Understand · explore · practise
Divide a population into non-overlapping strata, allocate a proportional sample, handle rounding and select randomly within each group. Original examples and worked practice.
Before you startSimple random sampling, fractions and percentages.
01 / Split the population into strata
Strata are mutually exclusive groups which together cover the target population.
Take a random sample separately within each stratum.
A college can divide current students into Year 12 and Year 13 if every student in the target population belongs to exactly one of those years. “Studies maths” and “studies physics” would overlap, because some students study both.
Proportional stratified sampling allocates sample places in the same proportions as the population, subject to necessary whole-number rounding.
Proportional targets: A = 10, B = 6, C = 4.
Allocate A: 10, B: 6, C: 4. Total = 20.
Sampling fractions: A 20%, B 20%, C 20%.
For fractional targets, this model takes floors, then gives remaining places to the largest fractional parts. Ties use A before B before C. This is one stated rounding rule; random selection still takes place separately within each group.
02 / Choose usable groups
Strata should be clearly defined, cover the population and be relevant to the investigation. An up-to-date register can identify year groups or departments. If a person belongs to several clubs, club membership is not automatically a non-overlapping classification.
A school divides students into “plays an instrument” and “plays a sport”. Explain the problem.
Could a student be in both groups, or neither?
Yes. The groups can overlap and fail to cover everyone. Use mutually exclusive categories such as instrument only, sport only, both and neither if these suit the investigation.
A company has day, evening and night shifts, with each employee assigned to exactly one. Why could these be useful strata for a survey of shift satisfaction?
Connect the grouping to both classification and the variable.
The groups are non-overlapping and cover the employees; experiences may differ by shift. Sampling each shift ensures its planned representation.
03 / Allocate proportionally
Target from stratum = (stratum size ÷ population size) × total sample size
When targets are whole numbers, each group has the same sampling fraction.
N = 180 + 120 + 60 = 360
Use the whole population as denominator.
Sampling fraction = 36/360 = 1/10
Select ten per cent from each group.
Targets: 18 morning, 12 afternoon, 6 evening
The total is 36.
Three strata contain 90, 60 and 30 units. Allocate a proportional sample of 24.
The total is 180.
24 × 90/180 = 12; 24 × 60/180 = 8; 24 × 30/180 = 4. Total 24.
04 / Select randomly within each group
After calculating group counts, create or use the frame for each stratum. Give its members unique labels and use a fair random method to select the required number without replacement. Repeat for every group.
Asking the first people you meet until each count is filled is quota sampling, not stratified random sampling.
Pause, replay or seek freely. The notes explain the same idea and stay in view.
A sample requires 18 day-shift, 12 evening-shift and 6 night-shift employees. What must happen next for stratified random sampling?
Explain how the individuals are chosen.
Use each shift’s eligible staff list and independently select its required number by a simple random method without replacement, with unique labels and repeats rejected if generating labels with replacement.
An interviewer fills the same three counts by asking convenient employees until each group is full. Is this stratified random sampling?
The counts are not the only distinguishing feature.
No. The within-group selections are non-random; this is quota sampling.
05 / Keep the total when rounding
N = 110; targets are 5.127…, 4.145…, 2.727…
Multiply each group’s proportion by 12.
Whole-number allocation: 5, 4, 3
This totals 12 and is close to the targets.
Actual sampling fractions are only approximately equal
Rounding prevents exact proportionality here.
Check the sum after rounding. Do not independently round every target and ignore an incorrect total. Use a stated adjustment rule when necessary.
Groups have sizes 35, 35 and 30 and a total sample of 5 is required. Explain why rounding all targets to the nearest whole number with halves up is unsuitable.
The targets are 1.75, 1.75 and 1.5.
This gives 2, 2 and 2, totalling 6 rather than 5. One allocation is 2, 2, 1: take floors 1, 1, 1 and assign the two remaining places to the largest remainders 0.75 and 0.75.
For group sizes 50, 30, 20 and sample size 15, the targets are 7.5, 4.5, 3. After taking floors, one place remains. What should you state?
The first two remainders are tied.
State the tie rule. For example, prioritising group A gives 8, 4, 3; prioritising B gives 7, 5, 3. Both total 15. The model declares its A-before-B tie rule.
06 / Why stratify?
A simple random sample can miss a small subgroup by chance. Stratification can ensure that each group receives a planned positive allocation and can improve precision when units within each group are similar for the measured variable. It does not guarantee that every estimate is better than a simple random estimate.
Group representation alone does not solve nonresponse or measurement errors. Small strata may receive zero places under an unadjusted proportional allocation, so inspect the plan if group-specific conclusions are needed.
A population of 1,000 includes a stratum of 10. Its proportional target in a sample of 20 is 0.2. Why is “every group will be represented” too strong?
The allocation must be a whole number.
Rounding may assign that stratum zero places. If representation is required, use a stated minimum allocation and acknowledge that the design is no longer exactly proportional.
07 / Practical limitations
Stratification requires clear group membership and frames or suitable records. Preparing and maintaining these can cost time. Too many tiny groups can make allocation and analysis awkward. A complete-looking allocation table also does not guarantee that all selected people reply.
A proportional plan selects 10, 6 and 4 people. Only 10, 3 and 4 respond. Are the responses still in the planned proportions?
Compare the middle group with the plan.
No. The middle group is underrepresented among the 17 responses. Report and address nonresponse; do not claim the respondent set preserves the original allocation.
Why is a register with unknown department for many staff a difficulty for stratified sampling by department?
You cannot allocate and select correctly without knowing group membership.
Group counts and within-department lists may be incomplete or wrong. Improve the records or adopt a design that explicitly handles unknown membership.
08 / Interpret the final sample
If group sizes differ, selecting the same number from each group generally overrepresents smaller groups in the combined sample. Such a design can be useful for comparing groups, but it is not proportional allocation. Estimating an overall population quantity may then need appropriate weights.
Two strata contain 800 and 200 people. A researcher selects 20 from each. Is the allocation proportional?
Compare the sampling fractions.
No. The fractions are 20/800 = 2.5% and 20/200 = 10%. A proportional sample of 40 would allocate 32 and 8.
09 / Groups, allocation, random selection
A college has 210 Year 12 and 140 Year 13 students. Describe a proportional stratified random sample of 30.
Calculate the allocation and explain the selection within each year.
Total 350. Allocate 30 × 210/350 = 18 to Year 12 and 30 × 140/350 = 12 to Year 13. Use complete current lists and a fair random method to select 18 distinct Year 12 and 12 distinct Year 13 students. Record nonresponse rather than assuming everyone replies.
Section 1 of 9 · Split the population into strata