Cochran Sample Size Formula and Finite-Population Correction

What Cochran’s formula estimates

Cochran’s large-population formula estimates how many independent observations you need to estimate a population proportion to a chosen absolute margin of error. Write it as n₀ = z² × p × (1 − p) / e². Here z is the normal critical value for the confidence level, p is the anticipated proportion expressed as a decimal, and e is the margin expressed as a decimal. The output is a number of completed, usable observations. It is not the number of people invited, the sample fraction, or a guarantee that the survey avoids bias.

A worked 95% confidence example

Suppose the quantity of interest is the fraction of customers who would select option A. You want an absolute margin of five percentage points and use 95% confidence. With no defensible prior estimate, p = 0.5 gives the largest p(1 − p), namely 0.25. Using z = 1.96 and e = 0.05 gives n₀ = 1.96² × 0.25 / 0.05² = 384.16. Round upward to 385 completed responses. Entering 5 directly into the formula would incorrectly treat the margin as five whole units. The site’s margin field is labelled percent, so enter 5 there.

Correcting for a finite population

If you will sample without replacement from a known finite population N, the common correction is n = n₀ / [1 + (n₀ − 1)/N]. For N = 1,000 and n₀ = 384.16, the unrounded corrected size is about 277.74, which rounds upward to 278. Use the unrounded n₀ in the correction; repeatedly rounding intermediate values adds unnecessary error. As N becomes very large, the denominator approaches one and the corrected size approaches the original 384.16. The population must be the people or units eligible for this particular question, not an unrelated national total.

Confidence, margin and prior information

The margin describes an absolute percentage-point distance. If an observed percentage is 40%, a five-point margin describes approximately 35% to 45%, not a five-percent relative change around 40. Holding other inputs fixed, halving e quadruples the large-population sample requirement because e is squared in the denominator. A stronger confidence requirement increases z and therefore increases the sample size. An evidence-based p away from 0.5 may lower the requirement, but choosing a convenient small p merely to reduce recruitment is not a defensible planning method.

Allowing for nonresponse and study design

For an expected usable response rate of 80%, a target of 385 completed questionnaires requires at least ceil(385/0.8) = 482 invitations under that planning assumption. The calculation does not predict which participants will reply or correct systematic nonresponse. Cluster samples, unequal selection probabilities and weighted estimates can require a design-effect adjustment. Keep the effective sample-size requirement separate from the operational recruitment target. For clustered or complex surveys, use a method that represents the design and obtain statistical advice. A convenient online sample cannot automatically support a probability-sampling margin of error just because its count exceeds 385.

Using the calculator without changing the question

Open the Sample Size Calculator and check its population, confidence and margin labels. Reproduce the large-population example first, then repeat with population 1,000 to inspect the correction. The simplified on-site form does not ask for an anticipated proportion and uses the conservative 0.5 assumption; it does not determine an HRA allowance, rent exemption or salary deduction. It also does not calculate power for a randomized trial, sample size for a mean, or sensitivity and specificity precision for a diagnostic study. Those questions need different inputs and sometimes different formulas.

Reporting the result

Record the outcome variable, target population, sampling design, assumed proportion, confidence level, margin, finite-population adjustment and planned nonresponse allowance alongside the final integer. For the example, a transparent statement is: 385 usable responses for a large population proportion, assuming independent random sampling, p = 0.5, a five-percentage-point margin and z = 1.96. Avoid reporting a decimal person or describing the estimate as an exact universal minimum. If the survey question, eligible population or recruitment method changes, revisit the sample design rather than only changing one number in the form.

Related guides

Sample Size When the Population Size Is Unknown

Plan a proportion sample without a finite population count, distinguish unknown N from unknown prevalence, and test precision and prior assumptions.

How Large Should a Sample Be for Useful Precision?

Compare precision and confidence targets, distinguish estimation from power, and plan subgroup, design and response needs instead of using a fixed rule.

Calculate Sample Size from a Confidence-Interval Target

Turn interval width into a margin, derive sample targets for proportions and means, and explain why a published interval may not reveal a unique count.