Sample size calculator
How many responses you need for your survey to be representative.
Calculator inputs
Results
Enter your values and press “Calculate” to see the result.
In short
- What it calculates
- How many responses you need for your survey to be representative.
- Formula used
n₀ = z² · p · (1 − p) / e²- Example
- Population of 10,000, 95% confidence, 5% error, 50% proportion:
Sample size calculator
It is the first figure any thesis panel or research client asks for: how many cases are needed for results to be representative.
The answer depends on three decisions: how much confidence you want, how much error you accept and what proportion you expect to find.
How it works
The size for an infinite population is computed first, then corrected for a finite population, which lowers the number when the universe is small.
Using 50% as the expected proportion is the conservative choice: it yields the largest possible sample.
Formula
n₀ = z² · p · (1 − p) / e²
Finite population correction:
n = n₀ / (1 + (n₀ − 1) / N)
z = 1.96 at 95% confidence
Worked example
Population of 10,000, 95% confidence, 5% error, 50% proportion:
n₀ = 1.96² × 0.25 / 0.05² = 384.16
n = 384.16 / (1 + 383.16/10,000) = 370
You need 370 responses.
Explanation
Population size barely matters
It is the most counter-intuitive result in sampling. A country of fifty million and a city of a hundred thousand need practically the same sample. What reduces uncertainty is the amount of information gathered, not the proportion it represents of the total. Below around twenty thousand individuals it is worth applying the finite population correction.
Why fifty per cent is used
It is the most conservative scenario: a proportion of fifty per cent maximises variance and therefore the required size. If you know in advance the result will be around ten or ninety per cent, the sample needed is smaller. When you have no idea, using fifty guarantees you will not fall short.
Quadruple to halve
The margin falls with the square root of the size. Going from a five per cent margin to two and a half requires four times as many interviews, not twice. It is what makes precision studies so expensive so quickly.
What the formula does not cover
It calculates random sampling error and nothing else. It does not correct a badly worded questionnaire, a biased recruitment channel or the problem of non-response. In practice those errors are usually larger than the calculated margin.
Frequently asked questions
Which margin of error should I choose?
5% is the standard in social and market research. Critical decisions drop to 3% or 1%, at the cost of much larger samples.
Why use a 50% proportion?
Because p(1−p) peaks at 0.5, giving the largest and therefore safest sample size when you do not know what to expect.
Does it work for very large populations?
Yes. Enter 0 for the population and the infinite population formula is applied.
Do I need a bigger sample for a huge population?
Practically no. Above around twenty thousand individuals, the required size barely changes however much the population grows.
Why use 50% as the expected proportion?
Because it is the scenario demanding the largest sample. Without a prior estimate, using it guarantees you will not fall short.
Need to calculate something else?
These tools are often used alongside this calculator.
Margin of error
How precise your survey is, given the sample you actually collected.
Standard deviation
Sample and population deviation, variance and coefficient of variation.
Z score
How many standard deviations a value sits from the mean and which percentile it occupies.