Mathematics

Stratified Samples

Take from each group in proportion to its size. 300 students in five year groups, a sample of 60, every stratum worked out, plus the rounding rule to use.

Stratified sample

A stratified sample splits the population into groups and takes from each group a number of members in proportion to that group's size.

Also called: Stratified random sample, Proportional sampling
Where students meet it: Grade 9 or 10 data handling, and in GCSE Statistics; the usual question form gives a table of group sizes and one total sample size.

The short answer
A stratified sample divides the population into groups, called strata, and takes from each a number in proportion to its size. In a school of 300 with a sample of 60, each stratum contributes one fifth of its members, so a year group of 90 supplies 18 and a year group of 30 supplies 6.

An example

60 ÷ 300 = 1/5 → Year 7: 90 × 1/5 = 18; Year 8: 75 × 1/5 = 15; Year 9: 60 × 1/5 = 12; Year 10: 45 × 1/5 = 9; Year 11: 30 × 1/5 = 6

One fraction does all five calculations. The sampling fraction is the sample size divided by the population size, here 60/300 = 1/5, and every year group is multiplied by it. The five answers come to 18 + 15 + 12 + 9 + 6 = 60, which is the check: if the parts do not add back to the sample size, one of the multiplications is wrong.

One fraction applied to every group

The whole method is a single proportion used repeatedly. Work out the sampling fraction — sample size divided by population size — and apply it to each stratum in turn. Some questions phrase it as a percentage and some as "one in five", but it is the same number doing the same job.

Setting the working out as a table with a column for group size and a column for the calculation makes it hard to lose a group and easy for a marker to follow. Show the fraction once at the top rather than rewriting 60/300 five times, and the five multiplications become one line each.

The marks are in the check and the rounding

Real data rarely divides so neatly. If the school had 301 students, the sampling fraction is 60/301 and the group sizes produce answers like 17.94, which is not a number of people. Round each stratum to the nearest whole number, then add them up.

The rounded totals will sometimes come to 59 or 61 rather than 60. When that happens, adjust one group by one — normally the largest, since one person changes its proportion least — and say in your answer which group you adjusted and why. An examiner is looking for a total that matches and a stated decision, not for a fudge left unexplained.

Why stratify at all

A simple random sample of 60 from 300 could, by chance, pick almost no Year 11 students. That would still be a fair method, but it would produce a poor sample for any question where year group matters — revision habits, screen time, subject choice. Stratifying removes that risk by building the shape of the population into the sample before the randomness starts.

The randomness has not gone anywhere, though, and this is the step students forget. Having decided that Year 7 contributes 18 pupils, those 18 must still be chosen at random from the 90. A stratified sample where the teacher picks which 18 is not a stratified random sample, and it carries every bias that a hand-picked sample carries.

Stratifying is only worth doing when the strata differ in the thing being measured. Splitting a school by year group to ask about homework is sensible, because the answer changes with age. Splitting it by the first letter of the surname would be arithmetic for its own sake.

Common questions

What is a stratum?

A stratum is one of the groups the population is divided into — a year group, a gender, a region, a shift. The plural is strata. The groups must not overlap and must together cover the whole population, so every member belongs to exactly one, which is why year group works and "plays a sport" does not.

How do I show my working for a stratified sample question?

Write the sampling fraction once, then one multiplication per stratum, then the total as a check. For 300 students and a sample of 60 that is 60/300, five lines of the form 90 × 60/300 = 18, and 18 + 15 + 12 + 9 + 6 = 60. Method marks are available even if one multiplication slips.

What if the numbers do not round to the right total?

Round each stratum first, then compare the total with the sample size. If you are one over or one under, change the largest stratum by one and state that you have done so. Never adjust several groups quietly to make the sum work, and never leave a decimal in the answer — you cannot survey 17.94 people.

Is a stratified sample better than a random one?

Usually, when the strata genuinely differ in what you are measuring, because it guarantees the sample matches the population's structure instead of leaving it to chance. It costs more effort and needs the group sizes known in advance. When the strata behave alike, it adds work without improving the estimate.

Last updated August 21, 2026