To calculate sample size for a survey, you need four numbers: the size of your population, your confidence level, the margin of error you can live with, and the proportion you expect. Plug them into Cochran’s formula, get a target number of completed responses, then divide by your expected response rate to find how many people to invite. Most student surveys land on 385 completed responses for 95% confidence and a 5% margin of error, and on roughly 1,284 invitations once you account for a 30% response rate. The whole thing takes about ten minutes with a calculator.
That last part is where most student projects go wrong. People calculate a target, treat it as a number of people to email, and then end up with a fraction of the responses they planned for. Fix the response rate issue before you field anything, and the calculation becomes genuinely useful.
Table of Contents
- 1What You Need
- 2Step-by-Step: How to Calculate Sample Size for a Survey
- 31. Define the Survey Population
- 42. Choose a Confidence Level
- 53. Set the Margin of Error
- 64. Estimate the Expected Proportion
- 75. Calculate the Initial Sample Size
- 86. Adjust for Nonresponse
- 97. Adjust for a Small Population
- 108. Round Up and Plan Distribution
- 11Common Mistakes
- 12Frequently Asked Questions
- 13What is the formula for determining sample size?
- 14How many responses do I need for a valid dissertation?
- 15Does a larger sample size reduce sampling error?
- 16Is a 7% margin of error acceptable?
- 17What happens when you decrease a sample size?
- 18Is there an app that can calculate sample size?
- 19Conclusion
What You Need
Before you touch a formula, write down four things. Skipping this step is how people end up defending a sample size they cannot explain.
- The population. Everyone your survey is meant to describe. Write down how many of them there are, and be honest about whether that number is exact or a rough estimate.
- The confidence level. How confident you want to be that your result would hold in the full population. 95% is the standard for general research.
- The margin of error. The amount of sampling error you can accept, written as a decimal. 0.05 means plus or minus 5 percentage points.
- The expected response proportion (p). If you have no prior data, use 0.5, which produces the largest and safest sample.
Two optional pieces of information change the answer: whether your population is small enough that you could survey everyone, and what fraction of people you expect to respond.
| Symbol | What it means | Typical value |
|---|---|---|
| Z | Z-score for your confidence level | 1.96 at 95% |
| p | Expected population proportion | 0.5 if unknown |
| e | Margin of error as a decimal | 0.05 |
| N | Total population size | Used only for small populations |
Here are the Z-scores you will need most often, along with the sample each one implies at a 5% margin of error with p = 0.5.
| Confidence level | Z-score | Sample needed at 5% margin of error |
|---|---|---|
| 80% | 1.282 | 165 |
| 90% | 1.645 | 271 |
| 95% | 1.96 | 385 |
| 98% | 2.326 | 540 |
| 99% | 2.576 | 664 |
Step-by-Step: How to Calculate Sample Size for a Survey
The calculation has three layers. First you find the minimum number of completed responses you need. Then you convert that into the number of people you must invite. Then, if your population is small, you check whether the formula is even asking for more people than exist.
1. Define the Survey Population
Your target population is the complete group the survey is meant to represent, not the group you can easily reach. A survey of 200 students at your university describes students at your university; it does not describe all undergraduates in the country.
Get a number for N even if you end up not needing it. If you are surveying all employees at a company with 450 staff, that is a finite population and the arithmetic later changes. If you are estimating opinions among all adults in a country, treat the population as effectively infinite and N drops out of the calculation.
2. Choose a Confidence Level
A 95% confidence level means that if you repeated the survey many times using the same method, about 95% of the results would land within a given distance of the true population value. The remaining 5% of results would miss. That is the plain meaning, and it is the one to write in your methods section.
Use 95% unless you have a reason not to. Going to 99% roughly doubles the sample needed, which usually means more invitations, more cost, or a longer field period. What does not change is the margin of error: confidence level and margin of error are separate controls, and you set both.
3. Set the Margin of Error
The margin of error is the amount of sampling error you are willing to accept in your estimate. At a 5% margin of error, a result of 60% becomes a reported range of 55% to 65%.
Smaller margins demand larger samples, and the effect is steep. Halving the margin of error from 5% to 2.5% quadruples the sample, because the formula divides by e squared. For most student work, 5% is defensible and 10% is acceptable for exploratory or internal work; 3% is worth it when you are making a decision that is expensive to get wrong.
Here is the lookup table for the common combinations, using p = 0.5 and an infinite population.
| Margin of error | 90% confidence | 95% confidence | 99% confidence |
|---|---|---|---|
| 10% | 68 | 97 | 166 |
| 5% | 271 | 385 | 664 |
| 3% | 752 | 1,068 | 1,844 |
| 2% | 1,692 | 2,401 | 4,148 |
If your population is under about 1,500 people, these numbers overstate what you need. Step 7 covers that.
4. Estimate the Expected Proportion
The p in the formula is the proportion of the population you expect to answer a particular question a particular way. If you expect 60% to say yes, p is 0.6.
When you have no prior data, use 0.5. That is not a claim that half your respondents will pick every option; it is the value that produces the largest possible sample, so it protects you if your guess turns out to be wrong. p multiplied by 1 minus p peaks at 0.25 when p is 0.5, and no other value gives a bigger number.
A pilot survey of 15 to 20 people, or a previous study on the same topic, lets you plug in a real estimate and cut the sample size. Be cautious about tightening p based on very little data. A pilot that happens to show 80% rather than 60% will noticeably shrink your target, and pilot samples of that size are noisy.
5. Calculate the Initial Sample Size
Cochran’s formula for a large or unknown population is n = (Z squared x p x (1 minus p)) divided by e squared.
Written out in plain text: n = (Z² × p × (1 − p)) ÷ e². Where Z is the Z-score for your confidence level, p is the expected proportion, and e is the margin of error as a decimal.
Here is the full arithmetic with 95% confidence, p = 0.5, and a 5% margin of error:
- Z = 1.96, so Z squared = 3.8416
- p x (1 minus p) = 0.5 x 0.5 = 0.25
- e = 0.05, so e squared = 0.0025
- n = (3.8416 x 0.25) ÷ 0.0025
- n = 0.9604 ÷ 0.0025
- n = 384.16
Always round up. The minimum number of completed responses is 385.
If you are estimating a mean rather than a proportion, the formula changes to n = (Z² x s²) ÷ e², where s is the standard deviation of the thing you are measuring. You cannot use 0.5 as a stand-in for s; you need a standard deviation from a previous study, published figures for your population, or a pilot survey. If you are stuck without a value, treat the variable as a proportion with a binary outcome and use the formula above.
6. Adjust for Nonresponse
This is the step most guides skip, and it is the one that decides whether you hit your target. Your calculation of 385 is the number of completed responses you need. Nobody answers every invitation, so divide by the response rate you expect.
Formula: invitations needed = completed responses required ÷ expected response rate.
Working the example: you need 385 completed responses. If you expect a 30% response rate, you need 385 ÷ 0.30 = 1,283.3, so 1,284 invitations. If your realistic response rate is 20%, you need 385 ÷ 0.20 = 1,925 invitations. If you have a list of only 900 people, then 900 x 0.30 = 270 expected responses, which falls short of 385, and you need to change something: a wider margin of error, a longer field period, or a different recruitment channel.
Be conservative. Students routinely estimate response rates far above what they get. A general email to staff often lands between 10% and 20%. A short survey to people you have already agreed to participate in can reach 60% or more. Assume the lower number, and release the survey in waves so you can chase non-responders rather than sending one wave and hoping.
7. Adjust for a Small Population
When you know the exact population size and it is small, the infinite-population formula can ask for more respondents than exist. The finite population correction fixes this:
n adjusted = n ÷ [1 + (n − 1) ÷ N]
For a population of 500 employees with n = 385: n adjusted = 385 ÷ [1 + (385 − 1) ÷ 500] = 385 ÷ [1 + 0.768] = 385 ÷ 1.768 = 217.8, so 218 completed responses.
You do not need to sample 218 out of 500 people to get a reliable read. There is simply less randomness to correct for when you are choosing from a small group. As a rule of thumb, the correction matters below roughly 1,500 and becomes negligible above about 10,000.
Below about 400 people, the honest answer is usually to run a census. Survey everyone and you eliminate sampling error entirely, along with non-response bias from people who were never asked. That also removes the need for the margin-of-error argument in your write-up.
8. Round Up and Plan Distribution
Write your final number down along with the assumptions that produced it. A one-line note in your methods section, such as: “A sample of 385 completed responses was targeted, based on 95% confidence, a 5% margin of error, and p = 0.5, inflated to 1,284 invitations assuming a 30% response rate,” will protect you in a viva or an ethics review.
Then plan recruitment around completed responses rather than invitations. Know your channels, your dates, and what happens if you are at 60% of target by the halfway point.
If you fall short, the margin of error gets worse, but it does not collapse. With p = 0.5 at 95% confidence, 150 completed responses gives a margin of error of about 8%, 270 responses gives about 6%, and 385 gives 5%. Recalculate and report the margin of error you actually achieved rather than the one you planned for. A study with 150 responses can still be perfectly useful; it just cannot claim a 5% margin of error.
Common Mistakes
Treating the calculated number as your invitation list. This is the big one. Fix: divide by your expected response rate before you field anything, and write the invitations number down separately from the responses number.
Guessing a round number like 100 and calling it a power calculation. Convenient targets such as 30 or 100 have no statistical basis and are easy to challenge. Fix: run the formula, then say in your methods section which values you used.
Mixing up confidence level and margin of error. They are different controls and both appear in your write-up. Fix: state them as a pair, for example “95% confidence at plus or minus 5%.”
Ignoring the population size when it is small. For a cohort of 100, a 385-response target is impossible. Fix: apply the finite population correction or survey everyone.
Choosing your sample size after seeing the data. Calculating the number needed once you already know the answer is not defensible. Fix: fix the target in your proposal or protocol, in advance.
Confusing invitations with responses in the write-up. Report both. “We invited 1,284 people and received 341 completed responses, giving a margin of error of 5.3%” is honest and holds up. “We surveyed 1,284 people” when only 341 answered is not.
A few threads on r/AskStatistics and r/UniUK make the same point from the other direction: students who lose marks over sample size usually lose them because they cannot justify the number, not because the number itself was small. Show the formula, name your assumptions, and the number becomes defensible on its own.
Three practical tips that cover most remaining cases. First, if your aim is to compare two subgroups rather than estimate a single proportion, the formula above understates what you need. Detecting a 10 percentage-point gap between two groups at 80% power takes roughly 390 completed responses per group, which is a power calculation rather than a margin-of-error calculation, and worth running before you promise the comparison in your aims.
Second, keep the recruitment target and the analysis target separate in your planning notes. Students routinely lose marks here because a methods section reads “384 participants took part” when 384 people were invited and 96 replied.
Third, watch how long the survey runs. A calculation with a 5% margin of error gives you a valid number of responses, not a valid sampling frame. If your completed responses come mostly from one class, one shift, or one signup window, the margin of error describes a group that is not your population.
Frequently Asked Questions
What is the formula for determining sample size?
The standard formula is n = (Z² x p x (1 – p)) / e², from Cochran’s 1977 sampling work. Z is the Z-score for your confidence level, p is the expected proportion (use 0.5 when unknown), and e is the margin of error as a decimal. It gives the minimum number of completed responses needed to estimate a population proportion. Multiply nothing, adjust nothing, and round the result up.
How many responses do I need for a valid dissertation?
There is no single number your department requires, but the usual target for a proportion is 385 completed responses at 95% confidence and a 5% margin of error with p = 0.5. If your committee allows a wider margin, 271 responses gives you 6%, and 167 gives you about 8%. What matters most is that you state the confidence level and margin of error you assumed and report the margin of error you actually achieved.
Does a larger sample size reduce sampling error?
Yes, but with sharp diminishing returns. Sampling error falls roughly in proportion to the square root of n, so doubling your sample cuts the margin of error by about 29%, not by half. Going from 100 to 400 responses helps a great deal. Going from 4,000 to 4,400 costs a lot of effort and buys very little precision. That is why oversizing a survey is usually wasted budget.
Is a 7% margin of error acceptable?
For exploratory, internal, or course-level work, yes, and you can reach it with roughly 196 completed responses at 95% confidence. It would not be enough for a published national estimate, where 5% or tighter is the usual expectation. Decide based on what your conclusion needs to support, then say so in your methods section rather than hoping nobody asks.
What happens when you decrease a sample size?
Your margin of error widens and your estimates get less stable. At 95% confidence with p = 0.5, 385 responses gives 5%, 150 gives about 8%, and 60 gives about 12.6%. Results remain usable, but subgroup or demographic breakdowns get thin quickly, and a small sample makes it easy to miss real differences between groups. If you are short, report the wider margin rather than pretending you hit the target.
Is there an app that can calculate sample size?
Yes, and free web calculators from survey platforms and academic sites will do the arithmetic in seconds. They are fine for checking a number, but most return a result with no explanation, which leaves you unable to defend the choice in a proposal or viva. Learn the formula and its variables first, then use a calculator as a cross-check rather than as the method.
Conclusion
Start by writing down your population size, picking 95% confidence and a 5% margin of error, and setting p to 0.5. That gives you 385 completed responses. Divide by a realistic response rate to get your invitation count, and only then decide how you will recruit that many people.
The number is a defensible starting point rather than a promise. Non-response, a rushed field period, or a narrow convenience sample can all leave you short. Record your assumptions at the start, track the response rate as it comes in, and report the margin of error you actually achieved. That is what turns a calculated number into a methods section your supervisor can sign off on. If your project covers more than one survey, work out how to calculate sample size for a survey once, write the assumptions down, and every later estimate on the project follows the same rule.
