Paper II · 7 Sampling Techniques · 7.1 Data, sampling, probability and presentation
7.1.1 Sampling techniques: census versus sampling, sample size, simple random, stratified (proportional and Neyman) and systematic sampling
A sample survey studies a part of the population chosen by a rule; the size of that part is set by the precision wanted, and the three basic designs are simple random (every sample equally likely), stratified (a random sample from each homogeneous group) and systematic (random start, then every k-th unit).
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Core facts
- Population size N, sample size n, sampling fraction f = n/N.
- Sampling frame: the list (or map) of all units from which the sample is drawn.
- Census: every unit is studied. No sampling error, but costly and slow; non-sampling errors are usually larger.
- Sample size for a mean: n₀ = (z σ / e)², where e is the margin of error. At 95% confidence z = 1.96 (often rounded to 2).
- Sample size for a proportion: n₀ = z² P Q / e², with Q = 1 − P. The largest n comes at P = 0.5.
- Finite population correction (fpc): (N − n)/N = 1 − f when the variance is written with S² (or (N − n)/(N − 1) when it is written with σ²). Ignore it when f is small (under about 5%).
- SRSWOR: number of samples ᴺCₙ; Var(ȳ) = (N − n)/(Nn) × S² = (1 − f) S²/n.
- SRSWR: number of ordered samples Nⁿ; Var(ȳ) = (N − 1)/(Nn) × S² = σ²/n.
- Here S² = Σ(Y − Ȳ)²/(N − 1) is the population mean square, and σ² = Σ(Y − Ȳ)²/N.
- The sample mean ȳ is unbiased for the population mean in SRSWR, SRSWOR, stratified (ȳst) and systematic sampling (when N = nk).
- Stratified: proportional allocation nᵢ = n Nᵢ/N; Neyman (optimum) allocation nᵢ = n NᵢSᵢ / ΣNᵢSᵢ.
- Systematic: interval k = N/n; random start i from 1 to k; units i, i + k, i + 2k, …, i + (n − 1)k.
Explained
1. Census or sample?
| Point | Census | Sample survey |
|---|---|---|
| Coverage | every unit | a selected part |
| Cost, time, staff | high | low |
| Sampling error | none | present, measurable |
| Non-sampling error | usually larger | smaller (trained staff) |
| Best when | small population, full detail needed (population census) | large population, destructive tests, quick results |
A sample is the only choice when testing destroys the unit (life of bulbs, crop-cutting) or the population is infinite.
2. How large a sample? Decide the margin of error e and the confidence level, then:
- Compute n₀ from the formula for a mean or a proportion.
- If N is known and n₀/N is not small, adjust: n = n₀ / (1 + n₀/N).
- Always round up.
Example 1 (mean). Population SD σ = 20. The mean must be estimated within ±4 with 95% confidence. Population N = 500. Solution.
- n₀ = (1.96 × 20/4)² = 9.8² = 96.04, so 97 for a large population.
- With N = 500: n = 96.04/(1 + 96.04/500) ≈ 80.6, so 81.
Answer: 97 (large population); 81 when N = 500
Example 2 (proportion). About 40% of households use LPG. Estimate the proportion within ±5 percentage points with 95% confidence. Solution.
- n₀ = 1.96² × 0.4 × 0.6 / 0.05² = 3.8416 × 0.24 / 0.0025 ≈ 368.8, so 369.
- If P were unknown, use P = 0.5: n₀ = 3.8416 × 0.25/0.0025 ≈ 384.2, so 385.
Answer: 369 (385 with no prior idea of P)
Note: n varies with 1/e². Halving the margin of error makes the sample four times larger.
3. SRSWR and SRSWOR.
Example 3. N = 50, n = 10, population mean square S² = 40. Find Var(ȳ) under SRSWOR and SRSWR. Solution.
- SRSWOR: (1/10 − 1/50) × 40 = 0.08 × 40 = 3.2.
- SRSWR: (49/50) × 40/10 = 3.92.
Answer: 3.2 (SRSWOR) and 3.92 (SRSWR)
So SRSWOR gives the smaller variance, i.e. it is more precise (precision = 1/variance). The ratio Var(SRSWOR)/Var(SRSWR) = (N − n)/(N − 1), which is below 1 whenever n > 1. In SRSWOR a unit cannot appear twice, so each draw adds new information.
4. Stratified sampling: the two allocations.
Example 4. Two strata: N₁ = 400 (S₁ = 6, mean 20) and N₂ = 600 (S₂ = 16, mean 40). Total sample n = 120. Solution.
- Proportional: n₁ = 120 × 400/1000 = 48; n₂ = 72.
- Neyman: N₁S₁ = 2,400 and N₂S₂ = 9,600; total 12,000. n₁ = 120 × 2,400/12,000 = 24; n₂ = 96.
Answer: proportional 48, 72; Neyman 24, 96
The more variable stratum (S₂ = 16) gets more units under Neyman allocation.
5. Systematic sampling.
Example 5. N = 120 shops, sample of 8, random start 4. Solution. k = 120/8 = 15. Shops 4, 19, 34, 49, 64, 79, 94, 109.
Answer: k = 15; shops 4, 19, …, 109
There are only k possible samples, one for each start, and each unit belongs to exactly one of them. So each unit has the same chance 1/k = n/N of selection.
Deeper
1. Variance in stratified sampling. Write Wᵢ = Nᵢ/N. The estimate is ȳst = ΣWᵢȳᵢ (weighted, not the plain mean of all sampled units unless allocation is proportional).
- Proportional: Var(ȳst)prop = (1/n − 1/N) ΣWᵢSᵢ².
- Neyman: Var(ȳst)opt = (ΣWᵢSᵢ)²/n − ΣWᵢSᵢ²/N.
- Ordering (ignoring small terms): Var(opt) ≤ Var(prop) ≤ Var(random).
Example 6. Use the data of Example 4 (N = 1,000, n = 120). Compare the three designs. Solution.
- W₁ = 0.4, W₂ = 0.6. ΣWᵢSᵢ² = 0.4 × 36 + 0.6 × 256 = 168. ΣWᵢSᵢ = 2.4 + 9.6 = 12.
- Overall mean = 0.4 × 20 + 0.6 × 40 = 32. Between-strata part ΣWᵢ(Ȳᵢ − Ȳ)² = 0.4 × 144 + 0.6 × 64 = 96. So S² ≈ 168 + 96 = 264 (large-N approximation).
- SRSWOR: (1/120 − 1/1000) × 264 ≈ 1.936.
- Proportional: (1/120 − 1/1000) × 168 ≈ 1.232.
- Neyman: 12²/120 − 168/1000 = 1.2 − 0.168 = 1.032.
Answer: 1.936 > 1.232 > 1.032
Why stratification gains. Total variation = within strata + between strata. Proportional allocation removes the between-strata part (96 above), because every stratum is represented in its true share. Neyman goes further by removing the part due to differences among the Sᵢ. So:
- Gain of proportional over SRS is large when stratum means differ a lot.
- Gain of Neyman over proportional is large when stratum SDs differ a lot. If all Sᵢ are equal, Neyman = proportional.
2. Optimum allocation with costs. With the linear cost function C = a + Σcᵢnᵢ (a = overhead, cᵢ = cost per unit in stratum i), the optimum is nᵢ ∝ NᵢSᵢ/√cᵢ. Take more units from a stratum that is larger, more variable or cheaper. Neyman allocation is the special case of equal cᵢ.
3. Allocation problems. Neyman can give nᵢ > Nᵢ for a small, very variable stratum. Then take the whole stratum (nᵢ = Nᵢ) and re-allocate the rest. Neyman also needs the Sᵢ in advance (from a pilot or past survey); proportional needs only the Nᵢ.
4. Systematic sampling: precision and periodicity.
- Var(ȳsys) = [(N − 1)/N] S² − [k(n − 1)/N] S²wsy, where S²wsy is the mean square within the systematic samples.
- Systematic is more precise than SRSWOR when S²wsy > S², i.e. when units within each systematic sample are more varied than the population as a whole. A list in rising or falling order (a linear trend) gives this, so systematic sampling then beats SRS.
- Periodicity: if the list repeats a cycle and k equals the cycle (or a multiple), every unit in the sample is alike. Example: daily sales with k = 7 picks the same weekday every time. Precision collapses and the estimate is biased.
- No unbiased estimate of its variance exists from a single systematic sample.
- If N is not a multiple of n, use circular systematic sampling: random start from 1 to N, take every k-th unit going round the list.
5. Comparison of the designs.
| Point | SRS | Stratified | Systematic |
|---|---|---|---|
| Needs | full frame | frame + stratum of each unit | ordered frame |
| Randomness | every draw | within each stratum | first unit only |
| Possible samples | ᴺCₙ (WOR) | product of ᴺⁱCₙᵢ | k |
| Precision | base level | higher (homogeneous strata) | high with trend; poor with periodicity |
| Ease | moderate | needs planning | easiest in the field |
Common traps
- ✗ "Var(ȳ) in SRSWOR = S²/n." ✓ It is (1/n − 1/N) S² = (1 − f)S²/n; S²/n ignores the fpc.
- ✗ "SRSWR is more precise than SRSWOR." ✓ SRSWOR has the smaller variance, so it is more precise.
- ✗ "The fpc is n/N." ✓ The fpc is 1 − f = (N − n)/N; f = n/N is the sampling fraction.
- ✗ "Neyman: nᵢ ∝ NᵢSᵢ²." ✓ nᵢ ∝ NᵢSᵢ (SD, not variance).
- ✗ "Proportional allocation: nᵢ = Nᵢ/n." ✓ nᵢ = n Nᵢ/N (same fraction n/N in every stratum).
- ✗ "For a proportion, P = 0.1 gives the safest sample size." ✓ P = 0.5 maximises PQ and gives the largest n.
- ✗ "Halving the margin of error doubles n." ✓ n ∝ 1/e², so it quadruples n.
- ✗ "A systematic sample has ᴺCₙ possible samples." ✓ It has only k = N/n.
Quick revision
- f = n/N; fpc = 1 − f; ignore fpc when f < 5%.
- Mean: n₀ = (zσ/e)²; proportion: n₀ = z²PQ/e²; worst case P = 0.5.
- Finite N: n = n₀/(1 + n₀/N); always round up.
- SRSWOR: ᴺCₙ samples, Var = (N − n)S²/(Nn).
- SRSWR: Nⁿ samples, Var = (N − 1)S²/(Nn) = σ²/n.
- SRSWOR more precise than SRSWR.
- Proportional nᵢ = nNᵢ/N; Neyman nᵢ = nNᵢSᵢ/ΣNᵢSᵢ; with costs nᵢ ∝ NᵢSᵢ/√cᵢ.
- Var(opt) ≤ Var(prop) ≤ Var(random).
- Systematic: k = N/n, random start 1 to k, k possible samples.
- Periodicity matching k ruins systematic sampling; a linear trend favours it.
Practise this topic. Questions on Census vs sampling & sample size, with answers explained.
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