Mean/median and variability | Year 11

01 / LEARN

Understand the idea.

Weight by observations

In a frequency table, a value with frequency three contributes three copies to the total. Calculate $\bar{x} = \frac{\sum f x}{\sum f}$ : add the weighted values, then divide by the number of observations, not by the number of rows.

The same idea combines group means: multiply each mean by its group size to recover its total. An ordinary average of two means works only when their group sizes are equal. Median instead follows ordered positions.

Choose the spread measure

Standard deviation measures spread around the mean. Squaring deviations prevents positive and negative deviations cancelling; taking a square root returns to the original units. A small SD means tight concentration, regardless of whether the mean is large.

Population SD divides the squared-deviation total by $n$ ; sample SD divides by $n - 1$ . Calculator labels often use $σ_{x}$ and $s_{x}$ . Select the requested convention and state whether the records represent a whole target group or a sample.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Frequency mean

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

Worked solution

1
Count the observations.
$n = 3 + 2 + 1 = 6$
Frequency counts how many times each value occurs.
2
Find the weighted total.
$\sum f x = 3 (2) + 2 (4) + 1 (7) = 6 + 8 + 7 = 21$
Each value contributes once for every observation carrying it.
3
Divide total by count.
$\bar{x} = \frac{21}{6} = 3.5$
Dividing by six averages the actual observations, whereas dividing by three would average table rows.

Answer

3.5.

EXAMPLE 02

Two SD conventions

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

Worked solution

1
Find the mean.
$\bar{x} = \frac{2 + 4 + 4 + 4 + 5 + 5 + 7 + 9}{8} = 5$
Deviations must be measured from the data set's own mean.
2
Calculate the squared-deviation total.
$\begin{aligned} \sum {(x - 5)}^{2} & = 9 + 1 + 1 + 1 + 0 + 0 + 4 + 16 \\ = 32 \end{aligned}$
Squaring makes all contributions non-negative and gives larger deviations more weight.
3
Calculate population SD.
$σ = \sqrt{\frac{32}{8}} = 2$
Population SD uses the full count as its divisor.
4
Calculate sample SD.
$s = \sqrt{\frac{32}{8 - 1}} \approx 2.14$
Sample SD uses one fewer than the count; the same numerator therefore gives a slightly larger result.

Answer

Population SD 2; sample SD about 2.14.

03 / YOUR TURN

Try it yourself.

Check your understanding

Add 10 to every observation. What happens to SD?

Show answer & reasoning

It stays the same.

The SD stays the same. The mean also increases by ten, so each deviation is unchanged: $(x + 10) - (\bar{x} + 10) = x - \bar{x}$ . The squared-deviation total and divisor remain unchanged.

Keep practising

6 worksheet questions · 12 marks

Open worksheet 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 $σ = 4.0$ 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.
$\frac{8}{3} \approx 2.67$ .
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; $\sqrt{\frac{16}{4}} = 2$ .
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 $σ = 4.0$ 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 ↗

In this lesson

Understand the idea.

Weight by observations

Choose the spread measure

See the method in action.

Frequency mean

Count the observations.

Find the weighted total.

Divide total by count.

Two SD conventions

Find the mean.

Calculate the squared-deviation total.

Calculate population SD.

Calculate sample SD.

Try it yourself.

Check your understanding

Keep practising