How do I calculate sample size for a research study?

Choose a confidence level (95% is standard), a margin of error (±5% is common), and an expected response proportion (use 50% if unknown for the safest estimate). The calculator applies Cochran's formula n₀ = z²·p·(1−p)/e². If your population is small and known, it also applies the finite-population correction so you don't over-sample.

What sample size do I need for a 95% confidence level and 5% margin of error?

385 respondents, assuming an unknown or very large population and maximum variability (p = 0.5). The working is z = 1.960, so n₀ = 1.960² × 0.5 × 0.5 / 0.05² = 384.16, rounded up to 385. This is the figure most general-population surveys quote.

What is Cochran's sample size formula?

Published by W. G. Cochran in Sampling Techniques (1977), the formula for a proportion is n₀ = z²·p·(1−p)/e², where z is the standard-normal critical value for your confidence level, p is the expected proportion, and e is the margin of error as a decimal. For a finite population N, you then apply n = n₀ / (1 + (n₀−1)/N).

Why is the recommended sample size often 385?

385 is the result for the most common combination — 95% confidence, ±5% margin, p = 0.5, and an unknown/infinite population. Because it assumes maximum variability and ignores population size, it is the safe default for surveys aimed at a large group such as a country or city.

How does population size affect the sample size I need?

For large populations it barely matters — sampling 385 people gives the same precision whether the population is 100,000 or 100 million. The finite-population correction only kicks in when the population is small relative to the sample. For example, at 99% confidence and ±3% margin, an infinite population needs 1,844 but a 2,000-person faculty needs only 960.

What expected proportion (p) should I use if I don't know it?

Use 50% (p = 0.5). The term p·(1−p) is largest at 0.5, which produces the biggest — and therefore safest — sample size. If a pilot study or prior research suggests the true proportion is closer to, say, 20% or 80%, using that value gives a smaller required sample.

Is this sample size valid for a dissertation or thesis?

Yes for simple random sampling of a single proportion, which covers most undergraduate and postgraduate questionnaire studies. Cite Cochran (1977) and state your confidence level, margin of error, and assumed proportion — the calculator gives a ready-to-paste sentence. It does not cover two-group comparisons, stratified or cluster designs, or statistical power for hypothesis tests.

When was this calculator's method last verified?

The formulas and z critical values were last cross-checked against Cochran (1977) and the NIST/SEMATECH e-Handbook on 2026-06-11. The required sample size is also reverse-checked against the margin of error it actually delivers, so rounding never undershoots your target precision.

Research · Statistics

Sample Size Calculator

Work out the minimum number of survey or experiment respondents you need for a chosen confidence level and margin of error. Uses Cochran's formula with the finite-population correction, shows the working, and gives you a sentence to cite. No signup, no ads.

By Induwara Ashinsana— Executive Director, Ryzera TechnologiesUpdated Jun 11, 2026

Calculate your sample sizeCochran's formula

NIST · Cochran 1977

Confidence level

Margin of error

± precision, e.g. 5

Expected proportion

Use 50 if unknown

Population size

Optional — leave blank if very large

Common scenarios

Required sample size

385

Rounded up to whole respondents

Unadjusted (n₀)

385

No population correction needed

z value used

1.960

Two-tailed standard normal

Delivered margin

±4.99%

At this rounded sample size

Your numbers in the formula

n₀ = z² · p · (1 − p) / e²

= 1.96² × 0.5 × 0.50 / 0.05²

= 3.8416 × 0.2500 / 0.0025 = 384.16

Required sample size = ⌈384.16⌉ = 385

Precision trade-off

Margin of error	Sample size needed
±1%	9,604
±2%	2,401
±3%	1,068
±5%	385
±10%	97

For your methodology section

A sample of 385 respondents was required for a 95% confidence level and a ±5.0% margin of error, assuming an expected proportion of 50% (Cochran, 1977).

Method: Cochran (1977), Sampling Techniques, with the finite-population correction; z critical values from the NIST/SEMATECH e-Handbook. Sources linked under the calculator. Assumes simple random sampling.

How it works

The calculator uses the method published by W. G. Cochran in Sampling Techniques (1977) for estimating a single population proportion under simple random sampling. It runs in four steps, each of which is shown back to you with your own numbers substituted in.

Pick the z critical value. Each confidence level maps to a two-tailed standard-normal critical value (NIST/SEMATECH e-Handbook §1.3.6.7.1): 80% → 1.282, 85% → 1.440, 90% → 1.645, 95% → 1.960, 99% → 2.576.
Convert to decimals. The margin of error e and proportion p are divided by 100, so a 5% margin becomes 0.05 and a 50% proportion becomes 0.5.
Compute the unadjusted size. Cochran's formula is n₀ = z² · p · (1 − p) / e². The term p·(1−p) is the variance of a proportion and is largest at p = 0.5, which is why 50% gives the safest (largest) sample when the true proportion is unknown.
Apply the finite-population correction. If you supply a population N, the tool applies n = n₀ / (1 + (n₀ − 1) / N), which lowers the requirement when the population is small relative to n₀. With no population entered, the infinite-population assumption applies and n = n₀.

The result is always rounded up to a whole respondent — you cannot survey a fraction of a person, and rounding down would undershoot your target precision. As an independent check, the tool feeds the rounded-up sample size back through the inverse of the formula, e = z · √(p·(1 − p) / n₀), and reports the margin of error that size actually delivers. That figure is always at or inside your requested margin, which confirms the rounding is on the safe side. The default proportion is 50% precisely because it maximises this safety margin.

Worked examples

95% confidence, ±5%, unknown population

The classic survey default

z = 1.960, p = 0.5, e = 0.05
n₀ = 1.960² × 0.5 × 0.5 / 0.05²
= 3.8416 × 0.25 / 0.0025
= 0.9604 / 0.0025 = 384.16
Rounded up → 385 respondents

99% confidence, ±3%, faculty of 2,000

Finite-population correction in action

z = 2.576, p = 0.5, e = 0.03, N = 2,000
n₀ = 2.576² × 0.25 / 0.03² = 1,843.27
n = 1,843.27 / (1 + (1,843.27 − 1) / 2,000)
= 1,843.27 / 1.92114 = 959.47
Rounded up → 960 respondents

95% confidence, ±10%, unknown population

Coarser precision needs far fewer people

z = 1.960, p = 0.5, e = 0.10
n₀ = 3.8416 × 0.25 / 0.01
= 0.9604 / 0.01 = 96.04
Rounded up → 97 respondents

Frequently asked questions

Sources & references

The formula and z critical values were last cross-checked against these sources on 2026-06-11. This version assumes simple random sampling of a single proportion; it does not cover two-group comparisons, statistical power, stratified or cluster designs, or sample size for estimating a mean.

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