← Data and chance

Ways to choose a sample

A sample can be chosen at random, every so many people on a list, a fair share from each group, or whoever is easiest to reach. Random ways give every member a chance; the easy way is often biased.

You need 20 students from a school of 200 for a sleep survey, and Year 7s sleep more than Year 10s. How do you pick 20 so the answer is fair?

How you choose a sample matters as much as how big it is. Each method below picks 20 students from the school of 200, listed in order by year group.

PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.

A fair share

You pick 20 of the 200 students completely at random. Will you get exactly 5 from each year group?

TryTry it on the board and compare the result with your prediction.

Four methods

Method How it chooses Fair?
simple random every member has the same chance (names from a hat, a random number generator) yes
systematic every kth member of a list, from a random start (k = population ÷ sample size) usually, unless the list has a pattern
stratified split the population into groups (strata), then choose at random within each, in proportion yes, and it guarantees each group its share
convenience whoever is easiest to reach often biased

A quota sample also fills a share for each group (5 per year), but lets the collector choose who, so it can be biased like a convenience sample.

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Take a simple random sample and read how many came from each year (you can take a few). Then take a stratified sample. Which promises 5 from each year?

Use the board. This ticks when you get there.

InvestigateWork out why it happens, one step at a time.

How each works

Here is the routine for a systematic sample.

Worked exampleBefore each step, say what you think comes next.
  1. Population 120, sample 15\text{Population } 120,\ \text{sample } 15Sizes.

Read Ava's method. Who decided which students were in the sample?

A systematic sample of 40 is taken from a list of 200. Every how many students is chosen?

Ava chooses a sample. Click the line where her method stops being random.

Year 7s sleep more than Year 10s. Why is a stratified sample (by year) a good choice for a sleep survey?

ModifyChange the problem on purpose and see what happens.

Use the methods

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Take a systematic sample of 20. Which students does it pick?

Use the board. This ticks when you get there.

A school has 300 Year 7s and 200 Year 8s. A stratified sample of 50 is taken in proportion. How many Year 7s are in it?

A list of 200 students alternates boy, girl, boy, girl. Why is a systematic sample with every 2nd student a bad idea?

An interviewer is told to ask 10 people from each age group, choosing whoever she likes in a shopping centre. What kind of sample is this?

PractiseA few on your own, to check you have it.

Choosing a method

Each student's name is put in a hat and 30 are drawn. What method is this?

A factory tests every 50th phone off the production line. What method is this?

A club has 80 junior and 40 senior members. A stratified sample of 15 is taken in proportion. How many seniors?

Review: a survey of people at a vegan festival asks how often they eat meat. What is the main problem?

MakeMake your own problem using what you have learned. Expect a few tries; that is normal.

Pick the best method

Then take 10 convenience samples and compare. Write which method you would use for the sleep survey, and why.

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Need a hint?

Press Take 10 samples for one method, then switch method and do it again.

Another hint

Look at where each method's results sit compared with the dashed line.

Take 10 samples of 20 with a fair method of your choice (not convenience). Do its results sit around the population's mean?

Use the board. This ticks when you get there.

For teachers

Students take simple random, systematic, stratified and convenience samples of a simulated school, compare them, and choose methods for real situations (AC9M8ST02, node 1). About 40 minutes.

Open the teacher guide: answers, solutions, slides and a worksheet.

Students will be able to

Before class

Do the Run board yourself first. Students need census, sample and bias (last two lessons).

The one idea: random methods (simple random, systematic from a random start, stratified) give every member a chance; convenience and quota let a person choose, so they can be biased. The common mistake is calling a hand-picked sample random (Ava's mistake).

The board lists the school by year group, so a systematic sample spreads through all years; a stratified sample guarantees 5 per year; a convenience sample is the front rows (Year 7).

Quota sampling (typical width in the curriculum's progression) is in the lesson's table and one Modify question. Wider: random number generators (=RANDBETWEEN(1,200)) to choose a real sample from a class list.

Running the lesson

  1. Predict: ask what could make a random sample uneven.
  2. Run: ask how many came from each year with each method.
  3. Investigate: ask Ava who chose the students.
  4. Modify: ask when a systematic sample goes wrong.
  5. Make: ask where each method's results sit.

The hard part (on purpose)

Random isn't the same as 'picked without thinking'. Ask: did every student have the same chance, and who decided?

When students get stuck

Stretch and support

Stretch: Take a real stratified sample of 12 from your class list (by a group of your choice) using a random number generator. Write each step.

Support: The four methods table as a card, with the board's picture for each.

Where it leads

This starts AC9M8ST02.

Where it leads: describing a sample's distribution (next), sampling variation, and investigations.

What each step is for

Now you can
Practise this skillNew questions, new numbers each timeGet 3 right in a row to mark it practised.Next lesson18. Describing a sampleDisplay a sample's data, then report its shape, centre and spread, and what it suggests about the population: an estimate, with the sampling method and size stated.