Chapter 1 of 8

All 8 chapters
  1. 1Samples and bias, readunderstood
  2. 2Averages from frequency tables, readunderstood
  3. 3Cumulative frequency, readunderstood
  4. 4Box plots, readunderstood
  5. 5Histograms with unequal widths, readunderstood
  6. 6Scatter graphs and correlation, readunderstood
  7. 7Data over time, readunderstood
  8. 8Telling the truth with data, readunderstood

M4.7.1 Chapter 1

Samples and bias

Ravi asks 20 people leaving the sports hall if they exercise every day. Do his answers tell us about the whole town?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Explain how a sample can be biased
  • Take a random sample using random numbers
  • Choose a sample that matches the groups in a population

The population is everything we want to know about: all the plants in a field, or all the people in a town. Asking every member takes too long, so we look at a few. Those few are the sample.

Ravi's town has a street with 60 people. 15 of them exercise most days. Ravi stands at the sports hall and asks 8 people leaving. 7 of them say yes.

The street has 60 people in a grid. The first 15 are the group Exercise most days, shown white. The other 45 are the group Not most days, shown dotted. Ravi's sample of 8 people is framed. 7 of the 8 are from the group Exercise most days.Exercise most daysNot most days
Framed: the 8 people Ravi asked. Almost all of them are from the small white group.

In the sample, 7 of 8 people exercise. In the street, only 15 of 60 do, which is 1 in 4. Ravi's sample gives the wrong picture. Here the cause is the place: people at a sports hall are more likely to exercise.

A sample is biased when some members of the population are more likely to be chosen than others. The results then lean one way. A bias comes from how members are chosen. A bigger sample taken the same way is just as biased.

A random sample gives every member of the population the same chance of being chosen. A fair way to make one is with random numbers. Number the members from 1 to the size of the population. Then let random numbers choose.

Draw it first

A random sample of 5 plants from a bed of 40

  1. Step 1: a bed of 40 plants numbered 1 to 40, in 4 rows of 10. Step 2: the plants chosen by random numbers are framed: 17, 36, 4, 29, 38. The label says sample of 5.12345678910111213141516171819202122232425262728293031323334353637383940

    1 Number every plant in the bed, from 1 to 40.

  2. Step 1: a bed of 40 plants numbered 1 to 40, in 4 rows of 10. Step 2: the plants chosen by random numbers are framed: 17, 36, 4, 29, 38. The label says sample of 5.12345678910111213141516171819202122232425262728293031323334353637383940sample of 5

    2 Read random numbers two digits at a time. Skip 00, any number above 40, and any repeat.

Suppose the random pairs are 82, 17, 36, 17, 04, 55, 29, 38. Plant 82 and plant 55 do not exist, so skip them. The second 17 is a repeat, so skip it. The sample is plants 17, 36, 4, 29 and 38.

Worked example

The pairs 52, 09, 17, 40, 09, 31 are random. A bed has 40 plants. Choose a sample of 4.

  1. Read the pairs in order. 52 is above 40, so skip it.
  2. 09 is plant 9. 17 is plant 17. 40 is plant 40.
  3. The second 09 is a repeat, so skip it.
  4. 31 is plant 31. The sample is plants 9, 17, 40 and 31.

Answer: Plants 9, 17, 40 and 31

A random sample can still be odd by luck. That is not bias. Bias is built into the way members are chosen. Luck changes from one sample to the next.

Sometimes a population has groups that differ. A stratified sample takes from each group in proportion to its size. A group that is half the population gives half the sample. The share from a group is group size ÷ whole population × sample size.

See it three ways

With things

A field has 60 plants: 36 in the sun, 15 in the shade and 9 by the pond. You want a sample of 20.

As a picture

A field of 60 plants in a grid, in three groups: Sun 36 plants, white. Shade 15 plants, dotted. Pond 9 plants, hatched. The 20 plants chosen are framed, in the same proportion: 12 from Sun, 5 from Shade and 3 from Pond.SunShadePond

In symbols

Sun: 36 ÷ 60 × 20 = 12. Shade: 15 ÷ 60 × 20 = 5. Pond: 9 ÷ 60 × 20 = 3. Together 12 + 5 + 3 = 20.

Inside each group, choose the members at random. Stratifying keeps the groups in proportion. Choosing at random keeps each member equally likely. If a share is not a whole number, round it and check that the total is still the sample size.

Worked example

A club has 60 members: 36 walkers and 24 cyclists. Choose a stratified sample of 10.

  1. Walkers: 36 ÷ 60 × 10 = 6.
  2. Cyclists: 24 ÷ 60 × 10 = 4.
  3. Check the total: 6 + 4 = 10, the size of the sample.
  4. Choose 6 walkers at random and 4 cyclists at random.

Answer: 6 walkers and 4 cyclists

The club has 60 members in a grid. Walkers 36 members, white. Cyclists 24 members, dotted. The 10 chosen are framed: 6 walkers and 4 cyclists.WalkersCyclists
Of the 10 chosen, 6 are walkers and 4 are cyclists, the same shares as the club.

Try it

Take a random sample with paper slips

You need: 20 small pieces of paper, a pencil, a bag or a bowl.

  1. Write the numbers 1 to 20 on the slips, one number on each. Fold them.
  2. Put the slips in the bag and shake well.
  3. Without looking, take out 5 slips one at a time and keep them out. Write the numbers.

What you will see: Every slip had the same chance. Nobody chose by looking. If you do it again you will get different numbers, and that is fine.

If you cannot try it, this is what happens.
Twenty numbered slips in 2 rows of 10, numbered 1 to 20. The slips 3, 7, 12, 16, 19 are framed: one possible sample of 5.1234567891011121314151617181920

A wrong idea many people have

A big sample cannot be biased.

Why it feels right: More people sounds like a better picture of the population.

The test that breaks it: Ravi asks 500 people at the sports hall. Most of them exercise, as before. More people from the same place repeat the same lean.

What is true: Bias comes from how members are chosen, not from how many. A smaller random sample can be fairer than a big biased one.

Key words

population
everything we want to know about.
sample
a part of the population that we look at.
biased
leaning one way, because some members are more likely to be chosen.
random sample
a sample where every member has the same chance of being chosen.
stratified sample
a sample that takes from each group in proportion to its size.

Back to your guess

The answer is B: No. People at a sports hall are more likely to exercise than other people.

People at a sports hall are more likely to exercise than the rest of the town, so his sample leans one way. Asking more people at the same place does not cure that. A fair sample gives every person in the town the same chance of being asked.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. Amina wants to know how many hours people in a town sleep. Which way of choosing 50 people is the best?

    Tap your answer.

    1. Not this one. Friends are often alike, in how they live and when they sleep. Some people in the town have no chance of being asked. Go to M4.7.1. Try another.

    2. Not this one. People up before dawn are more likely to sleep fewer hours. This sample leans one way. Go to M4.7.1. Try another.

    3. Yes, that is right. Every person on the list has the same chance of being chosen, so the sample is not biased. The other two ways make some people more likely to be asked.

    See the answer

    Choose names at random from a list of everyone in the town.

    Every person on the list has the same chance of being chosen, so the sample is not biased. The other two ways make some people more likely to be asked.

    • If you chose A: Friends are often alike, in how they live and when they sleep. Some people in the town have no chance of being asked. Go to M4.7.1.
    • If you chose B: People up before dawn are more likely to sleep fewer hours. This sample leans one way. Go to M4.7.1.

  2. Ravi asks 500 people at the sports hall. He says his sample is too big to be biased. What do you say?

    Tap your answer.

    1. Yes, that is right. Bias comes from how members are chosen. Ravi chose from a place where active people are more likely to be, so more of them only repeats the lean.

    2. Not this one. Size does not remove bias. All 500 people came from a place where people who exercise are more likely to be. Go to M4.7.1. Try another.

    3. Not this one. A sample does not need to be half the population. A random sample of a few hundred can be fair for a very large population. What counts is how it is chosen. Go to M4.7.1. Try another.

    See the answer

    A big sample can still be biased if the way of choosing favours some people.

    Bias comes from how members are chosen. Ravi chose from a place where active people are more likely to be, so more of them only repeats the lean.

    • If you chose B: Size does not remove bias. All 500 people came from a place where people who exercise are more likely to be. Go to M4.7.1.
    • If you chose C: A sample does not need to be half the population. A random sample of a few hundred can be fair for a very large population. What counts is how it is chosen. Go to M4.7.1.

  3. What makes a sample random?

    Tap your answer.

    1. Not this one. Choosing by eye lets the picker's own ideas decide. Some members are then more likely to be chosen than others. Go to M4.7.1. Try another.

    2. Yes, that is right. That is what random means in statistics. It does not depend on size or on looks.

    3. Not this one. In everyday talk "random" can mean careless. Picking fast tends to favour members that are quick to reach. A random sample gives every member the same chance. Go to M4.7.1. Try another.

    4. Not this one. A large sample can be biased. What makes a sample random is the equal chance for every member. Go to M4.7.1. Try another.

    See the answer

    Every member of the population has the same chance of being chosen.

    That is what random means in statistics. It does not depend on size or on looks.

    • If you chose A: Choosing by eye lets the picker's own ideas decide. Some members are then more likely to be chosen than others. Go to M4.7.1.
    • If you chose C: In everyday talk "random" can mean careless. Picking fast tends to favour members that are quick to reach. A random sample gives every member the same chance. Go to M4.7.1.
    • If you chose D: A large sample can be biased. What makes a sample random is the equal chance for every member. Go to M4.7.1.

  4. Plants in a bed are numbered 1 to 40. Random pairs are 47, 12, 38, 12, 05, 26. Which plants are in a sample of 4?

    A bed of 40 plants numbered 1 to 40, in 4 rows of 10. None is framed yet.12345678910111213141516171819202122232425262728293031323334353637383940
    Tap your answer.

    1. Not this one. There is no plant 47, because the bed has 40 plants. A number that is too big is skipped. A plant also cannot be chosen twice. Go to M4.7.1. Try another.

    2. Not this one. A plant cannot be in the sample twice. The second 12 is a repeat, so skip it. Go to M4.7.1. Try another.

    3. Not this one. The plants have two digit numbers, so read the digits two at a time, not one at a time. Go to M4.7.1. Try another.

    4. Yes, that is right. 47 is above 40, so skip it. Then 12 and 38 are chosen. The second 12 is a repeat. The pair 05 is plant 5, and 26 is plant 26, which fills the sample.

    See the answer

    Plants 12, 38, 5 and 26

    47 is above 40, so skip it. Then 12 and 38 are chosen. The second 12 is a repeat. The pair 05 is plant 5, and 26 is plant 26, which fills the sample.

    • If you chose A: There is no plant 47, because the bed has 40 plants. A number that is too big is skipped. A plant also cannot be chosen twice. Go to M4.7.1.
    • If you chose B: A plant cannot be in the sample twice. The second 12 is a repeat, so skip it. Go to M4.7.1.
    • If you chose C: The plants have two digit numbers, so read the digits two at a time, not one at a time. Go to M4.7.1.

  5. A population of 80 has two groups, A with 48 members and B with 32. A stratified sample of 10 is wanted. How many come from group A?

    A population of 80 in a grid. Group A 48 members, white. Group B 32 members, dotted.Group AGroup B
    Tap your answer.

    1. Not this one. You took the same number from each group. Group A is bigger than group B, so it must give more. Go to M4.7.1. Try another.

    2. Yes, that is right. Group A is 48 ÷ 80 of the population, so it gives 48 ÷ 80 × 10 = 6 of the sample.

    3. Not this one. That is the share for group B, 32 ÷ 80 × 10. Group A is the bigger group. Go to M4.7.1. Try another.

    See the answer

    6

    Group A is 48 ÷ 80 of the population, so it gives 48 ÷ 80 × 10 = 6 of the sample.

    • If you chose A: You took the same number from each group. Group A is bigger than group B, so it must give more. Go to M4.7.1.
    • If you chose C: That is the share for group B, 32 ÷ 80 × 10. Group A is the bigger group. Go to M4.7.1.

  6. A radio show asks listeners to phone in a vote on a new bus route. Why is the result likely to be biased?

    Tap your answer.

    1. Yes, that is right. People choose themselves. Those who feel strongly are more likely to phone, and people who do not listen cannot. So some members are more likely than others to be in the sample.

    2. Not this one. A miscount is a mistake in counting. Bias is different. Even a perfect count would lean one way, because only some people choose to vote. Go to M4.7.1. Try another.

    3. Not this one. Listeners decide for themselves whether to vote. That is not random: people who care a lot are more likely to take part. Go to M4.7.1. Try another.

    See the answer

    Only people who listen and care enough to phone take part.

    People choose themselves. Those who feel strongly are more likely to phone, and people who do not listen cannot. So some members are more likely than others to be in the sample.

    • If you chose B: A miscount is a mistake in counting. Bias is different. Even a perfect count would lean one way, because only some people choose to vote. Go to M4.7.1.
    • If you chose C: Listeners decide for themselves whether to vote. That is not random: people who care a lot are more likely to take part. Go to M4.7.1.

  7. A garden has 90 plants: 60 bean plants and 30 pea plants. A stratified sample of 15 is wanted. How many bean plants are in it?

    See the answer

    10

    Bean plants are 60 ÷ 90 of the garden. 60 ÷ 90 × 15 = 10. Check: peas give 30 ÷ 90 × 15 = 5, and 10 + 5 = 15.

  8. A field has 100 plants: 40 beans, 35 peas and 25 maize. A stratified sample of 20 is wanted. How many maize plants are in it?

    See the answer

    5

    Maize is 25 ÷ 100 of the field. 25 ÷ 100 × 20 = 5. Check: beans 8 and peas 7, and 8 + 7 + 5 = 20.

Keep it

Copy this into your exercise book.

A sample is biased if some members are more likely to be chosen. A random sample gives every member the same chance. A stratified sample takes group ÷ whole × sample size from each group.

Draw: A grid of squares with two groups and a few squares framed, and the share sum under it.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

No internet tomorrow? Save the whole book as one file (1.1 MB). It opens on any phone.