Sample size and precision calculator

What this tool calculates

Three views of the same statistics. Sample size finds the n needed to achieve a target margin of error. Margin of error gives the precision implied by a given n. The custom matrix takes any list of sample sizes and any list of proportions and returns the margin of error for every combination — useful for planning subgroup reporting on an existing study.

All three use the normal approximation to the binomial, appropriate for the sample sizes encountered in commercial and political research (n ≥ ~30 per cell).

Formulas

Z-value Φ⁻¹((1 + CL) / 2) Sample size n = p(1 − p) · Z² / e² With DEFF n_DEFF = n × DEFF With FPC n_FPC = n_DEFF / (1 + (n_DEFF − 1) / N) Margin of error e = Z · √( p(1 − p) · DEFF / n ) With FPC e_FPC = Z · √( p(1 − p) · DEFF / n · (N − n)/(N − 1) )

Choosing inputs

Incidence: use a prior estimate if available; otherwise 50% maximises p(1 − p) and gives a conservative sample. Confidence level: 95% for almost all commercial work; 99% only where false positives carry unusually high cost.

Design effect: 1.0 for unweighted simple random samples; 1.1–1.3 typical for RIM-weighted surveys; higher for clustered or stratified designs with poor allocation. After RIM/rake weighting, compute as n / n_effective, where n_effective = (Σw)² / Σw².

Population size: leave blank for general-population work. Populate only when the universe is small enough for FPC to make a material difference (rule of thumb: n / N above ~5%).

Subgroup analysis

For subgroup reporting, apply the formulas to the subgroup n rather than the total. Worked example: to report each of six segments at ±5% with p = 50%, each subgroup needs about 385 — implying a total sample of about 2,300 if segments are of equal size, more if any segment is under-represented.

References

Cochran, W. G. (1977). Sampling Techniques (3rd ed.). Wiley.

Lohr, S. L. (2019). Sampling: Design and Analysis (2nd ed.). CRC Press.

Kish, L. (1965). Survey Sampling. Wiley.

Latenta Ltd · Internal research tool v1.0