Sampling.pptx

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Sampling

Further statistics

Lesson 1 of 6

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Warm-up

Further statistics

Lesson 1 of 6

Answer each one, then check.

1

What is a percentage of an amount, e.g. 10% of 300?

2

What is the mean of 4, 6 and 8?

3

Simplify the ratio 20 : 30.

4

What does "random" mean?

5

What is a fraction of a total, e.g. ¼ of 80?

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Warm-up - Answers

Further statistics

Lesson 1 of 6

1

What is a percentage of an amount, e.g. 10% of 300?

30

2

What is the mean of 4, 6 and 8?

6

3

Simplify the ratio 20 : 30.

2 : 3

4

What does "random" mean?

Every item has an equal chance of being chosen

5

What is a fraction of a total, e.g. ¼ of 80?

20

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Learning Objectives

Further statistics

Lesson 1 of 6

1

Explain the difference between a population and a sample.

2

Recognise and describe bias in a sample.

3

Take a random, a systematic and a stratified sample.

4

Estimate a population size using capture-recapture (Higher).

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Further statistics

Lesson 1 of 6

SAMPLING

A sample should be a small group that fairly represents the whole population.

A bigger, random sample is more likely to be representative and give reliable results.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Population and Stratified Sample

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Lesson 1 of 6

A stratified sample keeps the same proportions as the population.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Types of Sample

Further statistics

Lesson 1 of 6

Know how each one is chosen.

Sample

How it is chosen

Good point

Random

Every member has an equal chance, e.g. numbers from a random number generator

Not biased

Systematic

Every kth member from a list, after a random start

Easy and quick

Stratified

Groups sampled in proportion to their size

Represents every group

Convenience

The people who are easiest to reach

Cheap, but often biased

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

A Stratified Sample

Further statistics

Lesson 1 of 6

A school has 200 pupils in Year 9, 160 in Year 10 and 240 in Year 11. A stratified sample of 60 pupils is taken. How many pupils come from each year?

1

Total pupils

200 + 160 + 240 = 600

2

Fraction sampled

60/600 = 1/10

3

Year 9

200 ÷ 10 = 20

4

Year 10

160 ÷ 10 = 16

5

Year 11

240 ÷ 10 = 24

ANSWER

20 from Year 9, 16 from Year 10 and 24 from Year 11.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

A Systematic Sample

Further statistics

Lesson 1 of 6

A list has 800 names. A systematic sample of 40 names is needed. Describe how to take it.

1

Interval

800 ÷ 40 = 20

2

Random start

Pick a random number from 1 to 20, e.g. 7

3

Then

Take every 20th name: 7, 27, 47, and so on

ANSWER

Choose a random start between 1 and 20, then take every 20th name.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Capture-Recapture (Higher)

Further statistics

Lesson 1 of 6

50 fish are caught, tagged and released. Later, 40 fish are caught and 8 of them are tagged. Estimate the number of fish in the lake.

1

Proportion tagged in the second sample

8/40

2

Set equal to the proportion tagged in the lake

8/40 = 50/N

3

Rearrange

N = (50 × 40)/8 = 250

ANSWER

There are about 250 fish in the lake.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Bias

Further statistics

Lesson 1 of 6

A sample is biased if some members are more likely to be chosen than others.

Where you ask

Asking only people at a football match about sport is biased.

Who answers

Only those who reply to an online poll may hold stronger views.

Sample too small

Small samples can be unrepresentative just by chance.

Fix it

Use a random method and a large enough sample.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Key Terms

Further statistics

Lesson 1 of 6

Population

The whole group you want to know about.

Sample

A smaller group taken from the population.

Bias

Unfairness that makes a sample unrepresentative.

Random sample

Every member has an equal chance of selection.

Stratified sample

A sample split into groups in proportion to the population.

Capture-recapture

A method to estimate the size of a wild population.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Further statistics

Lesson 1 of 6

YOUR TASK

Fair or Biased?

10 minutes

Decide whether each sample is fair or biased and explain why. (a) Asking 20 people leaving a gym how much they exercise. (b) Choosing 30 pupils at random from the register. (c) Asking the first 10 people to arrive at school about school start times.

1

Ask who might be missing.

2

Suggest a fairer method.

WHAT A GOOD ANSWER SHOWS

(a) Biased: gym users exercise more than most people. (b) Fair: random. (c) Biased: early arrivals may have different views (for example they may travel by bus).

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Can I...?

Further statistics

Lesson 1 of 6

Define population and sample.

Spot bias.

Take a random sample.

Take a systematic sample.

Work out a stratified sample.

Explain why bigger is better.

Use capture-recapture.

Give a reason for my choice.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Summary & Exam Focus

Sample = part of the population.

Random samples avoid bias.

Stratified: number in each group = group/population × sample size.

Capture-recapture: N = (first × second)/(tagged in second).

EXAM FOCUS

A school has 200 pupils in Year 9, 160 in Year 10 and 240 in Year 11. Alex takes a stratified sample of 60 pupils. How many pupils should he take from each year? (3 marks)

Find the fraction sampled, then multiply each group by it.

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