Sample standard deviation | Year 11

01 / LEARN

Understand the idea.

Use squared deviations to measure spread

Positive and negative deviations from the mean cancel, so their sum cannot measure spread. Square each deviation to make its contribution non-negative, add the squares and divide by $n - 1$ to obtain the sample variance.

Take the square root to obtain sample standard deviation $s$ in the data's original unit; variance has squared units. At least two observations are required. The denominator $n - 1$ accounts for estimating the mean from the same sample.

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

Choose the correct calculator output

Calculators may show both sample $s$ and population $σ$ . Choose the sample result for a question asking for sample spread or estimating population variability from a sample. The population formula instead divides squared deviations by the full population size.

Adding a constant moves every value and the mean equally, preserving deviations and $s$ . Multiplying all values by $k$ multiplies deviations by $k$ and standard deviation by $∣ k ∣$ . Keep unrounded values until the final reported result.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

A small sample

Find the sample standard deviation of 2,4,6.

Worked solution

1
Find the sample mean.
$\bar{x} = \frac{2 + 4 + 6}{3} = 4$
Deviations must be measured from the mean of these three observations.
2
Calculate and square the deviations.
$\begin{aligned} x_{i} - \bar{x} & : - 2, 0, 2 \\ \sum {(x_{i} - \bar{x})}^{2} & = {(- 2)}^{2} + 0^{2} + 2^{2} = 8 \end{aligned}$
Squaring prevents the two opposite nonzero deviations from cancelling.
3
Calculate sample variance.
$s^{2} = \frac{8}{3 - 1} = 4$
The sample formula uses n minus 1, which is 2 here.
4
Take the non-negative square root.
$s = \sqrt{4} = 2$
Standard deviation is non-negative and returns to the original data scale.

Answer

s=2.

EXAMPLE 02

A change of scale

A sample measured in centimetres has mean 30 and s=4. Convert to millimetres and then add a calibration offset of 2 mm to each reading. Find the new mean and s.

Worked solution

1
Express the transformation of a reading.
$y = 10 x + 2$
With $x$ recorded in centimetres and $y$ in millimetres, multiply by $10$ before adding the $2 mm$ offset.
2
Transform the mean.
$\bar{y} = 10 (30) + 2 = 302 mm$
The mean follows the same multiplication and addition as every individual reading.
3
Transform the standard deviation.
$s_{y} = ∣ 10 ∣ (4) = 40 mm$
Multiplication scales all deviations by 10; the constant offset cancels when deviations from the new mean are taken.

Answer

Mean 302 mm; s = 40 mm.

03 / YOUR TURN

Try it yourself.

Check your understanding

If all five observations equal 7, what is s?

Show answer & reasoning

The mean is $7$ , so each deviation is $7 - 7 = 0$ . Hence $s = \sqrt{\frac{0}{5 - 1}} = 0$ : there is no variation in the observations.

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6 worksheet questions · 13 marks

Open worksheet Worked solutions

Read the questions online

Find $s$ for 1,3,5.
2 marks
Find the sample variance and $s$ for 3,3,7,7.
2 marks
A sample has $s = 5$ . Every value increases by 20. Find the new $s$ .
2 marks
A sample has $s = 3$ . Values are multiplied by $- 2$ . Find the new $s$ .
2 marks
A calculator returns $S_{x} = 4.20$ and $σ_{x} = 3.76$ for a survey sample. Which should be reported as an estimate of population spread?
2 marks
For sample values 4,6,8,10, calculate the mean and $s$ to 3 decimal places.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Find $s$ for 1,3,5.
2 marks
Mean 3; squared deviations sum 8; sample variance $\frac{8}{2} = 4$ .
2.
Find the sample variance and $s$ for 3,3,7,7.
2 marks
Mean 5; squared deviations $4 + 4 + 4 + 4 = 16$ .
Divide by 3, then take the square root.
Variance $\frac{16}{3}$ ; $s \approx 2.309$ .
A sample has $s = 5$ . Every value increases by 20. Find the new $s$ .
2 marks
Shifting does not change deviations from the mean.
5.
A sample has $s = 3$ . Values are multiplied by $- 2$ . Find the new $s$ .
2 marks
Standard deviation scales by the absolute multiplier: $∣ - 2 ∣ \times 3 = 6$ .
6.
A calculator returns $S_{x} = 4.20$ and $σ_{x} = 3.76$ for a survey sample. Which should be reported as an estimate of population spread?
2 marks
The survey observations are a sample, so use the $n - 1$ calculation.
$S_{x} = 4.20$ , with the original unit.
For sample values 4,6,8,10, calculate the mean and $s$ to 3 decimal places.
3 marks
Squared deviations $9 + 1 + 1 + 9 = 20$ .
$s = \sqrt{\frac{20}{3}} \approx 2.581988897$ .
Mean 7; $s \approx 2.582$ .

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics: counting, binomial expansion, data, samples, summary measures and introductory normal distributions. These lessons provide the Counting and Statistics strand of the approved Methods preparation route.

Official curriculum source ↗

In this lesson

Understand the idea.

Use squared deviations to measure spread

Choose the correct calculator output

See the method in action.

A small sample

Find the sample mean.

Calculate and square the deviations.

Calculate sample variance.

Take the non-negative square root.

A change of scale

Express the transformation of a reading.

Transform the mean.

Transform the standard deviation.

Try it yourself.

Check your understanding

Keep practising