Stratified Sample Size Calculator

nₕ = n × Nₕ ÷ N — every stratum is sampled at the same rate, so the sample mirrors the population.

Results

Enter a total sample size and at least two strata, or load the example.

How to Use the Stratified Sample Size Calculator

1. 1. Choose the total sample size. Decide n first — the Sample Size Calculator works it out from a confidence level and margin of error — and enter it.
2. 2. List the strata. One per line: a name, the number of units in that part of the population, and optionally the standard deviation of your main variable within it. Paste from a spreadsheet; a header row is skipped.
3. 3. Pick the allocation. Proportional mirrors the population, equal gives every stratum the same number, and Neyman puts more units where strata are larger or more variable.
4. 4. Read and export. The table gives each stratum's exact quota, whole-number sample, sampling fraction and weight. Below it, all three allocations sit side by side, with their standard errors when every stratum has an SD.

Proportional, Equal and Neyman Allocation

Quotas are rarely whole numbers, and rounding each one separately can leave you a unit or two above or below n. The calculator uses the largest-remainder (Hamilton) method instead: every stratum first gets the whole part of its quota, then the units still unplaced go one at a time to the largest fractional parts, ties to the larger stratum. If a design asks for more units than a stratum holds — possible under equal or Neyman allocation when one stratum is small — the whole stratum is taken and the remainder is shared among the others.

Worked Example: Surveying Students' Paid Work

Press Load example. A university of 12,000 students wants a sample of 400 to estimate weekly hours of paid work, stratified by faculty, with standard deviations taken from a pilot survey:

The proportional quotas are 78.33, 129.33, 54.67 and 137.67. Their whole parts add up to 398, so the two leftover units go to the largest fractions — Engineering's and Science's, both 0.67 — giving 78 + 129 + 55 + 138 = 400. Every faculty is then sampled at about 3.3%, and each sampled student stands for about 30 others.

Neyman allocation moves 43 extra students into Business, the largest and most variable faculty, taking them from Engineering and Science, and the standard error falls from 0.305 to 0.296 hours. Equal allocation is the least precise overall at 0.332, but it gives 100 students per faculty, which is what you want if the real question is how the faculties differ.

Weights, Standard Errors and Practical Limits

Under anything but proportional allocation, sampled units stand for different numbers of people, so overall estimates must be weighted by wₕ = Nₕ ÷ nₕ, the weight column in the table. An unweighted mean of the Neyman sample above would over-represent Business. The standard errors use Cochran's (1977) formula with the finite population correction, SE² = Σ Wₕ² × Sₕ² ÷ nₕ × (1 − nₕ ÷ Nₕ), and assume simple random sampling within each stratum and SDs close to the truth.

• Neyman is only as good as its SDs. Guessed SDs can make it worse than proportional allocation, which needs none.
• One variable, one optimum. Neyman allocation is optimal for the variable whose SDs you entered; a survey with several key outcomes often settles for proportional allocation.
• Keep at least two units per stratum. With fewer, the stratum's variance cannot be estimated; the calculator warns you.
• Plan for non-response. Divide each nₕ by the response rate you expect in that stratum; the allocation itself assumes everyone selected takes part.

Frequently Asked Questions

How do you calculate the sample size for each stratum?

First choose the total sample size n for the precision you need, then allocate it. Proportional allocation gives each stratum n × Nₕ ÷ N; equal allocation gives n ÷ H; Neyman allocation gives n × NₕSₕ ÷ Σ NₖSₖ. Round so the strata still add up to n — the calculator does this with the largest-remainder method.

What is proportional allocation in stratified sampling?

Each stratum is sampled at the same rate, n ÷ N, so its share of the sample equals its share of the population. A faculty with 20% of the students gets 20% of the sample. Because every unit has the same chance of selection, the sample is self-weighting.

What is Neyman allocation?

Neyman (optimal) allocation sets each stratum's sample in proportion to its size multiplied by its standard deviation, NₕSₕ. For a fixed total n it gives the smallest possible standard error for the overall mean, as Neyman showed in 1934. It needs a reasonable SD estimate for every stratum and assumes the cost per unit is the same everywhere.

When should I use equal allocation?

Use it when the strata themselves are what you want to compare, such as men and women or public and private schools, and the overall figure is secondary. Equal groups give each comparison the same precision, but overall estimates then need weights.

Why don't my rounded stratum sizes add up to n?

Rounding each quota on its own can push the total up or down, for example 54.5 and 45.5 both rounding up to give 101. The largest-remainder method avoids this: take the whole parts, then give the units still unplaced to the strata with the largest fractional parts, which always adds up to n exactly.

What is the difference between stratified sampling and quota sampling?

Both divide the population into groups and fill a number per group. In stratified random sampling the units within each stratum are chosen at random from a sampling frame, so standard errors are meaningful; in quota sampling the interviewer chooses whom to include, which is quicker but open to selection bias.