

Problem 5

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

Problem 6

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

Problem 7

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

Problem 8

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

Problem 9

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

Problem 10

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

Problem 11

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

Problem 12

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

Problem 13

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

Problem 14

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

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

Problem 16

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

Problem 17

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

Problem 18

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

Problem 19

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

Problem 20

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

Problem 21

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

Problem 22

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

Practice

Standard problems matching the lesson.

Problem 23

Basketball team heights (in): 70, 72, 74, 76, 78. Find the standard deviation (round to 2 decimal places).

Problem 24

Daily highs (°F): 65, 70, 75, 80, 85, 90. Find the standard deviation (round to 2 decimal places).

Challenge

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

Problem 25

Daily step counts: 8000, 10000, 9000, 11000, 12000. Find the standard deviation (round to 2 decimal places).

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