Sample Size Determination
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Direct answer
Sample size grows when the difference worth detecting shrinks — that single inverse relationship organises the whole topic. For estimating a proportion, the classic formula is n = Z²pq/d², where Z is set by the confidence level (1.96 for 95%), p is the anticipated prevalence and d the precision or margin of error; for estimating a mean, n = (Z × s/d)², with s the expected standard deviation and d the tolerable error. Comparative studies add power: raising power from 80% to 90% (cutting beta) enlarges the sample, as does a smaller alpha, a more stringent one-tailed boundary, higher variability, and unequal group allocation. Cluster sampling multiplies the result by the design effect, and attrition allowances pad the final number upward. The WHO 30-cluster technique — 30 clusters, 7 children each, 210 subjects — remains the operational shorthand for immunisation coverage surveys and a favourite exam number.
What you must remember
- Proportion formula: n = Z²pq/d²; with p = 50% (q = 50%) and d = 10% at 95% confidence, n = 96 — the most quoted worked result in PSM.
- Mean formula: n = (Z × s/d)²; doubling the precision demanded (halving d) roughly quadruples the required sample.
- Power input: power = 1 − beta, conventionally 80% (Z_beta 0.84); smaller expected effect sizes demand larger samples to keep power intact.
- Alpha input: tighter alpha (0.01 versus 0.05) widens Z and enlarges n; two-tailed tests need larger n than one-tailed.
- Variability: greater SD or a prevalence nearer 50% increases required size — pq is maximal at p = 0.5.
- Design effect: cluster sampling imposes it (commonly taken as 2 in programme surveys); n is multiplied accordingly.
- Attrition and non-response: inflate the calculated number (dividing by the anticipated completion fraction) before recruitment begins.
- WHO 30-cluster survey: 30 clusters chosen by probability proportional to size, 7 children per cluster (210 total) for immunisation coverage — a number to know cold.
A worked calculation from protocol to field
A district wants to estimate full immunisation coverage expected near 50% within ±10% at 95% confidence. The formula returns n = 1.96² × 50 × 50 / 10² = 96 children. Tighten precision to ±5% and the same formula returns 384 — halving the error quadrupled the work, the inverse-square law that answers most "what happens to n" questions. Now field realities arrive: households will be sampled by clusters, so multiply by a design effect of 2 (192 at ±10%), and expect 10% non-response, so recruit about 215. Had the target been comparing coverage between two districts where a 15-percentage-point difference matters, the calculation shifts to the two-sample formula with power 80% and alpha 0.05, and the answer lands well above the estimation figure. This sequence — formula, precision choice, design effect, attrition — is exactly the reasoning an ethics committee reads in the sample-size justification, and its absence, not arithmetic error, is the commonest protocol flaw.
How the exam frames it
The dominant pattern is directional: "if precision is improved from 10% to 5%, the sample size will..." — become four times larger. The same logic covers "prevalence nearer 50%", "power increased to 90%", "alpha changed to 0.01" — all enlarge n. The second pattern is formula recall with plug-ins, nearly always the proportion formula with p = 50%; knowing that 96 answer converts the question into arithmetic. The third links to sampling methods: probability proportional to size within the 30-cluster design, stratification when subgroups must be represented, and systematic sampling with a random start. Expect one conceptual item too — an unnecessarily huge sample is not merely expensive but ethically questionable when it exposes more participants than needed to answer the question, the counterpoint that separates a statistician from a formula-user.
Frequently asked questions
What is the formula for sample size when estimating a proportion?
n = Z²pq/d², where Z reflects the confidence level, p and q the anticipated prevalence and its complement, and d the desired precision (absolute error).
What happens to sample size when the margin of error is halved?
It approximately quadruples, because n varies with the inverse square of d — the single most tested relationship in this chapter.
Why does a prevalence near 50% require the largest sample?
Because the product pq reaches its maximum at p = 0.5, maximising variability and therefore the number of observations needed for a given precision.
What is the design effect in cluster sampling?
The factor by which the sample must be inflated because observations within a cluster are correlated rather than independent; a value of about 2 is commonly assumed in programme surveys.
How does the WHO 30-cluster coverage survey work?
Thirty clusters are selected with probability proportional to size and seven eligible children sampled per cluster (210 total), giving rapid, operationally feasible immunisation coverage estimates.