What Is Cochran's Formula?
Cochran's formula gives the sample size n₀ needed to estimate a proportion within a chosen margin of error at a chosen confidence level for a very large population. For a proportion it uses n₀ = z² p (1 − p) ÷ e²; for a mean it uses n₀ = z² σ² ÷ e².
If your population is small, the finite population correction reduces the required sample, and the calculator also shows how many people to invite once you allow for non-response. Where you do not know the proportion, p = 0.5 is the conservative choice because it gives the largest n₀.
The sections below give the formulas, define each variable, work through an example and answer common questions such as when to apply the correction and how to choose p.
How to Use the Cochran's Formula Calculator
- Choose whether you are estimating a proportion or a mean.
- Set the confidence level and the margin of error you can accept.
- For a proportion enter an expected p or leave 50%. For a mean enter an estimate of the standard deviation from a pilot or earlier study.
- Enter the population size if it is limited; leave blank for a large population.
- Add a design effect and the response rate you expect to see how many to invite.
Cochran's Formula Formula
Cochran’s formula for proportions
Cochran’s formula for means
Finite population correction
Adjusting for design effect and non-response
Cochran's Formula Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| n₀ | Sample size for a very large population | |
| z | Critical value for the confidence level | 1.96 for 95 % |
| p | Estimated proportion | 0.5 is the safe choice |
| e | Margin of error | As a proportion for categorical data |
| N | Population size | Used in the finite population correction |
Cochran's Formula Worked Example
With the example values loaded in the calculator above, the steps are:
- z for 95% confidence = 1.96.
- n₀ = z²·p(1 − p) ÷ e² = 1.96² × 0.5 × 0.5 ÷ 0.05² = 384.15.
- Finite population correction: n = n₀ ÷ (1 + (n₀ − 1)/N) = 384.15 ÷ (1 + 383.15/2000) = 322.39.
Result: Required sample size (complete responses) = 323; People to invite (allowing for non-response) = 323.
Worked Example: From 385 to 323
For a 5% margin of error at 95% confidence and a 50% proportion, Cochran's formula gives n₀ = 1.96² × 0.5 × 0.5 ÷ 0.05² = 385 for a very large population. Here the population is N = 2,000, so the finite population correction applies: 385 ÷ (1 + 384 ÷ 2,000) comes to 323, which is 16.15% of the population.
These are completed responses, not invitations. If you expect only 60% of the people you invite to respond, divide 323 by 0.6 and invite about 539. Using 50% for the proportion is the cautious choice because it gives the largest sample. If you have a good reason to expect a value far from 50%, enter it and the answer will be smaller.
When does the finite population correction matter?
If your sample would be more than about 5% of the population, sampling without replacement tells you more about the population than the basic formula assumes, so the required sample shrinks. For a survey of 2,000 employees, the basic formula asks for 385; the correction cuts that to about 323. For a population of one million the correction is negligible.
| Population N | Sample size |
|---|---|
| 100 | 80 |
| 500 | 218 |
| 1,000 | 278 |
| 5,000 | 357 |
| 10,000 | 370 |
| 100,000 | 383 |
| 1,000,000 | 384 |
Choosing the expected proportion
The term p(1 − p) is largest at p = 0.5, so using 50% guarantees the margin of error you asked for whatever the true proportion turns out to be. If a reliable earlier study suggests the proportion is far from 50%, plugging that value in reduces the required sample, but if you are wrong you will not achieve the intended precision.
Adjusting for the real world
- Non-response: divide by the expected response rate to get the number of invitations.
- Design effect: cluster sampling, such as surveying whole classrooms, makes estimates less precise than simple random sampling; multiply by the design effect (often 1.5 to 2).
- Subgroup analysis: if you want to report results for subgroups, each subgroup needs enough respondents on its own.
- Stratified samples: allocate the total across strata in proportion to size or by Neyman allocation.
Limits of the formula
Cochran’s formula assumes a probability sample and a roughly normal sampling distribution. It does not fix bias from poor sampling or non-response, and for very small proportions or very small samples exact binomial methods are preferable. Treat the output as the minimum number of valid responses.
Where Cochran's Formula Is Used
Survey sample size
Find the number of responses needed for a chosen margin of error and confidence level.
Small populations
Apply a finite population correction when the population is not very large.
Means as well as proportions
Plan a sample for estimating a mean, using an estimated standard deviation.
Response rate
Adjust the number of invitations for the expected response rate.
Common Cochran's Formula Mistakes to Avoid
- Use 50% for p when unsure.
- State the confidence level and margin of error when reporting survey results.
- Round the sample size up, never down.
- Inflate for expected non-response before you start fielding the survey.
Cochran's Formula Terms and Definitions
- Margin of error
- The maximum expected difference between the sample estimate and the population value at the chosen confidence.
- Design effect
- The factor by which a complex sampling design inflates variance compared with simple random sampling.
Cochran's Formula FAQs
What is Cochran’s formula?
A formula for the sample size needed to estimate a population proportion within a given margin of error: n₀ = z²p(1 − p)/e².
What sample size does Cochran’s formula give for 95% confidence and ±5%?
About 385 for a very large population with p = 50%.
What is the finite population correction?
An adjustment that reduces the required sample when the sample is a large share of a small population.
What value of p should I use?
Use 0.5 when the true proportion is unknown. It gives the largest, safest sample size.
How do I use Cochran’s formula for a mean?
Use n₀ = (zσ/e)² with an estimate of the standard deviation and a margin of error in the same units.
How is this different from the Raosoft or Slovin formula?
Raosoft uses essentially the same maths with a finite population correction. Slovin’s formula n = N/(1 + Ne²) ignores confidence level and is a rougher shortcut.
What is the difference between Cochran's and Slovin's formula?
Slovin's formula, n = N ÷ (1 + N e²), uses only the population size and margin of error. Cochran's formula also uses the confidence level and the expected proportion, so it states its assumptions openly.
Why does a 5 % margin of error at 95 % confidence give 385?
With p = 0.5, n₀ = 1.96² × 0.25 ÷ 0.05² = 384.16, which is rounded up to 385 for a very large population.
When do I apply the finite population correction?
When the sample is a noticeable share of the population. n = n0 ÷ (1 + (n0 − 1) ÷ N). With N = 2,000 the 385 shrinks to 323.
Why use p = 0.5?
It maximises p × (1 − p), so it gives the largest and safest sample size when you do not know the true proportion.
How do I allow for non-response?
Divide the required complete responses by the expected response rate. If only 50% will reply, you need to invite twice as many people.