Comparing samples and sample sizes | Year 8

01 / LEARN

Understand the idea.

Use comparable proportions

Compare unequal samples using a proportion: the number with the feature divided by that sample's total. Raw counts can reverse the impression. Forty successes can represent a larger share than sixty-four successes when the two sample sizes differ.

Use a common scale such as percentages, and retain each sample's size in the report. Check that the populations, question wording, measurement units and collection periods are comparable before treating a numerical difference as a meaningful comparison.

Size is not everything

Even well-chosen random samples can produce different results because they contain different members. This sampling variation is expected. Larger random samples usually give more stable estimates, but no sample size guarantees that an estimate equals the population value exactly.

A large convenience sample can retain systematic bias. When comparing numerical samples, consider spread as well as centre: identical means can come from a tightly clustered sample or a widely scattered one. Report what the observed samples show without overstating the population conclusion.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Worked example 1

Compare 40 of 50 with 64 of 100 as proportions.

Worked solution

1
Convert the first sample to a percentage.
$\frac{40}{50} \times 100 % = 80 %$
The denominator must be the first sample's own size.
2
Convert the second sample to the same scale.
$\frac{64}{100} \times 100 % = 64 %$
Using percentages makes the unequal samples directly comparable.
3
Compare the observed shares.
$80 % > 64 %$
The first sample has the higher proportion despite its smaller success count.

Answer

80% versus 64%.

EXAMPLE 02

Worked example 2

Compare samples 10,12,14 and 11,12,13.

Worked solution

1
Calculate each sample mean.
$\begin{aligned} {\bar{x}}_{1} & = \frac{10 + 12 + 14}{3} = 12 \\ {\bar{x}}_{2} & = \frac{11 + 12 + 13}{3} = 12 \end{aligned}$
Both samples have three observations totalling 36.
2
Calculate each range.
$\begin{aligned} R_{1} & = 14 - 10 = 4 \\ R_{2} & = 13 - 11 = 2 \end{aligned}$
Subtract the smallest from the largest within each sample separately.
3
Describe centre and spread together.
${\bar{x}}_{1} = {\bar{x}}_{2}, R_{1} > R_{2}$
Both means are 12, but the first sample is more spread out by the range measure.

Answer

Both means 12; ranges 4 and 2.

03 / YOUR TURN

Try it yourself.

Check your understanding

Which proportion is larger: 18 of 30 or 25 of 50?

Show answer & reasoning

18 of 30.

The shares are $\frac{18}{30} = 60 %$ and $\frac{25}{50} = 50 %$ . Thus 18 of 30 is the larger proportion even though 18 is the smaller count.

Keep practising

6 worksheet questions · 15 marks

Open worksheet Worked solutions

Read the questions online

Compare 12 of 20 with 42 of 100 as percentages.
3 marks
A random sample of 25 from 500 students includes 10 cyclists. Estimate the population count and state a limitation.
3 marks
Why might 1000 responses from sports-club members still misrepresent all students’ exercise?
2 marks
An unusual value is included in samples of 10 and 100, with otherwise similar observations. In which does it influence the mean more?
2 marks
Name two details to match when comparing samples as evidence of a change over time.
2 marks
Which usually produces more stable proportions: random samples of 20 or 200? Is exact agreement guaranteed?
3 marks

Read the worked solutions

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

Compare 12 of 20 with 42 of 100 as percentages.
3 marks
$12 \div 20 \times 100 = 60$ ; $42 \div 100 \times 100 = 42$ .
60% and 42%; the first proportion is larger.
A random sample of 25 from 500 students includes 10 cyclists. Estimate the population count and state a limitation.
3 marks
$\frac{10}{25} = 0.4$ .
$0.4 \times 500 = 200$ .
About 200; sampling variation may affect the estimate.
Why might 1000 responses from sports-club members still misrepresent all students’ exercise?
2 marks
Large size cannot remove systematic under-coverage.
The frame excludes non-members.
An unusual value is included in samples of 10 and 100, with otherwise similar observations. In which does it influence the mean more?
2 marks
Each observation has weight $\frac{1}{10}$ rather than $\frac{1}{100}$ in the mean.
The sample of 10.
Name two details to match when comparing samples as evidence of a change over time.
2 marks
A changed group or question could explain the result instead of time.
For example, the target population and measurement definition.
Which usually produces more stable proportions: random samples of 20 or 200? Is exact agreement guaranteed?
3 marks
Larger random samples usually reduce variation, but chance differences remain.
200; exact agreement is not guaranteed.

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Curriculum reference & lesson scope

Australian Curriculum v9 and SA Curriculum

Population, census and sample; observational versus experimental data; random and non-random sampling; primary/secondary data; reviewing numerical displays and summary statistics; comparing samples and sample sizes; bias, variation and uncertainty; planning, conducting and reporting a fair investigation.

Official curriculum source ↗

In this lesson

Understand the idea.

Use comparable proportions

Size is not everything

See the method in action.

Worked example 1

Convert the first sample to a percentage.

Convert the second sample to the same scale.

Compare the observed shares.

Worked example 2

Calculate each sample mean.

Calculate each range.

Describe centre and spread together.

Try it yourself.

Check your understanding

Keep practising