Sample size calculator for a proportion in a finite population

Explanation

When the population size N is known and finite, Cochran's standard formula overestimates the required sample size. The finite population correction factor (FPC) corrects this: the larger the sampling fraction n₀/N, the greater the reduction obtained.

The correction is significant when the uncorrected sample represents more than 5% of the total population (n₀/N > 0.05). If N is very large (for example, > 100,000) and n₀/N < 5%, the correction practically doesn't change the result and you can use the standard formula for a proportion.

Sample size formula

\( n_0 = \frac{Z^2\,p(1-p)}{E^2} \)

\( n = \frac{n_0}{1 + \dfrac{n_0-1}{N}} \)

• n₀: uncorrected sample size (Cochran's formula for infinite population).
• n: final size corrected for the finiteness of the population.
• p: expected proportion (use 0.5 if you don't know it).
• E: maximum absolute margin of error.
• N: known total population size.

Effect of the correction factor

The FPC is \(\sqrt{(N-n)/(N-1)}\). For N = 500 and n₀ = 217, the correction reduces the sample to 145 (a 33% reduction). For N = 10,000 and n₀ = 385, it drops to 369 (less than a 5% reduction). The gain is larger the smaller N is and the larger n₀/N is.

Quick setup

• p: if you don't know the proportion, use 0.5 (conservative scenario).
• N: must be the real, known count of the population. If N is not well defined, use the uncorrected calculator.
• E: 0.05 (±5%) is common in organizational surveys; 0.03 if you need more precision.
• Confidence level: 95% is the standard.

Worked example

The People department of a services company wants to measure the proportion of employees satisfied with the internal training plan. The company has N = 2,000 employees on payroll, an exactly known and closed census, which makes it possible — and worthwhile — to apply the finite population correction.

There is no historical satisfaction data, so the most conservative estimate is adopted: p = 0.5 (maximum variance, which oversizes the sample). The acceptable margin of error is E = 0.05 (±5 percentage points) with a 95% confidence level (z = 1.96).

First, the sample size without correction is calculated using Cochran's formula:

\( n_0 = \dfrac{z^2 \cdot p\,(1-p)}{E^2} = \dfrac{(1.96)^2 \times 0.5 \times 0.5}{(0.05)^2} = \dfrac{3.8416 \times 0.25}{0.0025} = \dfrac{0.9604}{0.0025} = 384.2 \rightarrow 385 \)

Next, the finite population correction factor is applied:

\( n = \dfrac{n_0}{1 + \dfrac{n_0 - 1}{N}} = \dfrac{385}{1 + \dfrac{384}{2000}} = \dfrac{385}{1.192} \approx 322.9 \rightarrow \mathbf{323} \)

The correction reduces the sample from 385 to 323 surveys, a savings of 62 interviews (≈ 16%). The sampling fraction is \( f = 323/2000 \approx 16.2\,\% \), clearly above the 5% threshold above which the correction is recommended.

With the 323 surveys completed by randomly selected employees, it can be stated with 95% confidence that the true proportion of satisfied employees does not deviate by more than ±5 percentage points from the observed sample value. If greater precision is desired — for example, ±3 pp — the corrected sample would rise to approximately 638 employees.

Model assumptions

• Simple random sampling without replacement from a population of known size N.
• N is a fixed and well-defined number before sampling.
• The normal approximation is valid for n·p ≥ 5 and n·(1−p) ≥ 5.

Common uses

• Surveys of company or institution staff.
• Studies of students or patients at a specific center.
• Audits of client, supplier or closed record portfolios.
• Quality control on batches or inventories of known size.

How to interpret the result

The value \(n\) is the minimum sample size corrected for finite population to estimate a proportion with the specified margin of error \(E\) and confidence level. The finite population correction reduces \(n\) relative to the standard formula for infinite populations: if the sampling fraction \(f = n/N < 5\,\%\), the reduction is practically negligible; if \(f > 20\,\%\), the savings are substantial and the correction is necessary to avoid recruiting more individuals than needed. In extreme cases (\(f > 50\,\%\)), consider whether it would be more efficient to conduct a full census.

The proportion \(p\) remains the most uncertain parameter. If you have no prior information, use \(p = 0.5\) (conservative scenario). It is essential that \(N\) represents the actual, verified size of the target population: if you are studying satisfaction among the active customers of a business with 800 active accounts, \(N = 800\). Using an approximate or incorrect \(N\) eliminates the advantage of the FPC. Run a sensitivity analysis varying \(p\) by ±0.10 and \(N\) by ±10% to see how \(n\) changes; when \(N\) is large, the FPC is insensitive to small errors in \(N\), but with a small \(N\) it can be significant.

If you anticipate non-response rates, calculate the number of individuals to contact as \(\lceil n / (1 - \text{non-response rate}) \rceil\), verifying that the total does not exceed \(N\). Once the data has been collected, build the actual confidence interval with the CI calculator for a proportion; if the sampling fraction was significant (\(f > 5\,\%\)), also apply the FPC in the CI calculation to obtain a more precise estimate.

References and further reading

• Wikipedia (en): Finite population correction — derivation and when to apply it.
• Wikipedia (en): Simple random sample — basis of simple random sampling.
• Cochran, W. G. (1977). Sampling Techniques (3rd ed.). Wiley. — chapter 4 on sampling in finite populations.

Frequently asked questions

• When does the correction apply? When n₀/N > 5%. If the fraction is smaller, the correction reduces the sample size by less than 5% and is optional.
• What if N is unknown? Use the standard calculator (without correction). It will overestimate the sample but the result will be valid.
• Can the result be greater than N? No; the correction guarantees n < N. If n₀ ≥ N, you need to census the entire population.
• Is the formula exact? It is the standard normal approximation. For very extreme proportions or very small N, consider exact methods.