Sample Size Calculator
The planning question, solved backwards: how many observations does a margin of E cost? The algebra runs forward once and the ceiling does the rest — for a proportion and for a mean, with the worst case named.
Sample Size Calculator
Results recalculate instantly on every keystroke. Nothing you type is transmitted.
What this result does not account for
- Large-population answer — no finite-population correction
- One-sample estimation; two-variant comparison needs the A/B page (in build)
In short: To land within ±3 percentage points at 95% confidence, n = z*²p(1−p)/E² = 1.959964² × 0.25 / 0.0009 = 1,067.071895, and since a fraction of a respondent buys no margin, the answer is 1,068 — rounded UP, always up. Planning with p = 0.5 is the safe worst case: 0.5×0.5 = 0.25 is the largest p(1−p) can be, so 897 would do if you knew p̂ sits near 0.3. For a mean with σ = 5 and a ±3-unit target the bill is (1.959964×5/3)² = 10.670719 → 11. Precision is bought in whole observations, priced by the square.
Formula
n = z*² p(1−p) / E² · n = (z* σ / E)²
Solved algebraically from the margin formula, then rounded up: a fraction of an observation buys nothing.
Worked Example
- Type the confidence level and the margin you need.
- Give a planning proportion — 0.5 when unsure.
- Give σ to price the mean flavour on the same E.
- Read n, the raw algebra, and what half the margin would cost.
95%, E = 3 pp, p = 0.5: raw 1,067.071895 → 1,068. At p = 0.3 the same target needs 897. Halve E to 1.5 pp and the bill quadruples to 4,269. Mean flavour, σ = 5, E = 3: 11.
Strengths & Limits Of This Model
Where this engine is strong
- Ceiling applied and shown, not hidden
- Worst-case p = 0.5 priced beside your planning value
Where it stops
- No stratification or clustering design effects
- No power-based sizing for hypothesis tests
Practical Use Cases
Survey design
respondents per ±3 pp of margin
Experiment planning
the n a ±unit promise costs
Budget defence
precision priced before fieldwork, not after
Methodology & Editorial Standards
Computation runs in IEEE-754 double precision at full internal precision; rounding to two decimal places occurs strictly at the display layer, so no cumulative drift enters the result. All monetary outputs use accounting presentation — grouped thousands, two decimals, negatives in parentheses — so figures can be transcribed directly into a model or working paper. Division-by-zero and out-of-domain inputs return an em-dash rather than a misleading number.
This engine was reconciled against an independent reference implementation and hand-verified for the worked example above before release. Our full five-stage review process is published on the About Us page.
Disclaimer. This calculator is provided for informational and modelling purposes only and does not constitute financial, tax, legal, medical, or engineering advice. Verify all figures with a qualified professional before acting on them.
Sample Size Calculator — 8 Expert FAQs
8 analyst-written answers to the questions practitioners actually ask — optimised for voice and answer-engine retrieval.
Why is the answer always rounded UP?
Because the margin is a promise: n = 1,067.07 delivers slightly worse than ±3 pp, and nobody contracts for the 0.07. The ceiling is part of the algebra, not a rounding habit — every unfractional respondent short of the promise leaves the promise broken.
Why plan on p = 0.5 when I expect 30%?
Because the plan must survive the data. p(1−p) peaks at 0.25 when p = 0.5 and falls to 0.21 at p = 0.3, so planning at 0.5 buys insurance against the estimate landing further from your guess than you expected. If the guess is solid, type it — the worst-case card shows what the insurance cost.
Is this the same n an A/B test needs?
No — and the difference matters. This page sizes a one-sample ESTIMATE (how wrong can a single proportion be). Comparing TWO variants against each other is a different computation with a different owner: the A/B test page (in build) carries it, and its n is bigger because detecting a difference is harder than describing one proportion.
Does a big population change the n?
Only when the population is small enough that sampling without replacement matters — the finite-population correction, which shrinks n when the sample is a noticeable slice of the whole. This page quotes the standard large-population answer and says so.
What happens to n at 99% confidence?
z* grows from 1.959964 to 2.575829, and since n carries z*², the bill grows by (2.575829/1.959964)² ≈ 1.728 — seventy-three percent more respondents for the same margin. Confidence is bought with n.
Why does the mean flavour use raw units?
Because a mean lives in the data’s units. The same E = 3 that means three percentage points for a proportion means three whole units for a mean measured in that unit — the card labels which is which so the two are never silently confused.
Why does the margin barely move for every candidate in a poll?
Because the simple-random-sample margin depends only on n and the proportion — not on who is being asked. Real polls with clustering or stratification carry a design effect that widens the honest margin; this page quotes the simple-random answer and names the boundary.
What if the budget only covers 400?
Then the margin is the output, not the input: carry the 400 to the margin-of-error page and quote what it honestly buys. Planning runs both directions — the failure is reporting the margin you wanted beside an n that cannot price it.