Standard deviation | Year 10

01 / LEARN

Understand the idea.

Distance from the mean

Standard deviation measures spread around the mean. Calculate each deviation, square it, and add the squares. Squaring prevents positive and negative deviations from cancelling. For the entire population, divide by its size to obtain variance, then take the square root.

Variance has squared units; standard deviation has the original measurement unit. An SD of zero means every value equals the mean. Large deviations contribute especially strongly because of squaring, so extreme observations can have a substantial effect.

σ = \sqrt{\frac{\sum {(x - μ)}^{2}}{N}}

State which calculation you use

Choose the formula from the role of the data. A complete population uses denominator $N$ ; the usual sample variance estimating a larger population uses $n - 1$ . Calculators may display both, so select and label the appropriate output.

Adding a constant shifts every value and the mean together, leaving deviations unchanged. Multiplying values by $k$ multiplies the SD by $∣ k ∣$ . A negative multiplier reverses positions but cannot make a nonnegative measure of spread negative.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

A complete small group

The complete population is 2, 4, 4, 4, 5, 5, 7, 9. Find its population standard deviation.

Deviations from the population mean5
Value x	Deviation x−5	Squared deviation
2	−3	9
4	−1	1
4	−1	1
4	−1	1
5	0	0
5	0	0
7	2	4
9	4	16

Worked solution

1
Calculate the population mean.
$μ = \frac{2 + 4 + 4 + 4 + 5 + 5 + 7 + 9}{8} = \frac{40}{8} = 5$
The eight values comprise the whole population.
2
Add the squared deviations.
$\sum {(x - 5)}^{2} = 9 + 1 + 1 + 1 + 0 + 0 + 4 + 16 = 32$
The listed deviations are squared before being combined.
3
Find the population variance.
$σ^{2} = \frac{32}{8} = 4$
Population variance divides by the full population count.
4
Take the square root.
$σ = \sqrt{4} = 2$
The root converts variance into standard deviation.

Answer

Population standard deviation 2.

EXAMPLE 02

Population and sample outputs

For the complete population 1, 3, 5, find population SD. What would the sample SD be if these were a sample?

Worked solution

1
Find the mean and squared-deviation total.
$\begin{aligned} \bar{x} & = \frac{1 + 3 + 5}{3} = 3 \\ \sum {(x - 3)}^{2} & = 4 + 0 + 4 = 8 \end{aligned}$
Both formulas start from the same deviations about the mean.
2
Treat the three values as the population.
$σ = \sqrt{\frac{8}{3}} \approx 1.63$
The population formula divides by three.
3
Treat the values as a sample instead.
$s = \sqrt{\frac{8}{3 - 1}} = 2$
The usual sample formula divides by two, giving the larger output.

Answer

Population SD≈1.63; sample SD=2.

03 / YOUR TURN

Try it yourself.

Check your understanding

The population is 4, 4, 4, 4. What is its SD?

Show answer & reasoning

The mean is $4$ and every deviation is $4 - 4 = 0$ . Hence $σ = \sqrt{0 / 4} = 0$ .

Keep practising

6 worksheet questions · 15 marks

Open worksheet Worked solutions

Read the questions online

Find the population SD of 2, 4, 6.
3 marks
Find the population SD of 3, 3, 7, 7.
3 marks
All scores increase by 10. What happens to their SD?
2 marks
Lengths measured in metres have SD 0.4 m. Express the SD in centimetres.
2 marks
A sample of 5 measurements has squared deviations totalling 20. Find the sample SD.
3 marks
Population A has SD 2.1 min and B has SD 5.4 min. Which is less variable around its mean?
2 marks

Read the worked solutions

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

Find the population SD of 2, 4, 6.
3 marks
Mean 4; squared deviations total 8.
Divide by 3 and take the square root.
$\sqrt{\frac{8}{3}} \approx 1.63$
Find the population SD of 3, 3, 7, 7.
3 marks
Mean 5; four squared deviations are 4.
Variance $16 \div 4 = 4$ ; SD 2.
2
All scores increase by 10. What happens to their SD?
2 marks
The mean also increases by 10.
Every deviation from the mean stays equal.
It is unchanged.
Lengths measured in metres have SD 0.4 m. Express the SD in centimetres.
2 marks
Conversion multiplies every value by 100.
It multiplies SD by 100.
40 cm
A sample of 5 measurements has squared deviations totalling 20. Find the sample SD.
3 marks
Divide by $5 - 1 = 4$ .
Variance $20 \div 4 = 5$ ; SD $\sqrt{5}$ .
$\sqrt{5} \approx 2.24$
Population A has SD 2.1 min and B has SD 5.4 min. Which is less variable around its mean?
2 marks
The smaller standard deviation indicates less spread about the mean.
This does not determine which group has the lower mean.
A

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Curriculum reference & lesson scope

Australian Curriculum v 9 and SA Curriculum Mathematics

Year 10 continuous distributions, bivariate data, two-way categorical tables, bias and investigations, with the approved percentile and standard-deviation content included in the normal chapter.

Official curriculum source ↗

In this lesson

Understand the idea.

Distance from the mean

State which calculation you use

See the method in action.

A complete small group

Calculate the population mean.

Add the squared deviations.

Find the population variance.

Take the square root.

Population and sample outputs

Find the mean and squared-deviation total.

Treat the three values as the population.

Treat the values as a sample instead.

Try it yourself.

Check your understanding

Keep practising