Population versus sample
Suppose you want to know the average height of every student in your school. Measuring all of them would work, but it is slow. Instead you measure a smaller group and use that to estimate the answer for everyone. That smaller group is a sample, and everyone you are interested in is the population.
- Population
- The whole group you want to draw a conclusion about, for example every student in the school.
- Sample
- The smaller part of the population you actually collect data from.
- Sampling fraction
- The size of the sample divided by the size of the population, \(\dfrac{\text{sample size}}{\text{population size}}\).
A good sample is representative: its make-up mirrors the population, so what is true of the sample is likely true of everyone. A good sample is also large enough to be reliable but small enough to be practical. The three methods below are different ways of choosing who goes in.
Read the question twice
Exam questions often name the population and the sample in the same sentence. "The manager surveys 50 of the 600 members" tells you the population is 600 and the sample is 50. Spotting which is which is half the battle.
Random and systematic sampling
In a simple random sample, every member of the population has an equal chance of being chosen, and the choices do not affect one another. In practice you number the population and use a random source, such as a calculator or a random number generator, to pick who is in.
In a systematic sample, you order the population, choose a starting point at random, then pick every \(n\text{th}\) member after that. The gap \(n\) is found by dividing the population size by the sample size:
A factory has 800 items on a list and wants a systematic sample of 40. Describe how to choose them.
Stratified sampling
Sometimes the population splits naturally into groups, called strata, such as year groups or age bands. A stratified sample takes the same fraction from each stratum, so every group is represented in proportion to its real size. You use the same sampling fraction throughout:
A gym has 600 members: 240 aged under 18, 240 aged 18 to 40, and 120 aged over 40. Find a stratified sample of 50.
Where this is assessed
Carrying out the stratified calculation correctly is Criterion A (knowing the method) and Criterion D (applying it to a real context). Always finish by checking your group numbers add back to the target sample size, which is the clear Criterion C communication marker that shows your reasoning is complete.
Bias, and how to avoid it
Bias means a sample systematically favours some members over others, so it stops being representative and the results are misleading. It is not the same as random error; bias pulls every conclusion in one direction.
- Selection bias: only sampling people who are easy to reach, such as surveying at the gym door about exercise habits, so people who never come are missed.
- Non-response bias: the people who ignore a survey may differ from those who reply.
- Leading questions: wording that nudges an answer, which biases the data even from a fair sample.
You reduce bias by giving every member a fair chance of selection (random methods help), keeping questions neutral, and making the sample large enough that odd cases average out.
Convenience is not random
Asking the first 30 people you meet feels random, but it is a convenience sample and is often biased. Whoever happens to be in that place at that time is not a fair cross section of the whole population.
Check yourself
Work each one through before opening the answer.
1. A school of 1200 students surveys 60 of them. What is the population, the sample, and the sampling fraction? +
The population is all 1200 students; the sample is the 60 students surveyed. The sampling fraction is \(\dfrac{60}{1200} = \dfrac{1}{20}\), so one in twenty students is sampled.
2. A list of 900 customers is to be sampled systematically to give a sample of 45. What is the gap between chosen customers? +
The gap is \(n = \dfrac{\text{population}}{\text{sample}} = \dfrac{900}{45} = \) 20. After a random start between 1 and 20, you take every 20th customer.
3. A town of 900 people has 300 children and 600 adults. Find a stratified sample of 60. +
Sampling fraction \(= \dfrac{60}{900} = \dfrac{1}{15}\). Children: \(\dfrac{1}{15} \times 300 = 20\). Adults: \(\dfrac{1}{15} \times 600 = 40\). Check: \(20 + 40 = 60\). So take 20 children and 40 adults.
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