How to Calculate Sample Size for a Survey (Cochran's Formula, With Example)

Most questionnaire studies (surveys of doctors, nurses, patients or students) estimate a proportion: how many follow a guideline, hold an attitude or have a condition. The sample size for that question comes from Cochran's formula. It needs three choices, and the Methods section must state all three so reviewers can check the number.

The three choices

1. Confidence level: almost always 95% (z = 1.96).
2. Expected proportion (p): from earlier studies or a pilot. If you have no estimate, use 50%, which gives the largest (safest) sample.
3. Margin of error (e): how precise the estimate must be, often ±5 percentage points.

The formula

n0 = z² × p × (1 − p) / e²

If the population is small and known (for example, all 2,000 doctors in a region), apply the finite population correction:

n = n0 / (1 + (n0 − 1) / N)

Then divide by the expected response rate to get the number to invite.

A worked example

You plan to survey radiation oncologists in a region of 2,000 about their use of hypofractionation, with no prior estimate.

1. n0 = 1.96² × 0.5 × 0.5 / 0.05² = 384.2, rounded up to 385.
2. Finite population correction: 384.2 / (1 + 383.2 / 2,000) = 322.4, rounded up to 323.
3. With an expected 80% response rate: 323 / 0.8 = 403.8, so invite 404.

How to report it in the Methods:

The sample size was calculated with Cochran's formula for estimating a proportion, assuming an expected proportion of 50% (no prior estimate was available), a 95% confidence level and a margin of error of 5 percentage points. With a finite population correction for the 2,000 eligible oncologists, the required sample was 323; allowing for 20% non-response, 404 oncologists were invited.

Mistakes reviewers catch

• Not stating the assumptions. A sample size without p, e and the confidence level cannot be checked.
• Using this formula for a comparison. If the aim is to compare groups (for example, experienced vs junior doctors), you need a power calculation for that comparison, which usually needs more participants.
• Ignoring non-response, then falling short of the target.
• Sampling for the whole survey but analysing subgroups. Each subgroup estimate will be much less precise than ±5%.

Let KyroStat do the write-up for you

KyroStat analyses the survey once it is in: response rates, every question's distribution, subgroup comparisons and the tables, with the sample size paragraph checked against the numbers you report.

Frequently asked questions

Why use 50% when I have no estimate? p(1 − p) is largest at 50%, so it gives the largest sample and is never too small.

What if my population is huge or unknown? Skip the finite population correction; n0 applies.

Which reporting guideline covers surveys? STROBE covers cross-sectional studies in general (see the STROBE checklist guide); CHERRIES adds items for online surveys.

How do I describe the respondents? In a Table 1 of baseline characteristics.