What Slovin's Formula Does
Slovin's formula estimates how many people you need to survey when you already know the size of the group you are studying. It takes two inputs — population size and a chosen margin of error — and returns a required sample size. It is widely used in undergraduate and graduate research, market surveys, and quick planning studies because it is easy to explain and defend.
What it does not do matters just as much. Slovin's formula does not take a confidence level as input, does not use an expected proportion, does not model statistical power, and does not account for complex sampling designs. It is a screening-level estimate, not a full study design.
The Formula
n = N / (1 + N × e²)
- N — total population size, a whole number greater than zero.
- e — margin of error as a decimal. A 5% margin is 0.05, not 5.
- n — the required sample size. Round up to the next whole respondent.
As N grows large, n approaches a ceiling determined by e alone. At a 5% margin of error, the required sample caps out around 400 no matter how large the population becomes.
Worked Example
Population N = 1,000 and margin of error e = 5% (0.05):
- e² = 0.0025
- N × e² = 1,000 × 0.0025 = 2.5
- 1 + N × e² = 3.5
- n = 1,000 ÷ 3.5 = 285.714
- Required sample = 286 respondents
That is the number of completed responses needed, not the number of invitations to send. At an 80% response rate, you would invite roughly 358 people to end up with 286 completed responses.
Reverse Slovin: Population and Margin of Error
The formula rearranges in two directions, and both are supported by this calculator.
Solve for population: N = n / (1 − n × e²). This is only valid when 1 − n × e² is greater than zero. If the denominator is zero or negative, the sample and margin of error you entered cannot come from a valid Slovin calculation.
Solve for margin of error: e = √((N − n) / (n × N)). This is the error parameter Slovin's formula is producing for a given population and sample. It is the algebraically implied parameter, not a confidence interval computed from survey data.
Sampling Fraction and Response-Rate Planning
The sampling fraction is n / N, usually reported as a percentage. A sample of 286 from a population of 1,000 gives a sampling fraction of 28.6%. The unsampled population is N − n.
Response-rate planning is separate from Slovin's formula. The formula returns completed responses needed. If your expected response rate is 80%, invitations required = 286 ÷ 0.80 = 357.5, which rounds up to 358. This step does not change the statistical sample — it only estimates how many people you need to contact to end up with that many completed responses.
Comparison Tables
The results panel shows two comparison tables. The margin-of-error table gives the required sample at every level from 1% to 10% for the current population, so you can see how sensitive the result is to your chosen tolerance. The population table gives the required sample across a range of population sizes at the current margin, so you can see how little the answer changes once the population is large. Both tables use the same engine as the main calculation, so they always agree with it.
Assumptions and Limitations
Slovin's formula is often presented as a universal sample-size method. It is not. Five things it does not model:
- Confidence level — you cannot set one explicitly. Under the common assumptions of proportion 0.5 and 95% confidence, the standard sample-size formula simplifies to roughly 1 / e² for large populations, and Slovin's formula is a finite-population variant of that. The 95% figure is a side effect, not a design choice.
- Expected proportion — the formula assumes the worst-case proportion of 0.5, which maximizes the required sample. If you have a reliable prior estimate, a proportion-based method will give a smaller and still valid sample.
- Statistical power — Slovin's formula does not model power. If your study tests a hypothesis or compares groups, a power-based method is required.
- Effect size — no effect size is specified, so the result is not tied to what the study is trying to detect.
- Complex sampling designs — cluster, stratified, and multi-stage designs all require adjustments Slovin's formula does not include.
Slovin vs. Cochran and Yamane
Yamane's formula (1967) is usually written the same way — n = N / (1 + Ne²) — and gives the same number for the same inputs. The difference is that Yamane's is derived assuming 95% confidence and proportion 0.5, making its assumptions explicit, while Slovin's is often presented without derivation.
Cochran's formula takes a confidence level, a population proportion, and a margin of error as inputs, and does not include a finite-population correction by default. It is more flexible when you have a reliable prior estimate of the proportion. Each method answers a different question — pick the one that matches your design, not the one that gives the smallest number.
Common Mistakes
- Entering 5 instead of 0.05 for the margin of error.
- Forgetting to square e — the formula uses e².
- Rounding intermediate values instead of only the final result.
- Treating Slovin's formula as if it takes a confidence level as input.
- Confusing N (population) with n (sample).
- Reporting the required completed responses as if it were the number of invitations to send.
- Using Slovin's formula for a study that needs power, effect size, or a specified proportion.
Reporting the Result
A complete methodology sentence usually states the population, the margin of error, the formula, and the resulting sample size. For example: Using Slovin's formula with a population of 1,000 and a 5% margin of error, the calculated sample size was 285.714, resulting in a required sample of 286 respondents. Add the sampling fraction if useful, and note that Slovin's formula does not model power or proportion. Treating the result as a floor for a well-designed study — not as a ceiling — keeps the reporting honest.