Understanding your required sample size
The table below shows required sample sizes (unlimited population, p = 0.5) for common confidence levels and margins of error, computed from Cochran's formula.
| Confidence level | Margin of error 5% | Margin of error 3% |
|---|---|---|
| 90% (z = 1.6449) | 271 | 752 |
| 95% (z = 1.96) | 385 | 1068 |
| 99% (z = 2.5758) | 664 | 1844 |
- These figures assume simple random sampling and the conservative p = 0.5. If a reliable prior estimate of the proportion exists, the required sample can be smaller.
- The calculation addresses sampling error only. Non-response bias, poorly worded questions and unrepresentative sampling frames are not fixed by a larger sample.
- Halving the margin of error roughly quadruples the required sample size, because e enters the formula squared.
- For very small populations, the finite population correction can reduce the required sample substantially; in the limit, the sample approaches the whole population (a census).
What is a sample size calculation?
A sample size calculation answers the question: how many people (or items) must be measured so that a sample statistic — typically a proportion, such as the share of customers who are satisfied — estimates the population value to within a stated margin of error, at a stated level of confidence. Sampling more than necessary wastes resources; sampling fewer produces estimates too imprecise to be useful.
The margin of error is the half-width of the confidence interval around the estimate: a 5% margin means the true population proportion is expected to lie within 5 percentage points of the sample estimate. The confidence level (commonly 90%, 95% or 99%) describes how often intervals constructed this way would capture the true value across repeated sampling. Higher confidence or a smaller margin both require a larger sample.
This calculator uses the formula published in William G. Cochran's Sampling Techniques (3rd edition, 1977) with p = 0.5, the value of the population proportion that maximizes the required sample size. Using p = 0.5 is the standard conservative choice when the true proportion is unknown: the resulting sample is sufficient whatever the true proportion turns out to be. When the population is finite and known, the finite population correction reduces the required sample, because sampling a meaningful fraction of a small population yields more information per respondent.
How to use this sample size calculator
- Enter the population size — the total number of people or items you are studying. Enter 0 if the population is effectively unlimited or unknown.
- Choose the confidence level: 90%, 95% or 99%. Survey research most commonly uses 95%.
- Enter the margin of error as a percentage, commonly 5% or 3%. Smaller margins demand substantially larger samples.
- Read the required sample size (with finite population correction applied when a population was entered) and the unadjusted figure for an unlimited population. Plan for extra recruitment to offset expected non-response.
Cochran's formula and the finite population correction
Cochran's formula for estimating a proportion is n0 = z^2 x p(1 - p) / e^2, where z is the standard normal critical value for the confidence level (1.6449 for 90%, 1.96 for 95%, 2.5758 for 99%), p is the assumed population proportion, and e is the margin of error as a decimal. With the conservative choice p = 0.5, the term p(1 - p) takes its maximum value of 0.25.
Worked example at 95% confidence and a 5% margin: n0 = (1.96^2 x 0.25) / 0.05^2 = (3.8416 x 0.25) / 0.0025 = 0.9604 / 0.0025 = 384.16, which rounds up to 385 respondents for an unlimited population.
For a finite population of size N, the correction is n = n0 / (1 + (n0 - 1) / N). Continuing the example with N = 10,000: n = 384.16 / (1 + 383.16 / 10000) = 384.16 / 1.0383 = 369.98, which rounds up to 370 respondents. Sample sizes are always rounded up, since a fraction of a respondent cannot be surveyed.
Common mistakes
- Confusing the margin of error with the confidence level — the margin (e.g. 5%) is the interval half-width; the confidence level (e.g. 95%) is how often such intervals capture the true value.
- Believing the required sample scales with population size: for large populations the required sample barely changes, which is why 385 respondents suffice at 95%/5% whether the population is 100,000 or 100 million.
- Ignoring non-response: the formula gives the number of completed responses needed, so recruitment must exceed it by the expected non-response rate.
- Expecting a bigger sample to fix bias — sample size controls random sampling error, not systematic bias from an unrepresentative sample.
Questions fréquentes
How many respondents do I need for a 95% confidence level and 5% margin of error?
For a large or unlimited population, 385 respondents. Cochran's formula gives n0 = (1.96^2 x 0.25) / 0.05^2 = 384.16, rounded up to 385. If the population is smaller — say 10,000 — the finite population correction reduces this to 370.
What is the margin of error?
The margin of error is the half-width of the confidence interval around a survey estimate. A 5% margin means the true population value is expected to lie within 5 percentage points of the sample estimate, at the stated confidence level. Smaller margins require larger samples: halving the margin roughly quadruples the sample size.
Why does the calculator assume p = 0.5?
The required sample size depends on the unknown population proportion p through the term p(1 - p), which is largest when p = 0.5. Using 0.5 is the standard conservative convention from Cochran (1977): the resulting sample is large enough regardless of what the true proportion turns out to be. If a trustworthy prior estimate of p exists, a smaller sample may suffice.
Does a larger population need a much larger sample?
No — this is a common misconception. The unadjusted sample size does not depend on population size at all; the finite population correction only reduces the sample when the population is small relative to the sample. At 95% confidence and a 5% margin, roughly 385 respondents suffice for any large population, whether 100 thousand or 100 million.
What is the finite population correction?
When the population N is finite, sampling without replacement provides slightly more information per observation, so fewer respondents are needed. The correction is n = n0 / (1 + (n0 - 1) / N), where n0 is the unadjusted sample size. For N = 10,000 at 95%/5%, it reduces the requirement from 385 to 370; for very large N the correction is negligible.
Does this calculation account for non-response?
No. The formula gives the number of completed responses required. If you expect, for example, a 40% response rate, you must invite roughly n / 0.4 people to end up with n completions. Note also that non-response can introduce bias that no increase in sample size corrects.
Références
- Cochran WG. Sampling Techniques, 3rd edition. Wiley, 1977 (sample size for estimating proportions; finite population correction).
- National Institute of Standards and Technology (NIST). NIST/SEMATECH e-Handbook of Statistical Methods, Section 7.2.2: Sample sizes required. nist.gov.
- Kish L. Survey Sampling. Wiley, 1965 (design effects and practical survey sampling).