Mean/median and variability | Year 11

01 / LEARN

The idea, step by step

Weight by observations

For frequencies, multiply each value by its frequency, add, and divide by total frequency: $\bar{x} = \frac{\sum f x}{\sum f}$ . An unweighted mean of distinct values is generally wrong.

Combine group means by weighting with group sizes. A class of 10 and a class of 30 should not have equal influence. Median instead depends on ordered positions.

Choose the spread measure

Range uses extremes; IQR uses the middle half. Standard deviation measures spread around the mean using squared deviations and has the original units. Small SD means tighter concentration, not a small mean.

Calculators often label population SD $σ_{x}$ and sample SD $s_{x}$ . Population SD divides squared-deviation total by n; sample SD divides by n−1. Use the measure requested and explain whether the data describe the full target group or a sample.

A common trap

Average two means to combine any two groups.

Instead: Use group-size weights unless both groups have equal counts.

02 / WORKED EXAMPLES

See the thinking

Example 1

Frequency mean

Values 2, 4, 7 have frequencies 3, 2, 1. Find mean.

Weighted total $2 (3) + 4 (2) + 7 (1) = 21$ .
Count 6.
$21 / 6 = 3.5$ .

3.5.

Example 2

Two SD conventions

Find SDs for 2, 4, 4, 4, 5, 5, 7, 9.

Mean 5; squared-deviation total 32.
Population SD $\sqrt{32 / 8} = 2$ .
Sample SD $\sqrt{32 / 7} \approx 2.138$ .

Population SD 2; sample SD about 2.14.

Check your understanding

Add 10 to every observation. What happens to SD?

Show answer

It stays the same.

All deviations from the shifted mean are unchanged.

03 / WORKSHEET

Your turn

6 questions · 12 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Values 1, 3, 5 have frequencies 2, 3, 1. Find mean.
2 marks
Ten people average 6; thirty average 10. Find combined mean.
2 marks
Values 2, 5, 8 have frequencies 2, 3, 1. Find median.
2 marks
For the full population 3, 3, 7, 7, find mean and SD.
2 marks
All values multiply by 3. What happens to mean and SD?
2 marks
A calculator shows sigma=4.0 and s=4.2. Which is sample SD?
2 marks

Read the worked solutions

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

Values 1, 3, 5 have frequencies 2, 3, 1. Find mean.
2 marks
Weighted total 16 divided by count 6.
$8 / 3 \approx 2.67$ .
Ten people average 6; thirty average 10. Find combined mean.
2 marks
Total $10 (6) + 30 (10) = 360$ ; count 40.
9.
Values 2, 5, 8 have frequencies 2, 3, 1. Find median.
2 marks
Ordered data 2, 2, 5, 5, 5, 8.
5.
For the full population 3, 3, 7, 7, find mean and SD.
2 marks
Squared deviations total 16; $\sqrt{16 / 4} = 2$ .
Mean 5; population SD 2.
All values multiply by 3. What happens to mean and SD?
2 marks
Distances from the mean triple.
Both multiply by 3.
A calculator shows sigma=4.0 and s=4.2. Which is sample SD?
2 marks
Use s, with the original variable’s units.
4.2.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 General Mathematics: statistical investigation, univariate data, sampling, displays, summaries and interpretation.

Official curriculum source ↗

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