Sample size and precision calculator

Inputs

Confidence level

95% is the research default; 99% for high-stakes decisions

Z-value (calculated) 1.960

Expected incidence (p)

Use 0.5 if unknown — yields the most conservative sample

Target margin of error (±)

Enter as a decimal, e.g. 0.03 for ±3 points

Population size (optional)

Enables finite population correction when provided

Design effect (DEFF)

1.0 for SRS; 1.1–1.3 typical for RIM-weighted; higher for clustered

Result

Recommended sample size 1,068

Rounded up. Includes design effect and any finite population correction applied below.

Simple random sample1,067.1

With design effect1,067.1

With finite population correction1,067.1

Sensitivity — required sample size by precision and incidence

Inputs

Confidence level

95% is the research default; 99% for high-stakes decisions

Z-value (calculated) 1.960

Sample size (n)

Achieved or planned sample size

Reference incidence (p)

Use 0.5 for worst-case precision; otherwise the actual proportion

Population size (optional)

Enables finite population correction when provided

Design effect (DEFF)

1.0 for SRS; 1.1–1.3 typical for RIM-weighted; higher for clustered

Result

Achieved margin of error (±) ±3.10%

At the reference incidence, applying any design effect and finite population correction.

Simple random sample±3.10%

With finite population correction±3.10%

Worst-case (p = 50%)±3.10%

Sensitivity — margin of error by sample size and incidence

Shared inputs

Confidence level

Z-value (calculated) 1.960

Population size (optional)

Design effect (DEFF)

Margin of error — by sample size (rows) and proportion (columns)

Edit the sample sizes in the leftmost column and the proportions in the top row. Empty rows or columns display as blank cells. Finite population correction is applied automatically when a population size is provided.

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.