Probability and statistics · GCSE Maths
Stratified sampling
GCSE Maths stratified sampling: sample size proportional to each stratum, nᵢ = (Nᵢ/N) × n, and why subgroups are represented fairly.
Stratified sample: same proportion in the sample as in the population. (group size ÷ total) × sample size, round carefully.
The important bits
What you need to know
- 1
Stratified sampling divides the population into strata (groups) and samples from each group in proportion to its size.
- 2
Sample from stratum i: nᵢ = (Nᵢ / N) × n, where Nᵢ is stratum size, N is total population, n is total sample size.
- 3
Example: 120 students, 50 Year 10 and 70 Year 11. Sample 30: Year 10 take (50/120) × 30 = 12.5 → 12 or 13 depending on rounding rules.
- 4
Random selection within each stratum after calculating counts — stratified is not “pick 30 at random from everyone”.
- 5
Purpose: ensure subgroups (year groups, regions, gender) appear in the sample roughly as they do in the population.
- 6
Compare to simple random sampling: one big random pick. Compare to systematic: every kth item. Stratified when strata matter for comparison.
- 7
Rounding: if nᵢ must be integers, round and adjust so the stratum sample sizes sum exactly to n.
- 8
Stratified estimates of population mean combine stratum means weighted by proportion — extension on some Higher papers.
Quotations worth analysing
Short evidence. Real method.
“nᵢ = (Nᵢ / N) × n”
Proportion of population in stratum × total sample. Students who sample 30 from each stratum regardless of Nᵢ have ignored proportionality.
“Represent each subgroup fairly”
If Year 11 is 70% of the school, about 70% of the sample should be Year 11. Simple random might by luck miss a small group.
“Random within each stratum”
Calculate counts, then pick randomly inside each group. The formula alone is not a full method description without random selection.
Go deeper
Worked stratification
A factory has 200 on shift A, 300 on shift B, 500 on shift C. Sample 40 workers. Proportions: A 200/1000 = 0.2 → 8 workers. B 300/1000 = 0.3 → 12 workers. C 500/1000 = 0.5 → 20 workers. Sum 8 + 12 + 20 = 40. Then randomly choose 8 from shift A, 12 from B, 20 from C. If the formula gives 12.5, exam papers often say round to nearest integer and adjust: 12 + 13 = 25 if two strata each need half — state rounding. Check total matches n.
Go deeper
Why not simple random?
Simple random sample of 40 from 1000 might include zero shift A workers by bad luck if A is only 20% — 8 expected but variance exists. Stratified guarantees 8 from A, reflecting the population mix. Useful when comparing shifts, year groups, or regions in the analysis. GCSE questions focus on calculation, not critique, but “why stratified?” wants proportional representation of subgroups.
Go deeper
Rounding and totals
Three strata, sample 50: calculations 16.7, 16.7, 16.6 → round to 17, 17, 16 = 50. If you round all up to 17 you get 51 — adjust one stratum down. Show raw values before rounding. Some papers accept 12 and 13 for a 25 sample split 50–50 from 500 and 500 — sum must be 25. Fractional workers are impossible; integers are mandatory in the final answer.
See the idea in action
A school has 80 boys and 120 girls. A stratified sample of 50 is taken. How many boys and girls? Total N = 200. Boys: (80/200) × 50 = 20. Girls: (120/200) × 50 = 30. Check: 20 + 30 = 50. Randomly select 20 boys and 30 girls from respective lists.
Exam technique
Turn knowledge into marks
Show (Nᵢ/N) × n for each stratum. Check stratum samples sum to n. Mention random selection within strata if the question asks for the full method.
Common mistakes
Do not give these marks away
- 01
Taking equal sample sizes from each stratum regardless of population proportion.
- 02
Using n/N instead of Nᵢ/N — confusing total sample with stratum size.
- 03
Rounding each stratum without checking the total still equals n.
Population 400, stratum size 100, sample 20 total. Stratified sample from stratum is
A5
B10
C20
D100
Show the answer
5. (100/400) × 20 = 5. 10 would be half the sample for a quarter of the population. 20 is the whole sample.
Quick questions
If this is the bit you searched
What is a stratum?
A subgroup of the population with a common characteristic — year group, shift, region. Sampling is proportional within each.
How is stratified different from random sampling?
Random picks n from the whole population. Stratified picks nᵢ from each stratum in proportion to stratum size, then combines.
What if the calculation is not a whole number?
Round to nearest integer and adjust so the stratum sizes sum to n. Show the unrounded value first.
When would you use stratified sampling?
When subgroups should be represented fairly for comparison, and when simple random might miss small groups.