Stratified random sample
You already know which group each member of a population belongs to, and the groups answer differently. How do you make sure every group gets its fair share of the sample instead of leaving it to luck?
Where you are 4 in Random sampling
When you reach for it
A simple random sample lets chance alone choose who is asked, so every group of n members is equally likely. That fairness has a cost: by luck the sample may hold too many of one kind of member and too few of another, and if those kinds answer differently, the result swings with the mix.
Reach for a stratified random sample when, before sampling, you can sort every member of the population into groups by something you already know (class year, building, region), and that something is likely tied to the answer you are after. Then you stop leaving the mix to chance: you fix how many come from each group and let chance choose only which members.
The procedure
The definition. Split the population into groups called strata (one group is a stratum) so that every member is in exactly one stratum. Then take a separate simple random sample inside each stratum. The combined members form a stratified random sample. Write N_h for the size of stratum h, N for the whole population, n_h for the number sampled from stratum h and n for the whole sample.
Choosing the strata. Good strata are alike inside and different from each other on the thing being measured. If first-years mostly answer one way and seniors another, class year is a good stratifying variable: inside a stratum the answers vary little, so a modest sample pins that stratum down, and the differences between strata, the part an unlucky mix would distort, are handled by design. A variable unrelated to the answer, such as birth month, makes strata that all look like the population, and splitting by it changes almost nothing.
Proportional allocation. The plainest choice samples every stratum at the same rate, \displaystyle \frac{n}{N}:
Then every member’s chance is \displaystyle \frac{n_h}{N_h} = \frac{n}{N}, the same as in a simple random sample of size n, and the sample’s share from each stratum equals that stratum’s share of the population. If n_h is not a whole number, round, keeping the total at n.
The draw inside each stratum. Each stratum needs its own list, so the frame must record the stratifying variable for every member. Label each stratum’s members separately and draw from each exactly as for any simple random sample, by random digits, a calculator or slips. The draws are separate on purpose: one draw from the whole list would let the counts per stratum wander again.
Why the estimate varies less. A simple random sample’s mean moves for two reasons: which members of each group are drawn, and how many come from each group. Stratifying removes the second reason entirely. When the strata differ a lot, the second reason is the larger one, so the estimate varies much less from sample to sample. Example 3 counts every possible sample to show it. When the strata are alike, the second reason was small to begin with, and there is little to gain.
Combining stratum results. With proportional allocation the pooled sample result is already right, because each stratum fills its population share of the sample. With any other allocation, weight each stratum’s result m_h (a mean or a proportion) by its population share:
A pooled result that ignores the weights counts an oversampled stratum too heavily. Under proportional allocation \displaystyle \frac{n_h}{n} = \frac{N_h}{N}, so the pooled result and this weighted one are the same number.
Worked examples
Shares for four class years
A university’s housing office lists all 4{,}800 of its dormitory residents with their class year: 1{,}500 first-years, 1{,}300 sophomores, 1{,}100 juniors and 900 seniors (made-up numbers). Davit will ask a sample of 240 residents whether the dining hall should stay open until midnight. Explain why class year makes sensible strata, and find the proportional allocation and each resident’s chance.
SolutionResidents in different years keep different class schedules and different eating habits, and a senior may cook while a first-year relies on the dining hall, so the answer is plausibly tied to class year. Every resident’s class year is on the housing list, so each one falls in exactly one stratum.
The sampling rate is
so each stratum gives \displaystyle \frac{1}{20} of its members: \displaystyle \frac{1500}{20} = 75 first-years, \displaystyle \frac{1300}{20} = 65 sophomores, \displaystyle \frac{1100}{20} = 55 juniors and \displaystyle \frac{900}{20} = 45 seniors. As a check, 75 + 65 + 55 + 45 = 240. Each resident’s chance is \displaystyle \frac{1}{20} = 5\%, whatever the class year.
A simple random sample of 240 would also hold 75 first-years on average, but one draw might hold 90 or 60. Here it holds exactly 75, every time.
Drawing the seniors
Davit labels the 900 seniors 001 to 900, in a list separate from the other class years. Use this row of random digits, from its first digit, to choose the first 5 seniors.
The labels have three digits, so read blocks of three across the gaps: 931, 074, 829, 152, 000, 074, 865, 203. Block 931 is above 900 and 000 is no one’s label, so both are skipped; the second 074 repeats a senior already chosen and is skipped too. The first five seniors are seniors 074, 829, 152, 865 and 203; the last digit is not needed.
Davit continues until he has 45 different seniors, then makes three more draws: 75 from the first-years labeled 0001 to 1500, 65 from the sophomores and 55 from the juniors, each from its own list. A single label like 074 means a different person in each list, which is why the draws are kept apart.
Counting every possible sample
A small made-up hall has six residents. Three first-years eat 12, 13 and 14 dining-hall meals a week; three seniors eat 4, 5 and 6. Siranush will estimate the hall’s mean with a sample of 2. Compare a simple random sample of 2 with a stratified sample of one first-year and one senior by listing every possible sample mean.
SolutionThe population mean is \displaystyle \frac{12 + 13 + 14 + 4 + 5 + 6}{6} = \frac{54}{6} = 9 meals.
A simple random sample of 2 from 6 can be any of 15 pairs. The 9 pairs with one resident of each year have means from 8 (meals 12 and 4) to 10 (meals 14 and 6). The other 6 pairs come from one year only: 12.5, 13, 13.5 for two first-years and 4.5, 5, 5.5 for two seniors. So the means run from 4.5 to 13.5.
The stratified design can only produce the 3 \times 3 = 9 mixed pairs, so its means run from 8 to 10.
Both designs are right on average: the 15 means and the 9 means each average exactly 9. The difference is spread. Under simple random sampling only 9 of the 15 samples land within 1 meal of the truth, and 6 miss by 3.5 to 4.5 meals, all of them because the mix came out lopsided. Stratifying rules those samples out.
Weighting an equal allocation
Siranush wants each class year estimated equally well, so she takes 60 residents from each, again 240 in all, and asks about the midnight closing. In favor: 42 first-years, 36 sophomores, 27 juniors and 18 seniors. Estimate the percent of all 4{,}800 residents in favor.
SolutionThe stratum proportions are 0.70, 0.60, 0.45 and 0.30. Simply pooling gives
but that treats each year as a quarter of the residents. Seniors, who favor it least, are \displaystyle \frac{900}{4800} of the population yet a quarter of the sample, so pooling leans low. Weight each proportion by its stratum’s share:
About 54.1\% of the residents are estimated to favor it. The weights undo unequal chances: a first-year’s chance was \displaystyle \frac{60}{1500} = \frac{1}{25} and a senior’s \displaystyle \frac{60}{900} = \frac{1}{15}, so each sampled first-year speaks for 25 residents and each sampled senior for only 15.
Which variable to stratify by
Hovhannes wants to estimate the percent of residents who find their room too cold in winter. The four dormitory buildings have different heating systems, and every class year lives in every building. He considers three ways to stratify: by building, by class year, and by the first letter of the last name. Which should he choose?
SolutionHow cold a room is depends mostly on the building’s heating, so residents in one building tend to answer alike and the buildings tend to differ: strata by building are alike inside and different from each other. That is what makes stratifying pay off, and he should choose building.
Class year matters little here, since every year is spread through every building; those strata would each look much like the whole population. Last names have no tie to heating at all. Either choice still gives a fair sample, with every resident’s chance the same under proportional allocation, but the estimate would vary about as much as a simple random sample’s, so the work of splitting the list buys nothing.
Boundaries
Every member needs a known stratum. The frame must give each member’s stratum before the draw, and the strata must not overlap and must leave no one out. A member in two strata gets two chances; a member in none gets no chance.
Each stratum is sampled by chance. Fixing the counts is design; choosing the members within a stratum by convenience brings back the bias a random sample removes.
It is not a simple random sample. Under proportional allocation every member has the same chance, but many groups of n are impossible: Davit’s design can never yield 240 first-years. Methods built for simple random samples need adjusting for stratified data, and unequal allocation needs the weights.
It helps only when the strata differ. Strata that are alike on the response give about the same spread as a simple random sample. Stratifying never helps with a poor frame, nonresponse or a loaded question.
Where this goes next
A systematic sample is another way to spread a sample down a list. Sampling a few whole groups, such as entire floors, and asking everyone in them is a different design with a different purpose, the cluster sample. Members the frame misses are taken up in undercoverage and nonresponse. The same idea of grouping alike units first returns in experiments as blocking, where it keeps a known source of variation from hiding a treatment’s effect.
Exercises
Allocation and drawing
- For the 4{,}800 residents in Example 1, find the proportional allocation for a sample of 96, and each resident’s chance.
- Find the proportional allocation for a sample of 100 from the same four class years. Round so that the total stays 100.
- A smaller campus houses 1{,}200 residents in three buildings of 600, 450 and 150. Find the proportional allocation for a sample of 80 and each resident’s chance.
- The 1{,}300 sophomores are labeled 0001 to 1300. From the first digit, read the row 12985 \quad 06403 \quad 77120 \quad 00981 \quad 13004 in blocks of four to choose 4 sophomores.
- A calculator gives random whole numbers from 1 to 1100 to choose 4 juniors. They come out 812, 45, 1100, 812, 377, 690. Which juniors are chosen?
Combining with weights
- Hovhannes takes 50 residents from each class year and asks whether their room is too small. Yes: 20 first-years, 16 sophomores, 12 juniors and 10 seniors. (a) Find the pooled proportion that ignores the weights. (b) Find the weighted estimate for all 4{,}800 residents.
- The 4{,}800 residents are split into 2{,}800 first-years and sophomores and 2{,}000 juniors and seniors, and 40 are sampled from each. The sample means of weekly spending on food outside the dining hall are 18.50 dollars and 41.00 dollars. Estimate the mean for all residents, and compare it with the plain average of the two means.
- In exercise 6, find each class year’s chance of being sampled, and the number of residents each sampled member speaks for.
Why it varies less
- A made-up population has four residents. Stratum A eats 2 and 4 meals a week; stratum B eats 10 and 12. Take samples of 2. (a) List all 6 simple random samples and their means. (b) List all stratified samples of one from each stratum and their means. (c) Compare the ranges.
- Repeat exercise 9 with stratum A eating 2 and 10 meals and stratum B eating 4 and 12. What has changed, and why?
- For each question, say which stratifying variable is better, and why. (a) Weekly hours worked at a campus job: class year, or birth month. (b) Whether the residents’ Wi-Fi is too slow, when each building has its own network: building, or major.
Reasoning
- In Example 1 every resident’s chance is \displaystyle \frac{1}{20}. Explain why Davit’s sample is still not a simple random sample.
- A residents’ list puts student staff in a stratum of their own and also in their class-year stratum. Explain what goes wrong and how to fix it.
- Explain in which direction an equal-allocation sample’s pooled result is pulled when the smallest stratum favors something least, and why proportional allocation needs no weights.
- Explain why stratifying by class year makes Davit’s estimate vary less from sample to sample, if class years really do answer differently.
- Siranush says: “A stratified sample always gives an estimate closer to the truth than a simple random sample.” Explain what is right and what is wrong in this.
Answers
- Rate \displaystyle \frac{96}{4800} = \frac{1}{50}: 30 first-years, 26 sophomores, 22 juniors and 18 seniors, total 96. Each chance is \displaystyle \frac{1}{50} = 2\%.
- Exact shares 31.25, 27.08, 22.92 and 18.75; rounded, 31, 27, 23 and 19, which total 100.
- Rate \displaystyle \frac{80}{1200} = \frac{1}{15}: 40, 30 and 10. Each chance is \displaystyle \frac{1}{15}.
- Sophomores 1298, 0377, 1200 and 0981. Blocks: 1298 pick, 5064 skip (above 1300), 0377 pick, 1200 pick, 0981 pick; 1300 is not needed.
- Juniors 812, 45, 1100 and 377. The second 812 is a repeat and is ignored, and 690 is not needed.
- (a) \displaystyle \frac{58}{200} = 0.29. (b) The proportions are 0.40, 0.32, 0.24 and 0.20, so the weighted parts are 1500(0.40) = 600, 1300(0.32) = 416, 1100(0.24) = 264 and 900(0.20) = 180, with total 600 + 416 + 264 + 180 = 1460, and the estimate is \displaystyle \frac{1460}{4800} \approx 0.304, about 30.4\%.
- 2800(18.50) = 51800 and 2000(41.00) = 82000, so the estimate is \displaystyle \frac{133800}{4800} = 27.875, about 27.88 dollars. The plain average, 29.75 dollars, counts the smaller, higher-spending group as half the residents.
- First-years \displaystyle \frac{50}{1500} = \frac{1}{30}, sophomores \displaystyle \frac{50}{1300} = \frac{1}{26}, juniors \displaystyle \frac{50}{1100} = \frac{1}{22}, seniors \displaystyle \frac{50}{900} = \frac{1}{18}. Each sampled member speaks for 30, 26, 22 and 18 residents.
- (a) \{2, 4\}: 3; \{2, 10\}: 6; \{2, 12\}: 7; \{4, 10\}: 7; \{4, 12\}: 8; \{10, 12\}: 11. (b) 6, 7, 7, 8. (c) Simple random: 3 to 11; stratified: 6 to 8. Both average 7, the population mean.
- Stratified means: \{2, 4\}: 3; \{2, 12\}: 7; \{10, 4\}: 7; \{10, 12\}: 11, a range of 3 to 11, no narrower than a simple random sample’s. In fact half of the 4 stratified means sit at the extremes 3 and 11, against 2 of the 6 simple random means, so here stratifying spreads the estimate slightly more. Each stratum now holds a low and a high value, so the strata are alike, and fixing the mix removes nothing worth removing.
- (a) Class year: hours of work plausibly change as students advance, while birth month has no tie to work. (b) Building: Wi-Fi speed is set by the building’s network, so residents of one building answer alike.
- Some groups of 240 residents can never be chosen, for instance any group with 76 first-years. A simple random sample makes every group of 240 equally likely.
- Staff members are in two strata, so they have two chances, and one could be drawn twice. Put each in exactly one stratum (staff, or class year), then draw.
- The smallest stratum fills more than its share of the sample, so pooling pulls the result toward its answer: since it favors the idea least, the pooled result comes out too low. Under proportional allocation each stratum’s share of the sample equals its share of the population, which are exactly the weights.
- A simple random sample’s mean also moves with how many residents of each year happen to be drawn; a lopsided draw shifts it toward one year’s answer. The stratified sample always has 75, 65, 55 and 45, so only the luck within each year remains.
- Right: when strata differ on the response, its estimates vary less, so they tend to be closer. Wrong: any one stratified sample can still miss, and a lucky simple random sample can beat it; with strata unrelated to the response there is no gain at all.