The Formula
Cochran's formula gives the number of completed interviews needed to estimate a proportion within a chosen margin of error: n0 = Z² x p x (1 - p) / e². Use it for a simple random survey, correct it when the population is small, and round once. This page shows how, with three fictional examples. Before you size anything, settle the design with sampling methods, and ask whether a census rather than a sample is cheaper for a small population.
The Formula in Plain Terms
Three choices drive the answer.
| Symbol | Meaning | Common value |
|---|---|---|
| Z | Score for your confidence level | 1.645 (90%), 1.96 (95%), 2.576 (99%) |
| p | Expected proportion with the characteristic | 0.5 when unknown (largest sample) |
| e | Margin of error you accept, as a proportion | 0.05 for plus or minus 5 points |
n0 = Z² x p x (1 - p) / e²
How n0 moves with the margin of error (95% confidence, p = 0.5, large population):
| Margin of error | n0 (full decimals) | Rounded up |
|---|---|---|
| plus or minus 10 points | 96.04 | 97 |
| plus or minus 7 points | 196.00 | 196 |
| plus or minus 5 points | 384.16 | 385 |
| plus or minus 3 points | 1,067.11 | 1,068 |
| plus or minus 2 points | 2,401.00 | 2,401 |
Halving the margin roughly quadruples the sample. Choose the margin for the decision you will make, not from habit.
Worked Example 1: Large Population
A fictional agricultural extension program wants the share of farmers using an improved seed, with 95% confidence and a 5-point margin. Nobody knows the current rate, so p = 0.5.
n0 = 1.96² x 0.5 x 0.5 / 0.05² = 3.8416 x 0.25 / 0.0025 = 384.16, so 385 completed interviews.
If an earlier survey suggests about 20% use it, and you want plus or minus 3 points: n0 = 3.8416 x 0.2 x 0.8 / 0.03² = 0.614656 / 0.0009 = 682.95, so 683. A p far from 0.5 lowers the sample, but only use it when the earlier figure is credible for the same population.
Finite Population Correction
Worked Example 2: Small Population and the Correction
A fictional women's savings network has N = 1,200 members. The large-population answer, 384.16, is almost a third of the whole group. Apply the finite population correction:
n = n0 / (1 + (n0 - 1) / N)
n = 384.16 / (1 + 383.16 / 1,200) = 384.16 / 1.3193 = 291.18, so 292 completed interviews.
Planning for 85% response: 292 / 0.85 = 343.5, so contact 344 members.
The correction matters when the sample would be more than about 5% of the population. For a population of 100,000 it changes almost nothing. It applies to simple random sampling without replacement. If you also apply a design effect for clustering, do not stack the two corrections without stage-specific information; see sample size guidance and design effect.
If the group is this small, also ask whether a census is easier than a sample: see census vs sample.
Rounding Rule
One Rounding Convention
Use the same rule every time, and the same one the uses.
| Step | Do this | Example (N = 10,000, 95%, 5 points, 10% non-response) |
|---|---|---|
| 1. Base n0 | Keep full decimals | 384.16 |
| 2. Finite population correction | Apply to the unrounded n0, keep decimals | 384.16 / (1 + 383.16 / 10,000) = 369.98 |
| 3. Design effect | Multiply the unrounded value (only when it applies) | not used here |
| 4. Completes | Round up once | 370 |
| 5. Contacts | Divide completes by the response rate, round up | 370 / 0.90 = 411.1, so 412 |
Why not round n0 to 385 first? Because 385 / (1 + 384 / 10,000) = 370.76, which rounds to 371. Rounding early inflates the answer by one and makes different analysts disagree. Round up once to completes, once to contacts, and nowhere else.
Precision per Domain
Worked Example 3: Precision for Each District
A fictional nutrition project runs in four districts and wants the share of households with adequate food variety, plus or minus 5 points at 95% confidence, in each district. This is a domain requirement: the precision applies to every reporting subgroup, not only the total.
- Each district needs its own n0 = 384.16, so 385 completes (districts are large, so no correction).
- Four districts: 4 x 385 = 1,540 completes, not 385.
- With 15% non-response, size each district: 385 / 0.85 = 452.9, so 453 contacts; 4 x 453 = 1,812 contacts.
What 385 split across four districts buys: about 96 per district gives a margin of 1.96 x sqrt(0.25 / 96) = 0.100, or plus or minus 10 points per district. Say that plainly to the funder if the budget caps the total: the overall estimate is tight, the district estimates are loose.
Two further points. If the question is whether district A differs from district B, the sample for each must be sized for that difference, which needs more than a single-district margin; use the Detect a Change mode of the . And domains that are clustered need a design effect on top; see the intraclass correlation entry. See also disaggregation.
Limits
Where the Formula Stops Working
- Clustered designs: Multiply by a design effect. Cochran's formula assumes simple random sampling.
- Means instead of proportions: Replace p x (1 - p) with the outcome variance, which you must estimate from a prior survey.
- Comparisons over time or between groups: A margin of error is not the same as power to detect a change.
- Non-response and coverage bias: Inflating the sample by the response rate plans for completes; it does not remove bias.
- Rare outcomes and tiny samples: The normal approximation behind Z is weak when expected counts are very small.
Frequently Asked Questions
Is Cochran's Formula the Same as the Standard Sample Size Formula?
For a proportion, yes. The formula you see as n = Z² p (1 - p) / e² is Cochran's. The finite population correction is a separate second step.
Why Is the Answer 385 and Not 384?
The calculation gives 384.16. You cannot interview a fraction of a person, and rounding down would leave the margin slightly wider than planned, so you round up.
Do I Use p = 0.5 Every Time?
Use 0.5 when you do not know the proportion; it gives the largest sample. Use a different value only when you have a credible estimate for the same population.