The idea, step by step
Use squared deviations to measure spread
Deviations from the mean sum to zero, so adding them directly hides spread. Square the deviations, add them, divide by n−1 for a sample variance, then take a square root. The final standard deviation has the original unit; variance has squared units. A sample of at least two observations is needed.
Choose the correct calculator output
A calculator may show both sample s and population sigma. Use the sample version when estimating population variability from a sample, unless the question explicitly requests the spread of a complete finite population. Adding a constant to all values shifts their mean but leaves deviations unchanged. Multiplying all values by k multiplies standard deviation by the absolute value of k. Keep full precision until the final result.
A common trap
Dividing by n when the question asks for sample standard deviation.
Instead: Select s or Sx on your calculator and use denominator n−1; do not confuse it with population sigma.
See the thinking
A small sample
Find the sample standard deviation of 2,4,6.
- The mean is 4; deviations are−2,0,2.
- Squared deviations sum to 8.
- Divide by n−1=2 to obtain variance 4; take its square root 2.
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.
- Multiplying by 10 gives mean 300 mm and s = 40 mm.
- Adding 2 mm changes the mean to 302 but not the spread.
Check your understanding
If all five observations equal 7, what is s?
Show answer
0.
Every deviation from the mean is zero.
Your turn
6 questions · 13 marks · Work at your own pace.
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 Sx=4.20 and sigma-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 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 16/3; s≈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|×3=6.
6. - A calculator returns Sx=4.20 and sigma-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.
Sx=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(20/3)≈2.581988897.
Mean 7; s≈2.582.
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 ↗