6 Statistics: Data, Sampling & Presentation · 6.1 Data, sampling, probability and presentation
6.1.2 Sampling methods: random (simple, stratified, systematic, cluster) and non-random; sampling and non-sampling errors
A random (probability) sample gives every unit a known chance of selection (simple, stratified, systematic, cluster); a non-random sample is picked by judgement or convenience; the gap between a sample result and the true value is the sampling error, while mistakes in collecting and processing are non-sampling errors.
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Core facts
- Sampling rests on two laws:
- Law of statistical regularity: a reasonably large sample picked at random tends to have the characteristics of the population.
- Law of inertia of large numbers: large aggregates are more stable than small ones (errors cancel out).
- Random (probability) sampling: each unit has a known, non-zero chance of selection. Types: simple random, stratified, systematic, cluster, multistage.
- Simple random sampling (SRS): every unit has an equal chance; every sample of size n is equally likely.
- Methods: lottery method, random number tables (Tippett, Fisher and Yates, Kendall and Babington Smith, RAND), computer random numbers.
- SRSWR (with replacement) and SRSWOR (without replacement).
- Stratified sampling: divide a heterogeneous population into homogeneous strata, then draw a random sample from each stratum.
- Systematic sampling: pick a random start from the first k units, then every k-th unit; k = N ÷ n (sampling interval).
- Cluster sampling: divide the population into groups (clusters, e.g. villages, blocks); select whole clusters at random and study all units in them.
- Multistage sampling: sampling in stages (districts → villages → households). NSS uses it.
- Non-random (non-probability) sampling: judgement (purposive), quota, convenience, snowball.
- Sampling error: difference between a sample statistic and the population parameter, because only a part was studied. It falls as the sample size rises; it is absent in a census.
- Non-sampling errors: errors in collection, recording, processing (wrong answers, non-response, bias of enumerators, tabulation mistakes). Present in both census and sample, and usually larger in a census.
- Standard error (SE): standard deviation of a statistic over all possible samples; SE of the mean = σ ÷ √n (large population or SRSWR).
Explained
1. Simple random sampling.
- Lottery method: write each unit's number on identical slips, mix well, draw n slips.
- Random number table: number the units 1 to N, read digits from any point in a fixed direction, take numbers within 1 to N, skip repeats (for SRSWOR) and out-of-range numbers.
- Number of possible samples:
- SRSWOR: ᴺCₙ (unordered samples), each with probability 1/ᴺCₙ.
- SRSWR: Nⁿ (ordered samples).
- In SRSWOR the chance that a given unit enters the sample is n/N.
Example 1. A population has 5 units. How many samples of size 2 are possible (a) without replacement (b) with replacement (ordered)? Solution. (a) ⁵C₂ = 10. (b) 5² = 25. Answer: 10 and 25
2. Stratified sampling: allocation.
- Proportional allocation: nᵢ = n × Nᵢ / N (each stratum in proportion to its size).
- Optimum (Neyman) allocation: nᵢ = n × NᵢSᵢ / Σ NᵢSᵢ, where Sᵢ is the stratum's SD (more variable strata get more units).
Example 2. A college has 600 B.Com, 300 B.A. and 100 B.Sc. students. Draw a proportional stratified sample of 50. Solution. N = 1,000. B.Com 50 × 600/1000 = 30; B.A. 15; B.Sc. 5. Answer: 30, 15, 5
3. Systematic sampling.
Example 3. A list has 500 ration cards. A sample of 25 is wanted; the random start chosen from 1 to 20 is 7. Which cards are picked? Solution. k = 500/25 = 20. Cards 7, 27, 47, … up to 7 + 24 × 20 = 487. Answer: 7, 27, 47, …, 487 (25 cards)
- Simple and quick; spreads the sample over the whole list.
- Danger: if the list has a periodic pattern matching k (e.g. every 20th house is a corner house), the sample is badly biased.
- Only the first unit is random, so it is called quasi-random sampling.
4. Cluster versus stratified.
| Point | Stratified | Cluster |
|---|---|---|
| Groups are | internally homogeneous | internally heterogeneous (mini-populations) |
| Sample taken from | every stratum | only selected clusters |
| Units studied | a sample within each stratum | all units in chosen clusters |
| Main gain | precision | low cost, no full list of units needed |
5. Non-random methods.
| Method | How units are picked | Weakness |
|---|---|---|
| Judgement (purposive) | investigator chooses "typical" units | personal bias |
| Quota | each interviewer fills a quota (e.g. 20 women, 30 men) as convenient | bias inside quotas |
| Convenience | whoever is easiest to reach | not representative |
| Snowball | each respondent names others (rare groups) | chain bias |
Non-random samples are cheap and quick, but their error cannot be measured by probability theory.
6. Sampling and non-sampling errors.
- Sampling error comes from studying only a part; it can be reduced by a larger sample and measured (standard error).
- Non-sampling errors sources: faulty definitions or questionnaire, non-response, wrong answers (memory, prestige), enumerator bias, errors in coding, tabulation and printing, incomplete coverage.
- Biased error works in one direction and accumulates with more units (cumulative); unbiased error is random, tends to cancel out (compensating).
Deeper
Standard error in SRSWOR. For a finite population, SE of the mean = (σ/√n) × √((N − n)/(N − 1)). The factor √((N − n)/(N − 1)) is the finite population correction (fpc); it can be ignored when the sampling fraction n/N is small (under about 5%).
Example 4. N = 100, n = 25, population SD σ = 10. Find the SE of the sample mean (SRSWOR). Solution. σ/√n = 10/5 = 2. fpc = √(75/99) ≈ 0.8704. SE ≈ 2 × 0.8704. Answer: about 1.74 (SRSWR would give 2)
Example 5. Population SD 12, sample size 36 (large population). Find the SE of the mean. To halve it, what sample size is needed? Solution. SE = 12/√36 = 2. SE ∝ 1/√n, so halving SE needs 4 times n. Answer: SE 2; n = 144
Example 6. Strata sizes 600, 300, 100 with SDs 5, 10, 30. Allocate a sample of 60 by Neyman allocation. Solution. NᵢSᵢ = 3,000, 3,000, 3,000; total 9,000. Each stratum gets 60 × 3,000/9,000 = 20. Answer: 20, 20, 20 (proportional allocation would give 36, 18, 6)
Example 7. Which statement is correct? (a) A census has larger sampling error than a sample. (b) Non-sampling errors are absent in a sample survey. (c) Sampling error can be reduced by increasing the sample size. (d) Cluster sampling studies a few units from every cluster. Solution. (a) census has no sampling error; (b) false; (d) describes stratified, not cluster. Answer: (c)
Merits of sampling over census: less time and cost, more detailed and intensive study, better-trained staff hence fewer non-sampling errors, the only method for destructive tests or infinite populations, and its error can be measured.
Precision versus bias. A large sample reduces random error but does not remove bias (a badly framed list or a leading question). The 1936 Literary Digest poll (millions of replies from telephone and car owners) wrongly predicted the US election: size cannot cure a biased frame.
Sample size depends on: variability of the population (more variable → bigger sample), precision wanted, method of sampling, cost and time available, and the nature of the study.
Common traps
- ✗ "In stratified sampling, whole strata are selected." ✓ That is cluster sampling; stratified takes a sample from every stratum.
- ✗ "Strata should be heterogeneous inside." ✓ Strata are homogeneous inside; clusters are heterogeneous inside.
- ✗ "Systematic sampling interval k = n/N." ✓ k = N/n.
- ✗ "Sampling error is present in a census." ✓ A census has no sampling error (but non-sampling errors remain).
- ✗ "Doubling the sample size halves the standard error." ✓ SE ∝ 1/√n: you need four times the sample.
- ✗ "Quota sampling is a random method." ✓ Quota, judgement, convenience and snowball are non-random.
- ✗ "A bigger sample removes bias." ✓ It reduces random error only; bias stays.
Quick revision
- Laws: statistical regularity; inertia of large numbers.
- Random = known non-zero chance; SRS = equal chance.
- SRSWOR samples ᴺCₙ; SRSWR samples Nⁿ.
- Stratified: homogeneous strata, sample from each; proportional nᵢ = nNᵢ/N.
- Neyman: nᵢ ∝ NᵢSᵢ.
- Systematic: k = N/n, random start, every k-th unit.
- Cluster: pick whole groups, study all units in them.
- Non-random: judgement, quota, convenience, snowball.
- SE of mean = σ/√n; fpc √((N − n)/(N − 1)) for SRSWOR.
- Sampling error falls with n; non-sampling errors in both, bigger in census.
Practise this topic. Questions on Data collection, census & sampling, with answers explained.
All Data, sampling, probability and presentation notes · APPSC Paper I syllabus · APPSC Paper I Mock 1 with answers · Start free
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