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This page sets out the method for sampling & bias in the order you would use it, then walks through the full lesson explanation. For three complete worked examples with the slips that cost marks, see the Sampling & bias topic page.
When a question asks 'give a reason why the sample may be biased' or 'comment on the method'.
When asked to describe how to choose a random sample from a list.
When the population is split into groups (year groups, departments) and the sample should reflect them.
The population is everyone you want to know about.
The sample is the group you actually ask or measure.
In a simple random sample, every member has an equal chance of being chosen.
Number the population and use random numbers, not whoever is nearby.
Asking only your friends, or only people who walk past one shop at 9am, leaves people out.
The sample no longer looks like the population.
A larger random sample usually gives a better estimate.
A large biased sample is still biased.
If 10 out of 40 people in a fair sample like the plan, that is 25%.
You can use that percentage as an estimate for the population, and you should say it is an estimate.
Capture-recapture estimates a population you cannot count.
This topic is about whether the sample was chosen fairly.
A random sample gives every member the same chance. Number the population and draw numbers, or draw from a hat. Asking the first 30 people through one gate is an opportunity sample. People who use another gate are less likely to be picked. That is bias.
Asking more people in the same unfair way does not remove the bias. A survey of 200 friends is still a survey of friends. If the question asks why a sample is biased, name who was left out. If it asks for a stratified count, calculate that count.
Method: 1. Ask whether every member had the same chance. 2. Name the method: random, opportunity, or stratified. 3. If some people could not be chosen, the sample is biased. 4. Do not call a sample unbiased just because it is large.
A stratified question wants each group in proportion. Add the group sizes if the population is not given. The count is group size over population, times the sample size. For 96 juniors out of 240, in a sample of 30, that is 96/240 × 30 = 12.
Cancel first when it helps: 96/240 = 2/5, then 2/5 × 30 = 12. The population is the denominator. 96/30 uses the sample size underneath, which is the wrong fraction.
Method: 1. Add the group sizes to get the population. 2. Write the named group over that population. 3. Multiply the fraction by the sample size. 4. Check that the answer is a whole number.
Work out every group, even if only one is asked for. The counts must add to the sample size. Juniors 12 and seniors 18 give 12 + 18 = 30. If the total is wrong, the denominator or a multiplication has slipped. This check is about the arithmetic, not about bias.
The right count can still be collected unfairly. Taking 12 juniors only from one queue is an opportunity sample inside that group. Give the stratified number when the question asks for it. Explain who was left out when the question asks about bias.
Method: 1. Calculate the count for every group. 2. Add those counts. 3. Check the group counts add up to the sample size. 4. Keep bias comments separate from the stratified number.
A common exam line is "Give one reason why this sample may not be representative". A mark needs two things: who was asked, and who was missed. "Not enough people" on its own rarely earns the mark. A better answer is "She only asked people at the gym, so people who do not like exercise could not be picked".
Look at where, when and how the people were chosen. A sample taken outside a cinema on a Saturday night leaves out people who stay at home. A sample of friends is likely to share the same age and interests. A phone-in or a website poll only hears from people who choose to reply. Name the missing group in your answer.
Method: 1. Say who the population is, for example all pupils in the school. 2. Say how the sample was picked and where the people were found. 3. Name a group that could not be picked, or had a smaller chance. 4. Finish by saying why that group might answer differently, so the result could be too high or too low.
Tip: If you are asked for two reasons, give two different groups that are missed, not the same idea reworded.
Many questions give a table from a sample and ask you to estimate for everyone. Find the scale factor first: population size divided by sample size. Then multiply the frequency you need by that factor. For a sample of 40 from 600 people, the scale factor is 600 ÷ 40 = 15, so a group of 14 in the sample becomes 14 × 15 = 210 people.
These questions usually have a second part: "State an assumption you made". The assumption is that the sample is representative, so the same proportion in the sample appears in the population. You can add that the estimate could be too high or too low if the sample was taken from only one place or time. Say it in one clear sentence.
Method: 1. Check the table adds up to the sample size. If it does not, find the missing frequency first. 2. Work out the scale factor: population divided by sample size. 3. Multiply the group frequency by the scale factor, and write the units. 4. State the assumption: the sample is representative of the whole population.
Tip: Work out the scale factor once and reuse it. It saves time when the table has several rows.
Exams sometimes ask how a sample could be improved. A random sample gives everyone the same chance. In practice you can number every person on a list and use a random number generator or a calculator to pick the numbers. Another fair method is to choose every tenth name from an alphabetical list, starting from a random place.
A larger sample is more reliable because one odd result matters less, but size alone does not fix a bad method. Ask 500 people outside a football match about their favourite sport and the answer is still biased. When the population has clear groups, a stratified sample makes sure each group is included in the right proportion.
Method: 1. Make a list of the whole population if one exists. 2. Number the list and pick numbers at random, or take every nth item from a random start. 3. Increase the sample size to make the result more reliable. 4. For clear groups, use a stratified sample so every group is represented in proportion.
Harder stratified questions hide the population total or the group sizes. You may have to add the group sizes first, or work backwards from a sample count to the population. The relationship is always dfracsample groupsample = dfracpopulation grouppopulation, so one unknown can be found from three known values. Write the equation first, then rearrange it carefully.
Some questions use two steps: first split the sample between two big groups, then use a percentage inside one group. Show every stage. Check that the group counts add to the sample size. In a real exam the numbers are chosen so that every count is a whole number. If you get a decimal, recheck the total, because a slip in adding the group sizes is the usual cause.
Method: 1. Add the group sizes to find the population, unless it is given. 2. Write the fraction group over population and multiply by the sample size. 3. For a reverse question, set the two fractions equal and solve for the unknown. 4. Add the counts to check they equal the sample size.
Tip: Cancel the fraction before you multiply: 150/450 = 1/3, and a third of 30 is easy.
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