Concepts · 3 min read

Standard Deviation Explained With a Worked Example

Published October 3, 2026Written and reviewed by Mateuss M.

In short: Standard deviation measures how spread out data is. For 2, 4, 4, 4, 5, 5, 7, 9 the mean is 5 and the population standard deviation is 2.

The steps

Find the mean.
Subtract the mean from each value and square the result.
Average those squares (÷ n for a population, ÷ n − 1 for a sample): this is the variance.
Take the square root.

Full working for 2, 4, 4, 4, 5, 5, 7, 9

AnswerMean 5 · Median 4.5

Mode: 4
Range: 7
Sample SD: 2.1381
Population SD: 2

Step-by-step working 7 steps

1
Sort the data
```
2, 4, 4, 4, 5, 5, 7, 9
```
8 values.
2
Mean = sum ÷ count
```
Sum = 40
Mean = 40 ÷ 8 = 5
```

Median = middle value

Average of values 4 and 5: (4 + 5) ÷ 2 = 4.5

4
Mode = most frequent value
```
4 (appears 3 times)
```
5
Range = largest − smallest
```
9 − 2 = 7
```

Squared deviations from the mean

(2 − 5)² = 9
(4 − 5)² = 1
(4 − 5)² = 1
(4 − 5)² = 1
(5 − 5)² = 0
(5 − 5)² = 0
(7 − 5)² = 4
(9 − 5)² = 16
Sum of squares = 32

Sum these for every value to get the total used below.

7
Variance and standard deviation
```
Population: σ² = 32 ÷ 8 = 4,  σ = 2
Sample:     s² = 32 ÷ 7 = 4.5714,  s = 2.1381
```
Use the sample version (÷ n−1) when your data is a sample from a larger group; use the population version when it is the whole group.

Sample versus population

Use ÷ n when your data is the whole group. Use ÷ (n − 1) when it is a sample from a larger group: a sample tends to understate spread, and dividing by n − 1 corrects for that.

The 68-95-99.7 rule

For roughly bell-shaped data, about 68% of values lie within 1 standard deviation of the mean, about 95% within 2 and about 99.7% within 3.

Compare two sets

Same mean, different spread — see the effect in mean vs median, or use the mean, median and mode calculator.

A tighter data set with the same mean

AnswerMean 5 · Median 5

Mode: 5
Range: 2
Sample SD: 0.8165
Population SD: 0.7559

Step-by-step working 7 steps

1
Sort the data
```
4, 4, 5, 5, 5, 6, 6
```
7 values.
2
Mean = sum ÷ count
```
Sum = 35
Mean = 35 ÷ 7 = 5
```
3
Median = middle value
```
Value 4 of 7 = 5
```
4
Mode = most frequent value
```
5 (appears 3 times)
```
5
Range = largest − smallest
```
6 − 4 = 2
```

Squared deviations from the mean

(4 − 5)² = 1
(4 − 5)² = 1
(5 − 5)² = 0
(5 − 5)² = 0
(5 − 5)² = 0
(6 − 5)² = 1
(6 − 5)² = 1
Sum of squares = 4

Sum these for every value to get the total used below.

7
Variance and standard deviation
```
Population: σ² = 4 ÷ 7 = 0.5714,  σ = 0.7559
Sample:     s² = 4 ÷ 6 = 0.6667,  s = 0.8165
```
Use the sample version (÷ n−1) when your data is a sample from a larger group; use the population version when it is the whole group.

Frequently asked questions

Can standard deviation be negative?

No. It is a square root, so it is zero or positive; zero means every value is identical.

What is variance?

The standard deviation squared — the average squared distance from the mean.

Written and reviewed by Mateuss M.

Mateuss M. writes and reviews mathematical content for CalcSolver, focusing on online calculators, formulas, equations, and practical math tools. He reviews calculator functionality, calculation methods, formulas, examples, and explanations to help ensure that each tool is clear, useful, and easy to understand.

Keep reading

Concepts

Standard Deviation Explained With a Worked Example

The steps

Sample versus population

The 68-95-99.7 rule

Compare two sets

Frequently asked questions

Written and reviewed by Mateuss M.

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