Population estimates and uncertainty | Year 9

01 / LEARN

Understand the idea.

Scale the proportion, not the count

To estimate a population count, first calculate the proportion in a reasonably representative sample. If $x$ out of $n$ observations have the characteristic, the estimated proportion is $\frac{x}{n}$ . Multiply this proportion by the population size.

The result is an estimate because only part of the population was measured. Report it with “about” or an approximate sign. If the calculation gives a fractional person or household, round the estimated count sensibly rather than claiming an exact fraction.

estimated count = \frac{sample count}{sample size} \times population size

Samples vary

Fair random samples can still give different answers because they contain different people. This is sampling variation, not necessarily a calculation error. Larger fair samples usually vary less, while a larger biased sample can remain consistently misleading.

Report the sample size, selection method and result alongside limitations. Repeated estimates reveal how results vary, but agreement does not establish an exact population total. Do not invent an exact margin of error from the observed range of a few samples.

Sample	Support out of 100	Estimate out of 500
1	52	260
2	58	290
3	55	275

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Estimate a school count

In a random sample of 60 students, 36 usually bring lunch from home. Estimate the number among 900 students.

Worked solution

1
Calculate the sample proportion.
$\hat{p} = \frac{36}{60} = 0.6 = 60 %$
Divide students who bring lunch by everyone in the sample.
2
Scale to the school population.
$\hat{N} = 0.6 \times 900 = 540$
Apply the estimated proportion to the $900$ students in the population.
3
Report an estimated count.
$About 540 students$
The random sample supports an estimate, rather than an exact count of all students.

Answer

About 540 students.

EXAMPLE 02

Read repeated estimates

Samples of 100 give 52%, 58% and 55% support. Estimate the corresponding counts in a population of 500.

Worked solution

1
Convert percentages to proportions.
$52 % = 0.52, 58 % = 0.58, 55 % = 0.55$
Each sample percentage must be expressed as a fraction of one before scaling.
2
Calculate the three population estimates.
$\begin{aligned} 0.52 (500) & = 260 \\ 0.58 (500) & = 290 \\ 0.55 (500) & = 275 \end{aligned}$
Each sample proportion is applied separately to the same population of $500$ .
3
Describe the variation.
$290 - 260 = 30$
The three estimated counts span $30$ people because different samples can contain different support proportions.
4
State the limit of the evidence.
The estimates are $260$ , $290$ and $275$ ; none has been established as the exact population count, and their span is not a guaranteed error bound.

Answer

260, 290 and 275; none is established as the exact population count.

03 / YOUR TURN

Try it yourself.

Check your understanding

18 out of 30 sampled residents support a proposal. Estimate the count among 600 residents.

Show answer & reasoning

About 360.

The sample proportion is $\frac{18}{30} = 0.6$ . Applying it to $600$ residents gives $0.6 \times 600 = 360$ , so the estimate is about $360$ residents.

Keep practising

6 worksheet questions · 12 marks

Open worksheet Worked solutions

Read the questions online

A sample of 40 trees has 14 gum trees. Estimate the gum trees among 600 trees.
2 marks
52 of 80 surveyed students walk or cycle. Estimate the count among 500 students.
2 marks
A sample suggests 27% of 880 households own a dog. Give a sensible estimated count.
2 marks
Two fair samples give 42% and 47%. Must one be wrong?
2 marks
A biased sample of 1,000 gives 70%. A fair random sample of 150 gives 48%. Which method is preferable for the target population?
2 marks
Samples give 61%, 58%, 63% and 60%. Find their range in percentage points.
2 marks

Read the worked solutions

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

A sample of 40 trees has 14 gum trees. Estimate the gum trees among 600 trees.
2 marks
$\frac{14}{40} = 0.35$ .
$0.35 \times 600 = 210$ .
About 210.
52 of 80 surveyed students walk or cycle. Estimate the count among 500 students.
2 marks
$\frac{52}{80} = 0.65$ .
$0.65 \times 500 = 325$ .
About 325.
A sample suggests 27% of 880 households own a dog. Give a sensible estimated count.
2 marks
$0.27 \times 880 = 237.6$ .
Round to a whole household and retain about.
About 238 households.
Two fair samples give 42% and 47%. Must one be wrong?
2 marks
The selected people differ.
The sample proportions need not match exactly.
No. Ordinary sampling variation is possible.
A biased sample of 1,000 gives 70%. A fair random sample of 150 gives 48%. Which method is preferable for the target population?
2 marks
Representativeness matters.
Large size does not remove selection bias.
The fair sample of 150, assuming a suitable frame and response quality.
Samples give 61%, 58%, 63% and 60%. Find their range in percentage points.
2 marks
The largest is 63% and smallest 58%.
$63 - 58 = 5$ percentage points.
5 percentage points.

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Curriculum reference & lesson scope

Australian Curriculum v9 / SA Curriculum Mathematics

Year 9 sampling, comparative distributions, centre/spread/outliers, displays, critique and statistical investigations.

Official curriculum source ↗

In this lesson

Understand the idea.

Scale the proportion, not the count

Samples vary

See the method in action.

Estimate a school count

Calculate the sample proportion.

Scale to the school population.

Report an estimated count.

Read repeated estimates

Convert percentages to proportions.

Calculate the three population estimates.

Describe the variation.

State the limit of the evidence.

Try it yourself.

Check your understanding

Keep practising