Probability sampling: how to calculate sample size step by step
What probability sampling is, its types, and a calculator that takes you step by step to the number of cases you need.
- With the formula in view
- Free, no sign-up
- Calculated in your browser
Calculate your sample, step by step
Change any box and the formula and the result update on their own. The formula is the one for simple random sampling to estimate a proportion.
Step 1
Population (N)
How many cases there are in total in the group you want to study. It is optional: if you leave it empty, the calculation is for an infinite population.
Step 2
Confidence level (z)
How sure you want to be that the real result falls within the margin of error. The most common is 95%.
Step 3
Margin of error (e)
How far you accept the sample result to be from the population's. You can also do it the other way around: enter the sample you already have and the margin of error is calculated.
Step 4
Expected proportion (p)
What proportion you expect to find. If you don't know, leave 50%: it is the most conservative case, the one that gives the largest sample.
q = 1 − p =
Step 5
Result
This is the number of valid responses you need, not of invitations: if not everyone responds, you will have to invite more people.
Step 6
Proportional stratification (optional)
If you want each group to weigh the same in the sample as in the population, enter the strata and their size. The sample is allocated with rounding that keeps the total. With percentages, you don't need step 1.
| Stratum | Size | % | Sample |
|---|---|---|---|
| Total |
Your calculation is ready
A paragraph ready for the methodology of your thesis or report, with the values you chose.
What probability sampling is
In probability sampling, every case in the population has a known, greater-than-zero probability of entering the sample, and chance makes the choice. That is why the margin of error can be calculated and the results generalized to the whole population.
Types of probability sampling
Simple random
All cases have the same probability. Example: from a list of 5,000 students, you draw 300 with a random number generator.
Systematic
You pick one case at a fixed interval along the list, from a random start. Example: from 5,000 students, you take every 16th starting at number 7.
Stratified
You divide the population into groups and draw within each one. Example: you draw by degree program, in proportion to the size of each. Step 6 does this allocation.
Cluster
You draw whole groups instead of people. Example: you draw 20 schools and survey all their students. It usually needs a larger sample than this calculator gives.
It depends on the confidence, the margin of error, and the population. With 95% confidence, a 5% margin of error, and p = 0.5, you need 385 cases if the population is very large and 278 if it is 1,000.
Define the population, the confidence level, the margin of error, and the expected proportion, and apply them to the formula. This calculator does it step by step and shows you the formula with your values.
The sample size is the number of valid responses you need. Divide it by the proportion of people you expect to respond: if you need 385 responses and expect 40% to respond, you will have to invite 963 people.
The most common is 5%. A smaller margin gives more precision but asks for many more cases: with 95% confidence and a very large population, 5% asks for 385 cases, 3% for 1,068, and 10% for 97.
Leave step 1 empty and the calculation is for an infinite population. The result is the most demanding case: if the real population is small, the sample you need would be smaller.