Survey sample size calculation: formula, examples and planning worksheet

For a simple random sample estimating a population proportion, start with the confidence level, margin of error and expected proportion. At 95% confidence, ±5 percentage points and the conservative assumption p = 0.50, the initial target is 385 completed responses for a large population. Apply a finite population correction when the population is small, then separately inflate the completed-response target for expected nonresponse.

What is survey sample size?

Survey sample size is the number of units whose usable responses contribute to an estimate. The required size depends on the statistic, precision, confidence level and sampling design. This worksheet covers one common case: estimating a proportion from a probability-style simple random sample.

What formula calculates sample size for a proportion?

For a large population, use n₀ = z² × p × (1 − p) ÷ e². Here, z is the critical value for the chosen confidence level, p is the expected proportion and e is the desired half-width of the confidence interval expressed as a decimal.

Using p = 0.50 is conservative because p × (1 − p) is largest at 0.25. If a reliable prior estimate supports another value, record the evidence and run sensitivity calculations rather than choosing a value to make the target smaller.

Worked example for a large population

With z = 1.96, p = 0.50 and e = 0.05:

n₀ = 1.96² × 0.50 × 0.50 ÷ 0.05² = 384.16.

Round up, not to the nearest whole number. The target is 385 usable completed responses. Rounding down would leave the planned precision just outside the stated target.

How does the finite population correction work?

If sampling without replacement from a known finite population, adjust the initial result with n = n₀ ÷ (1 + (n₀ − 1) ÷ N). The United Nations survey-sampling handbook describes the finite multiplier and also emphasizes that real sample design must account for design effects and nonresponse.

For N = 10,000 and n₀ = 384.16:

n = 384.16 ÷ (1 + 383.16 ÷ 10,000) = 369.98, so the completed-response target is 370.

Response-rate planning: how many invitations are needed?

Calculate invitations separately: invitations = completed-response target ÷ expected usable response proportion, rounded up. If the finite-population target is 370 and a defensible planning assumption is a 60% usable response proportion, plan 370 ÷ 0.60 = 616.67, or at least 617 invitations.

This is operational planning, not a promise. Use experience from the same audience, mode and contact process, and run scenarios. Do not substitute an arbitrary web benchmark. After fieldwork, report the actual response rate with a stated denominator and disposition rules.

Reusable planning worksheet

When this formula is not enough

• Clustered or multistage samples: inflate for a defensible design effect and use survey-design analysis.
• Subgroup estimates: size each important analysis domain, not only the overall sample.
• Rare outcomes: p = 0.50 may be conservative for absolute precision but still yield too few positive cases.
• Comparisons and experiments: use a power calculation based on the minimum difference that matters.
• Open-link or convenience surveys: a large response count does not create a probability sample or justify a classical margin-of-error claim.

Limitations

The calculation describes sampling variability under its assumptions. It does not measure frame coverage, nonresponse bias, measurement error, fraud or data quality. The normal approximation can also be unsuitable for small samples or proportions near zero or one. A survey statistician should review consequential, regulated or complex designs.