Sample Size Calculator

Frequently Asked Questions

What is the standard deviation to use for a mean calculation?

Use a prior estimate from historical data, a pilot study, or expert knowledge. In the absence of any estimate, a conservative approach is to set e ≈ σ/2 and accept a wider confidence interval, then refine after collecting some data.

When does the finite population correction matter?

When your sample exceeds about 5% of the total population. Sampling 400 of 1,000 employees gives a meaningful correction (from 385 to ~278); sampling 400 of 1,000,000 citizens gives essentially no correction.

Why does the result need to be rounded up?

You cannot collect a fraction of an observation. Rounding down would give a sample slightly too small to meet the desired precision. Always round up to guarantee the margin of error requirement is met.

Does sample size tell me how many to recruit or how many to analyze?

The formula gives the number of completed, usable observations needed in the final analysis. Adjust upward for expected dropout, nonresponse, or data quality issues. If you expect 20% attrition, recruit n ÷ 0.80 participants.

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Estimates for informational purposes only.

Important Disclaimer: Estimates for informational purposes only.

This calculator provides estimates for informational purposes only. Results are based on assumptions and may not reflect actual outcomes. Consult qualified professionals in relevant fields before making important decisions based on these results.

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How This Calculator Works

This calculator determines the minimum number of observations needed to estimate a population proportion or mean within a specified margin of error at a chosen confidence level. The formulas are:

For a proportion: n = z² × p(1−p) ÷ e²

For a mean: n = z² × σ² ÷ e²

where z is the critical value (1.645 at 90%, 1.96 at 95%, 2.576 at 99%), e is the desired margin of error, and σ is the population standard deviation. If you enter a known population size N, the finite population correction reduces the required sample: n_adj = (n × N) ÷ (n + N − 1). Results are always rounded up to the next whole observation.

Worked Example

A 95% confidence survey with p = 0.5 (most conservative) and e = 0.05 (margin of 5 percentage points): n = 1.96² × 0.5 × 0.5 ÷ 0.05² = 3.8416 × 0.25 ÷ 0.0025 = 384.16, rounded to 385 respondents.

With a known population of N = 10,000, the adjustment gives n_adj = (385 × 10,000) ÷ (385 + 10,000 − 1) = 3,850,000 ÷ 10,384 ≈ 371. The finite correction saves 14 interviews - modest here, but larger when sampling a higher fraction of the population.

How to Enter the Margin of Error

For a proportion, enter e as a decimal matching the scale of p. If p = 0.5 and you want a 5% margin of error, enter e = 0.05 (not 5). The result is the number of observations needed for the interval p̂ ± 0.05 (i.e. ±5 percentage points).

For a mean, enter e in the same units as the standard deviation. If σ = 15 (as in IQ scores) and you want your estimate of the mean to be within 3 IQ points, enter e = 3. The standard deviation for the mean calculation must be estimated from prior data or a pilot study.

Why p = 0.5 Is the Conservative Default

The term p(1−p) is maximized at p = 0.5, which gives the maximum variance 0.25. Using p = 0.5 when the true proportion is unknown guarantees the required sample is large enough regardless of the actual outcome. If you have a reliable prior estimate, using that instead of 0.5 can substantially reduce the required n. For example, p = 0.1 gives p(1−p) = 0.09, requiring only 36% of the n needed for p = 0.5.

Cost of Higher Confidence and Smaller Margins

Sample size scales with z²/e². Moving from 95% to 99% confidence (z from 1.96 to 2.576) increases n by a factor of (2.576/1.96)² ≈ 1.73 - about 73% more observations. Halving the margin from 5% to 2.5% quadruples the required sample (since n ∝ 1/e²).

These relationships explain why national polls settle near n = 1,000: getting from ±3.1% to ±2.2% MoE would cost double the budget while delivering marginal precision improvement for most decisions.

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