Calcylator
Sample Size

Sample size for a survey:
confidence, margin of error and the 385 rule

Plan a survey that is big enough to trust. Learn what drives the number of responses, why 385 appears so often, and how to adjust for a small population.

Calcylator Editorial Team

Updated · 4 min read

What sample size is supposed to do

A survey asks a few people to speak for many. The question is how many voices you need before the percentages you report are reliable enough to act on. Too few and the results swing from sample to sample. Too many and you spend time and money on precision nobody needed.

Sample size sits between three choices you make: how sure you want to be, how close you need the result to be to the truth, and how varied you think the answers are. The formula translates those three choices into a count of completed responses.

The formula for a proportion

Sample size (large population) =n = z² × p × (1 − p) ÷ e²
z:
Z-score for the confidence level: 1.645 for 90%, 1.96 for 95%, 2.576 for 99%
p:
Expected share answering yes; use 0.5 if unknown, since it gives the biggest n
e:
Margin of error as a decimal; ±5% is 0.05
Always round up to the next whole response.

The p × (1 − p) term is largest at 0.5, which is why 0.5 is the safe default. If previous studies suggest only 10% will say yes, the term shrinks and so does the sample you need.

A useful way to see the formula is to look at the standard error. The share you observe wobbles from sample to sample by about the square root of p(1 − p) ÷ n. At p = 0.5 and n = 385, that is about 2.55 percentage points, and 1.96 of those is the ±5% margin of error. More responses shrink the wobble, but only with the square root of n, which is why the cost of precision climbs so quickly.

Worked example: ±5% at 95% confidence

  • Confidence

    95%, so z = 1.96

  • Expected share p

    0.5 (unknown)

  • Margin of error e

    0.05

  • z²

    1.96² = 3.8416

  • p(1 − p)

    0.5 × 0.5 = 0.25

  • e²

    0.05² = 0.0025

Required sample

3.8416 × 0.25 ÷ 0.0025 = 384.16, so 385 responses

Rounding down to 384 would fall slightly short of the stated margin.

This is where the 385 you see in many survey guides comes from. It assumes a large population and simple random sampling, and it counts completed, usable responses, not invitations sent.

Choosing confidence and margin sensibly

Ninety-five percent confidence and a ±5% margin are conventions, not laws. They suit most opinion and satisfaction surveys. A quick internal poll that only needs to show which of two options is ahead can use 90% and ±7%. A survey that will inform a costly decision or be published deserves a tighter margin.

Try this test: if the true answer were at either edge of your margin, would you decide differently? If the decision is the same for 45% and 55%, a ±5% margin is enough. If a 3-point swing flips the decision, you need a narrower margin and a bigger sample, or you should accept that the survey cannot settle the question.

Remember also that margin of error applies to each reported percentage. When you break results by gender, age band or city, each group has its own, larger margin, because each group has fewer respondents.

How margin and confidence change the answer

Responses needed for a large population, p = 0.5
Margin of error90% confidence95% confidence99% confidence
±10%6897166
±5%271385664
±3%7521,0681,844
±2%1,6922,4014,148

The relationship is a square: halving the margin of error quadruples the sample. Going from ±5% to ±2.5% takes you from about 385 to about 1,537. Think carefully about whether the extra precision changes any decision.

When the population is small

The formula assumes the population is very large. If you are surveying 2,000 employees, you can sample a smaller fraction because each response removes more uncertainty. Apply a finite population correction.

Adjusted sample size =n₀1 + (n₀ − 1) ÷ N
n₀:
Sample size from the large-population formula
N:
Population size

  • n₀

    384.16

  • Population N

    2,000

  • Denominator

    1 + 383.16 ÷ 2,000 = 1.1916

Adjusted sample

384.16 ÷ 1.1916 = 322.4, so 323 responses

For N below roughly 5% of... no correction is needed once N is well over 20,000.

The correction matters most when the sample would be a large share of the population. At a population of 100,000 the adjustment trims only a couple of responses.

As a rough check, the correction factor is small when the sample is less than about 5% of the population. A survey of 385 people out of 50,000 needs essentially no adjustment, while the same 385 out of 1,000 needs a big one.

Sample size is not the same as responses received

Planned sample size is the number of usable answers you want. Not everyone you ask will reply. If you expect a 30% response rate, send the survey to at least 385 ÷ 0.30 = 1,284 people. Incomplete forms, duplicates and failed attention checks will also trim the usable count.

  • Inflate the number invited by the expected response rate.
  • Plan for subgroups: if you want ±5% for each of four regions, you need roughly 385 in each, not 385 in total.
  • Check that the invited list reflects the population; a big sample from a biased list stays biased.

What the formula cannot fix

The margin of error describes random sampling variation only. It says nothing about wording that leads respondents, people who self-select into answering, or those who cannot be reached. A sample of 385 from an enthusiastic mailing list is still a sample of enthusiasts.

For medical, legal or high-stakes research, a statistician should design the sample, and the formulas change for means, comparisons between groups and clustered designs. This calculation is the right starting point for a simple opinion or satisfaction survey.

Think also about how respondents are chosen. The formula assumes a random sample, where each person in the population has an equal chance of being picked. Convenience samples such as a social media poll do not meet that assumption, so the stated margin of error is optimistic. Report the method honestly next to the result.

Quick planning checklist

  1. Write down the decision the survey supports and the margin that would change it.
  2. Pick the confidence level and compute the sample for a large population.
  3. Apply the finite population correction if the population is below a few thousand.
  4. Divide by the expected response rate to get the number to invite.
  5. Multiply by the number of subgroups you need to report separately.
  6. Check that the invited list actually covers the people you want to describe.

Common questions

What is the formula for sample size?

For a proportion, n = z² × p × (1 − p) ÷ e². With a 95% confidence level (z = 1.96), p = 0.5 and a margin of 0.05, n is 384.16, which you round up to 385 responses.

Why is 385 the usual sample size?

It is the result for 95% confidence, a ±5% margin of error and an unknown proportion of 0.5, in a large population. Different confidence levels or margins give other numbers, and smaller populations need fewer.

How does margin of error affect sample size?

Sample size grows with the inverse square of the margin. Moving from ±5% to ±3% raises the requirement from about 385 to about 1,068 at 95% confidence, so tighter precision becomes expensive quickly.

Do I need a bigger sample for a bigger population?

Only up to a point. Once the population is above roughly 20,000, the required sample barely changes. For small populations, apply the finite population correction and you will need fewer responses than 385.

What value of p should I use?

Use 0.5 when you do not know the likely answer, because it produces the largest and safest sample size. If earlier research suggests a lower or higher share, using that value reduces the required responses.

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