The idea, step by step
Distance from the mean
Subtract the mean from each value. Squaring makes negative and positive deviations contribute positively, so spread does not disappear when deviations cancel. Average the squared deviations to obtain the population variance, then take its square root.
We use population standard deviation when the listed values are the entire group of interest. Variance has squared units, but standard deviation returns to the data's original units.
State which calculation you use
When a sample estimates a larger population's variance, a common sample formula divides the squared-deviation total by n−1. A calculator may show both population and sample outputs. Choose the one matching the question and label it.
Adding a constant shifts every value and the mean equally, so standard deviation stays the same. Multiplying all values by k multiplies standard deviation by the absolute value of k. Standard deviation is sensitive to extreme observations.
A common trap
The mean of the deviations is the standard deviation.
Instead: Signed deviations cancel. Standard deviation uses squared deviations and a final square root.
See the thinking
A complete small group
The complete population is 2, 4, 4, 4, 5, 5, 7, 9. Find its population standard deviation.
| 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 |
- The mean is 40÷8=5.
- Squared deviations are 9, 1, 1, 1, 0, 0, 4, 16; their sum is 32.
- Population variance=32÷8=4; standard deviation=√4=2.
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?
- The mean is 3 and squared deviations total 4+0+4=8.
- Population SD=√(8/3)≈1.63.
- Sample SD=√(8/2)=2.
Check your understanding
The population is 4, 4, 4, 4. What is its SD?
Show answer
0
Every deviation from the mean 4 is zero.
Your turn
6 questions · 15 marks · Work at your own pace.
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.
- Find the population SD of 3, 3, 7, 7.3 marks
Mean 5; four squared deviations are 4.
Variance 16÷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÷4=5; SD√5.
- 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
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 ↗