Sampling Error Calculator

Every sample lies a little — not from dishonesty but from arithmetic: a part, however fairly drawn, is not the whole. Sampling error is the size of that necessary lie, and it is computable before a single interview is conducted. This calculator prices it three ways: the margin of error for a proportion at your sample size and confidence level, the sample size a target margin demands, and the plain standard error underneath both. The pollster’s ±3%, demystified into a square root and a critical value. Bring any survey’s headline claim to this page and it will be priced — generously where the design was honest, and revealingly where it was not.

The Formula and Its Two Levers

For a proportion, the standard error is √(p(1−p)/n) and the margin of error is a critical value times it: MoE = z*·√(p(1−p)/n). Two levers, and only two, move the result. The confidence level enters through z* (1.645 at 90%, 1.96 at 95%, 2.576 at 99% — the hub’s critical-value page computes the family): raising confidence widens the margin honestly, because a promise kept more often must be made more loosely. The sample size enters under the square root, with the diminishing returns that dominate survey economics: the margin halves only when n quadruples. The proportion p itself is the third, quieter input: p(1−p) peaks at p = 0.5, so 0.5 is the conservative planning value — a poll about a lopsided question (p near 0.1) enjoys a margin barely 60% of the headline figure, a refinement the calculator prices when you enter the actual p.

Worked Example: Where ±3% Comes From

The defaults reproduce the most quoted number in polling: p = 0.5, n = 1,067, 95% confidence. SE = √(0.25/1067) ≈ 0.0153, and MoE = 1.96 × 0.0153 ≈ ±3.0 percentage points. The third output prices ambition: to halve that margin to ±1.5 points, the sample-size formula demands n = 1.96² × 0.25 / 0.015² ≈ 4,268 respondents — the quadratic tax in full view, and the reason professional polls cluster near n ≈ 1,000: it is the knee of the curve, where precision is cheapest per respondent. At n = 400 the margin is ±4.9 points; at 10,000, ±0.98. And the same formula, run with the true p of a one-sided race (say 0.75), narrows the honest margin to ±2.6 points at n = 1,067 — the headline ±3% is the maximum, at the dead-heat proportion where uncertainty is greatest.

What the Margin Does Not Cover

The margin of error carries a permanent asterisk, and intellectual honesty requires it in the same font size: it prices sampling error only — the variability of drawing this sample from that frame. The errors that actually sink polls live outside the formula: coverage (the frame misses people — no phones, no panels, no fixed abode), non-response (the reached differ from the refusers, increasingly and unmeasurably), measurement (question wording, ordering, and the shy respondent’s polite lie), and weighting (the corrections applied to the raw sample import their own assumptions). A poll can be a perfect sample of the wrong population answering a leading question, with a margin of error of ±3.0 computed flawlessly. This is why methodologists insist the margin be reported as the sampling margin, and why this calculator’s note repeats the warning under every result: the formula is exact about the one error it owns and silent about the rest — which are, empirically, the larger country.

Means, Differences, and the Finite Population

Three extensions complete the working kit. For a mean rather than a proportion, the standard error is s/√n with the sample’s own spread in the numerator — the t-statistic’s denominator, and the same square-root law. For a difference between two independent proportions (the horse-race gap everyone actually reads), the standard error combines both samples’ — roughly √2 times a single margin at equal n and p near 0.5, which is why a 2-point lead inside a ±3 poll is a coin flip wearing a headline. And when the sample is a substantial fraction of a finite population (a survey of 400 of a company’s 900 employees), the finite population correction √((N − n)/(N − 1)) shrinks the error — at n = N it reaches zero, as it must: a census has no sampling error, only all the other kinds. Each extension is the same law — variability of an average falls as the root of the count — wearing the design’s own clothes.

Design Effects: When the Simple Formula Understates

Real surveys rarely draw simple random samples: they cluster (interviewing whole neighbourhoods for cost), stratify, and weight — and clustering in particular inflates the true error above this page’s figure, because neighbours resemble each other and contribute less independent information than strangers. The inflation is summarised by the design effect: an effective sample size of n/DEFF. At a modest DEFF of 1.5, the defaults’ 1,067 respondents carry the information of about 711 simple-random ones, and the honest margin widens from ±3.0 to 1.96 × √(0.25/711) ≈ ±3.7 points. Stratification, done well, can push the other way (DEFF below 1, a genuine free lunch of design). The calculator prices the simple design that anchors the family; a professional report states its design effect beside its margin, and a reader who knows this paragraph knows to ask for it.

Subgroups Carry Wider Margins

The headline margin describes the whole sample; the numbers readers actually argue about are usually subgroups — young voters, rural respondents, owners of a brand — and every subgroup re-runs this page’s formula with its own, smaller n. A subgroup of 200 inside the defaults’ poll carries 1.96 × √(0.25/200) ≈ ±6.9 points, more than double the headline figure, and the gap between two such subgroups carries a wider margin still. The discipline follows immediately: a subgroup “surge” inside a poll whose subgroup margin is ±7 is weather, not climate — and the calculator, re-run per subgroup, is the instrument that tells the two apart before the headline is written.

Related Tools

More from the Statistics Calculators hub: the Normal Approximation Calculator — the theorem this page spends — the Critical Value Calculator for the z* family, and the live Descriptive Statistics Calculator for the p̂ and s your sample produced.

Frequently Asked Questions

Why does everyone quote ±3%?

Because the industry’s standard design — roughly 1,000 to 1,100 respondents at 95% confidence, reported at the worst-case proportion 0.5 — manufactures it: 1.96 × √(0.25/1067) ≈ 0.030. It is not a law of nature but a budget equilibrium, which is why telephone-era polls of 400 quoted ±5 and why a poll of 2,400 can honestly claim ±2. When a number is quoted without its n and confidence level, it is marketing; this page exists so you can recompute the claim.

Does a bigger population need a bigger sample?

Barely — the great counterintuition of sampling. The formula contains n and p but no population size (until the sampled fraction grows large enough for the finite-population correction): a sample of 1,067 prices a city of 100,000 and a nation of 100 million to the same ±3, because what governs precision is the absolute count of respondents, not their share. A spoonful tastes the soup, however large the pot — provided the soup is stirred, which is the random-sampling assumption doing all the work in that metaphor.

What confidence level should I choose?

Match the convention of your audience: 95% for published social research (the z* = 1.96 standard), 90% for internal business tracking where decisions are reversible, 99% where a false precision claim is expensive. The calculator’s select spans the three because the trade is real — at fixed n, moving 95% → 99% widens the margin by a factor of 2.576/1.96 ≈ 1.31 — and it should be made with eyes open rather than inherited by default.

Why plan with p = 0.5 when I expect 0.3?

Planning is insurance: p(1−p) is maximised at 0.5, so a design sized at 0.5 meets its margin promise whatever the true proportion turns out to be, and if the outcome lands at 0.3 the delivered margin is simply better than promised (±2.75 at the defaults’ n, by the formula). Size at the expected p only when it is known with confidence from prior waves of the same instrument — otherwise the conservative input is the professional one.

Can sampling error explain a poll being “wrong”?

Within its own terms, rarely as often as blamed: a result outside the margin occurs by design 5% of the time at 95% confidence (that is what the level means), so one miss in twenty is the formula working, not failing. Systematic misses beyond that rate indict the unpriced errors — frame, non-response, late swing, likely-voter modelling — which is why post-mortems of famous polling misses almost never conclude that the square root betrayed anyone.

How does weighting affect the margin?

Weighting a sample to match population margins (age, sex, region) improves representativeness while typically increasing the true sampling variance — extreme weights concentrate the effective sample into fewer respondents. Statisticians summarise this with the design effect and an effective sample size smaller than the nominal n. The calculator prices the unweighted design; a heavily weighted poll’s honest margin is wider than its headline, by a factor its methodology statement should disclose.