Sample Size Calculator

About the Sample Size Calculator

Before running a survey, experiment or study, you need to know how many people or measurements to collect. Too few, and the results are too imprecise to be useful; too many, and time and money are wasted. The sample size you need depends on how precise you want the answer to be (the margin of error), how sure you want to be (the confidence level), how variable the thing you are measuring is, and sometimes the size of the population.

This sample size calculator works for estimating a percentage — the share of customers who are satisfied, voters who support a candidate — or a mean, such as average spending or blood pressure. It applies the finite population correction when you are sampling from a small group, and shows how many people to invite once you allow for non-response. The answer is always rounded up, since you cannot survey a fraction of a person and rounding down would miss the target precision.

How to Use the Sample Size Calculator

Choose For a percentage or For a mean.

For a percentage, enter the margin of error you want, in percentage points, and the expected percentage — use 50% if you have no idea.

For a mean, enter an estimated standard deviation and the margin of error in the same units.

Optionally, enter the population size, and choose the confidence level.

The Formulas

  percentage:  n₀ = z² × p(1 − p) ÷ E²
  mean:        n₀ = (z × σ ÷ E)²

  finite population:  n = n₀ ÷ (1 + (n₀ − 1) ÷ N)

  z = 1.645 (90%),  1.960 (95%),  2.576 (99%)

Step-by-Step Example: A Survey

A ±5 point margin at 95% confidence, no prior estimate.

  n₀ = 1.96² × 0.5 × 0.5 ÷ 0.05²
     = 3.8415 × 0.25 ÷ 0.0025
     = 384.1  → 385

You need 385 completed responses. For a ±3 point margin the answer is 1,068, and for ±1 point it is over 9,600.

Step-by-Step Example: A Small Population

The same survey of a company with 2,000 employees.

  n = 384.1 ÷ (1 + 383.1 ÷ 2,000) = 384.1 ÷ 1.1916 = 322.4 → 323

Because the sample is a noticeable share of the population, 323 responses are enough. If you expect only 30 percent of those invited to reply, invite about 1,077.

Step-by-Step Example: A Mean

Estimating average spending, with SD about 15 and a desired margin of ±2.

  n₀ = (1.96 × 15 ÷ 2)² = 14.70² = 216.1 → 217

You need 217 measurements.

Why Population Size Barely Matters

It surprises many people that a survey of a country of 50 million needs no more respondents than a survey of a city of 500,000. For a ±5 point margin, the country needs 385 responses and the city 384. Precision depends on the number of people asked, not the fraction of the population they represent — just as a spoonful of soup tells you how salty the pot is, whatever the size of the pot. The correction only matters when the sample is more than about 5 percent of the population.

Choosing the Inputs

Margin of error: ±5 points is common for general surveys, ±3 for polls that need to detect modest changes. Halving the margin quadruples the sample.

Confidence level: 95 percent is the convention. Moving to 99 percent increases the sample by about 73 percent.

Expected percentage: if a previous survey found 20 percent, using 20 rather than 50 cuts the sample needed by more than a third, from 385 to 246 for ±5 points.

Standard deviation: for a mean, take it from a pilot study, previous research or the likely range divided by about four.

Allowing for Non-Response and Subgroups

The calculated size is the number of completed responses. Response rates for email and online surveys are often 10 to 30 percent, so the number invited must be much larger. If you plan to report results for subgroups, such as separate regions, each subgroup needs to be large enough on its own, which can multiply the total. And remember that a large sample does not fix a biased one: 10,000 responses from an unrepresentative group can be less accurate than 400 from a well-chosen random sample.

Sample Size for Experiments

Comparing two groups — a new treatment against a placebo, or two versions of a web page — needs a power calculation, which also depends on the smallest difference worth detecting and the chance of detecting it if it exists, usually 80 percent. Those calculations generally need larger samples than a simple survey, because the difference between two estimates is less precise than either estimate alone.

Understanding Your Result

The headline is the number of completed responses or measurements needed.

The what it gives line restates the precision that sample achieves.

The population effect line shows the size with or without the population correction.

The allowing for non-response line suggests how many to invite at a 30 percent response rate.

When Should You Use This Calculator?

Use it to plan customer, employee or market research surveys.

Use it to size pilot studies and quality checks.

Use it to justify a sample size in a research proposal.

Use it for coursework on sampling.

Common Mistakes

Confusing invited and completed responses. Allow for non-response.

Rounding down. Always round up to reach the target margin.

Ignoring subgroups. Each needs enough responses for its own estimate.

Assuming a big sample removes bias. Only random sampling does that.

Using an unrealistically small standard deviation. The sample will be too small.

Planning with the wrong confidence level. Decide on 90, 95 or 99 percent before collecting data, and report the level alongside the results.