Standard Deviation Calculator with Steps: Mean, Population and Sample (Year 10)

Standard deviation calculator, with the working out

Paste or type your data. You’ll get the mean and the standard deviation, both population and sample, with every step shown so you can follow it or copy the method.

By Ryan, founder of Bayside Academics, 15 years in private tutoring · Updated 6 November 2026

Calculate the standard deviation

Your data (separate with commas or spaces)

Mean 5 · σ ≈ 2.00 (population) · s ≈ 2.14 (sample)

Count the values: $n = 8$
Find the mean: $\bar{x} = \dfrac{40}{8} = 5$
Square each value's distance from the mean:
$2$ : $(2 - 5)^2 = 9$
$4$ : $(4 - 5)^2 = 1$
$4$ : $(4 - 5)^2 = 1$
$4$ : $(4 - 5)^2 = 1$
$5$ : $(5 - 5)^2 = 0$
$5$ : $(5 - 5)^2 = 0$
$7$ : $(7 - 5)^2 = 4$
$9$ : $(9 - 5)^2 = 16$
Add them up: $\sum (x - \bar{x})^2 = 32$
Population (divide by $n$ ): $\sigma = \sqrt{\dfrac{32}{8}} = \sqrt{4} \approx 2.00$
Sample (divide by $n - 1$ ): $s = \sqrt{\dfrac{32}{7}} = \sqrt{4.5714} \approx 2.14$

Not sure which to use? If the data is the whole group you care about, use the population value σ. If it’s a sample from a bigger group, use s. Your teacher or calculator will say which.

What standard deviation means

Standard deviation measures how spread out data is around the mean: roughly, how far a typical value is from the average.

Two classes might both average 70% on a test. If one has a standard deviation of 5 and the other 15, the first class's marks are bunched close to 70 and the second's are spread from much lower to much higher.

For data shaped like a bell curve, about 68% of values lie within one standard deviation of the mean, and about 95% within two.

The formula

\sigma = \sqrt{\frac{\sum (x - \bar{x})^2}{n}}

s = \sqrt{\frac{\sum (x - \bar{x})^2}{n - 1}}

Find the mean $\bar{x}$ .
Subtract the mean from each value and square the result.
Add the squares.
Divide by $n$ (population) or $n - 1$ (sample).
Take the square root.

Worked example

Data: 2, 4, 4, 4, 5, 5, 7, 9. The mean is $\dfrac{40}{8} = 5$ .

Squared deviations: 9, 1, 1, 1, 0, 0, 4, 16, which add to 32.

\sigma = \sqrt{\frac{32}{8}} = \sqrt{4} = 2

s = \sqrt{\frac{32}{7}} \approx 2.14

By year level

Following the Victorian Curriculum 2.0. Schools sequence topics differently, so check your course.

Year 10

Statistics

Comparing data sets with box plots, the interquartile range and scatterplots.

Worksheets and topic test · Free PDF with answers

Year 10A

Mean and standard deviation

Calculating and interpreting the mean and standard deviation.

Worksheets and topic test · Free PDF with answers

Standard deviation: questions

What does standard deviation tell you?

How spread out the data is around the mean. A small standard deviation means the values are close to the mean; a large one means they're more spread out.

What's the difference between σ and s?

σ (population) divides by n and is used when your data is the whole group. s (sample) divides by n − 1 and is used when your data is a sample from a larger group. s is always a little bigger.

Which one does my calculator show?

Scientific calculators usually show both: σx (or σn) is the population value and sx (or σn−1) is the sample value. Check which one your teacher or textbook uses.

What year level learns standard deviation?

In the Victorian Curriculum 2.0 it's a Year 10A statistics topic; some schools and states introduce it in Year 10. It's used a lot in senior maths.

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