Sample Size Calculator

Precision is priced by the square

The margin of error appears squared in the denominator, and every surprising thing about survey budgets follows from that.

At 95% confidence with the conservative p = 0.5, the requirement runs: 385 responses for ±5 points, 601 for ±4, 1,068 for ±3, 2,401 for ±2 and 9,604 for ±1.

Look at the last two steps. Going from ±3 to ±2 costs more than double. Going from ±2 to ±1 costs four times again. The final percentage point of precision costs more than everything before it combined.

This is why 1,000 is the standard polling number. It buys about ±3.1 points, which is enough to distinguish a 55% result from a 45% one, and the next meaningful improvement costs several times as much for a difference that rarely changes any decision.

The corollary is more useful than it looks: if your current sample gives a margin you can live with, adding to it is usually poor value. The money is almost always better spent on reducing non-sampling error — a better sampling frame, a higher response rate, more careful question wording — because those are not subject to a square-root law and are frequently the larger source of error anyway.

Why the default proportion is one half

The formula needs p, the proportion you are trying to measure — which is the thing you do not know yet. That looks circular, and the way out is neat.

The term p(1−p) is a downward parabola. It peaks at p = 0.5, where it equals 0.25, and falls away symmetrically: 0.21 at p = 0.3, 0.09 at p = 0.9, 0.0475 at p = 0.95.

So p = 0.5 requires the largest sample. Planning with it means you cannot be caught short: whatever the true proportion turns out to be, you will achieve at least the margin you designed for.

If you do have prior information — a previous wave, a pilot, a known base rate — using it is legitimate and can save a great deal. At p = 0.1 the requirement for ±3 points falls from 1,068 to 385, a two-thirds reduction.

The risk is stated plainly by the tool: if the true proportion comes out closer to half than you assumed, your achieved margin is wider than planned and you cannot go back for more. A sensible compromise is to plan with a value part-way toward 0.5 from your prior estimate, which buys most of the saving and most of the protection.

One case where it genuinely matters: rare events. Estimating a 1% incidence rate to ±0.5 points needs about 1,522 observations — but you must be confident the rate really is near 1%, because using 0.5 there would demand 38,415.

When the population size actually helps

The intuition that a bigger population needs a bigger sample is wrong, and the correction that does exist works in the opposite direction to what people expect.

The finite-population correction reduces the required sample to n·N/(n+N−1). It exists because sampling without replacement from a small closed group gives you more information per observation: at the limit, if you survey everyone, there is no sampling error at all.

The effect is negligible unless the sample is a substantial fraction of the population. For ±3 points you need 1,068 responses, and:

From a population of 300 million the requirement stays 1,068 — the correction changes it by less than one person. From 10,000 it drops to 966. From 2,000 it drops to 697. From 500 it drops to 341, because you are now sampling most of the group.

So: ignore it for anything resembling national or market-wide research, and apply it when surveying a defined closed population — one company’s employees, one school’s pupils, one hospital’s patients, the members of an association. In those cases it is a real and sometimes large saving.

The other sample-size question, which this is not

There are two entirely different questions people mean by “what sample size do I need?”, and conflating them is the commonest planning mistake.

Estimation. “I want to know what proportion supports this, to within ±3 points.” That is what this page answers. Inputs: the margin you will accept, the confidence level, and a guess at the proportion.

Detection. “I want to detect a 5-point difference between two groups, if it exists.” That is a power calculation and it needs different inputs entirely: the smallest effect worth detecting, the significance level, the power you want (conventionally 80% or 90%), and the variability. It also gives a different, usually larger, answer — and it gives a size per group.

The mistake is running an estimation calculation, getting 385, and using it to design a two-group experiment. Detecting a small difference between two groups of 385 is a much weaker proposition than estimating one proportion from 385, and a study sized this way is frequently underpowered from the start.

A related and equally common error: sizing for the whole sample when the analysis will be by subgroup. If you intend to report separately on four regions, each region needs to be large enough on its own. A survey of 1,000 split four ways gives ±6.2 points per region, not ±3.1.

Why the number you calculate is not the number you field

The calculated n is the number of usable responses you need at the end. Fielding exactly that many is a plan to miss.

Non-response. The dominant factor, and often severe: telephone polling response rates are now in the low single digits, and even well-run online panels lose a substantial share. If you expect 30% to respond and need 1,068, you must approach about 3,560.

Ineligibility. Screening questions remove people who do not qualify. If a fifth of contacts fail the screener, inflate accordingly.

Incomplete and unusable responses. Abandoned surveys, straight-lining, failed attention checks. Budget several percent.

Attrition in anything longitudinal. A study with three waves needs to start large enough to still be adequate at the third.

The arithmetic is a division: required contacts = n / (response rate × eligibility rate × completion rate). Those rates multiply, so three modest-looking losses compound quickly.

And a warning that belongs here rather than in the arithmetic: raising the number contacted does not fix bias from non-response. If the people who reply differ systematically from those who do not, a larger sample gives a narrower interval around the same wrong answer.

What the formula assumes about how you sample

The arithmetic assumes simple random sampling: every member of the population equally likely to be selected, and selections independent of one another. Real studies rarely do that, and the departures change the required size.

Cluster sampling — selecting schools then pupils, or districts then households — makes observations within a cluster correlated, so each one carries less independent information. The correction is the design effect, and required sizes are commonly inflated by a factor of 1.5 to 3. Ignoring it is a genuine and frequent error.

Stratified sampling works the other way: dividing the population into homogeneous strata and sampling within each reduces variance, so you can need fewer responses for the same precision.

Weighting to correct demographic imbalance also costs precision. Heavily weighted data has a smaller effective sample size than its raw count — sometimes much smaller — and the margin should be computed from the effective size.

The general rule: this page gives the size for an idealised design. Any real design that is not simple random sampling needs that number adjusted, almost always upward.

Choosing the confidence level, and what it costs

The confidence level enters squared as well, through z², so it is not a free parameter either.

At 90% the multiplier is 1.645; at 95% it is 1.960; at 99% it is 2.576. Because those are squared, the requirements for ±3 points are 752, 1,068 and 1,844 responses respectively.

Moving from 95% to 99% therefore costs about 73% more data for the same margin. That is a real decision, not a formality, and it should be driven by the consequences of being wrong rather than by convention.

95% is the default almost everywhere, and it is worth remembering that it is a convention with no derivation behind it — the same one that gives 0.05 its status as a significance threshold. For most commercial research it is a sensible balance. For a decision with serious safety or financial consequences, 99% is defensible and cheap relative to the downside.

One thing not to do: choose 95% because it is standard, then report a result whose interval barely excludes the value you care about, and treat that as settled. The confidence level should be set before the data exists, alongside the margin.

Sources and methodology

Standards and references for survey planning.

• AAPOR disclosure standards — what a study is expected to report about its design and response rates
• NIST/SEMATECH e-Handbook — sample sizes required for a given margin of error
• Pew Research Center on survey methodology — design effects, weighting and effective sample size
• Margin of Error Calculator — the same formula run forwards
• Confidence Interval Calculator — what the resulting sample will actually give you
• How we calculate — methodology

Method. Sample sizes are computed from the normal multiplier at the chosen confidence level, defaulting to p = 0.5 because it maximises p(1−p) and therefore the requirement — the suite asserts that no other value of p across ninety-nine tested demands more. The result is rounded up, and the pre-rounding figure is shown so the arithmetic is checkable. The finite-population correction is applied only when a population is supplied, and the suite asserts it can never increase the required size. The planning figures quoted in the text — 385, 1,068, 2,401, 9,604 — are asserted rather than transcribed: the ±5 figure is 385 and not the 384 that circulates widely, because the exact multiplier is 1.959964 rather than 1.96, and an earlier draft of this page repeated the folklore value. That engine is verified on every change against 155 assertions. The count and the per-case breakdown are published on the formula verification page.

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