Sample Size for a Population of 4000: 95%, ±5 Margin

Sample size for a survey, a mean, an A/B test or a two-group comparison

How many people do you actually need? For a poll with a ±5 point margin at 95% confidence the classic answer is 385, and it barely moves whether the population is 50,000 (382) or 50 million. For an A/B test hoping to lift a 3% conversion rate to 3.6% the answer is 13,914 visitors per variant. This sample size calculator covers four planning questions in one form: a survey percentage with an optional finite population and response rate, the mean of a measurement, two conversion rates, and two means compared with a t-test. The two-means mode uses the exact noncentral t distribution, so it returns 64 per group for an effect of d = 0.5, the figure R and G*Power give, where the plain normal formula says 63.

Four decisions to make before you type anything

1

Which question is the sample for?
Estimating one number (a share, an average) or detecting a difference between two groups. Estimation is planned with a margin of error, comparison with power.

2

How precise, or how small an effect?
For a survey, the margin in percentage points. For a mean, the margin in the units of the measurement. For a comparison, the smallest difference that would change a decision, not the difference you hope to see.

3

How sure?
Confidence 95% is standard for estimates. For comparisons, alpha 0.05 and 80% power are the usual pair; 90% power costs about a third more participants.

4

Who drops out?
Response rates for email surveys are often a fraction of those invited, and trials lose participants along the way. Enter the rate you expect and the calculator returns the number to invite or recruit.

The form, one field at a time

What is the sample for - survey percentage, estimated mean, A/B test on two rates, or two means. Only the fields for that goal appear.
Confidence level (estimates) - 90%, 95% or 99%. It sets the z value: 1.645, 1.960 or 2.576.
Margin of error - for a survey in percentage points (5 means ±5 points), for a mean in the units you measure (±10 minutes, ±2 lb).
Expected percentage - your best guess of the answer. Use 50 when you have none; it gives the largest, safest sample.
Population size and response rate - optional. Leave the population empty when it is large or unknown; fill in the response rate to get the number of invitations.
Standard deviation - for a mean or two means, taken from a pilot, earlier data, or roughly the range divided by 4.
Baseline and target rate, or the difference in means - the effect you want to be able to detect.
Alpha, power, one- or two-sided, dropout - for the two-group goals. Dropout is optional.
Read the result: the headline n, the tiles with the alternative formulas, the table of how n changes with the margin or the power, and the step table.

Normal formula, exact t or continuity correction: which number to trust

Textbooks and online tools disagree by a few percent. Here is where each figure comes from.

Question	Normal (z) formula	The refinement shown beside it	Who should use the refinement
Two means, d = 0.5	63 per group	exact t-test power: 64	anyone who will analyze with a t-test, which is almost everyone
Two means, d = 0.333 (4 points, SD 12)	142 per group	exact t: 143	the gap is always one or two people, larger for small groups
A/B test, 3% to 3.6%	13,914 per group	Fleiss continuity correction: 14,245	teams whose analysis uses R's prop.test with its default correction
Mean, SD 40, margin 10	62 measurements	t-based interval: 64	anyone reporting a t confidence interval, which is standard for means
Verdict	the textbook answer	the one to plan with	plan with the larger number; the difference is cheap insurance

The formulas behind each goal, with three worked cases

    survey: n0 = z² p(1 - p) / e², then n = n0 / (1 + (n0 - 1) / N)

    mean: n0 = (z sigma / e)², then stepped up until t(n - 1) sigma / sqrt(n) fits e

    two rates: n = [z_a sqrt(2 p(1 - p)) + z_b sqrt(p1(1 - p1) + p2(1 - p2))]² / (p1 - p2)²

    two means: smallest n with noncentral t power at least the target, starting from 2 ((z_a + z_b) / d)²

A customer survey. A company with 2,400 customers wants the share who would recommend it within ±4 points at 95% confidence, expects about 30%, and gets roughly 20% of emails answered. Without the population correction 1.96² × 0.3 × 0.7 / 0.04² = 504.19, so 505. With N = 2,400 the sample shrinks to 417, and at a 20% response rate 2,085 invitations go out.

Average time on a task. Earlier tests suggest a standard deviation of 40 seconds and the goal is ±10 seconds at 95%. The z formula gives (1.96 × 40 / 10)² = 61.46, so 62. With 62 people, though, the t-based interval is ±10.1581 seconds; 64 people bring it to ±9.9917.

Two teaching methods. Test scores have an SD of 12 and a 4-point gain would justify switching (d = 0.333). The exact t calculation asks for 143 students per group, 286 in total. Expecting 10% to leave before the final test, enroll 159 per group.

Survey sizes for an expected 50%, by margin and confidence

The worst case p = 50% for an unlimited population. The 95% row at ±5 is the famous 385; halving the margin to about ±2.5 quadruples the sample, because the margin enters the formula squared.

Confidence	±1	±2	±3	±4	±5	±10
90%	6,764	1,691	752	423	271	68
95%	9,604	2,401	1,068	601	385	97
99%	16,588	4,147	1,844	1,037	664	166

A finite population matters only when the sample is a noticeable slice of it. At ±5 points and 95% the answer is 218 for 500 people, 323 for 2,000 and 382 for 50,000.

Where sample size plans usually go wrong

Mistake: powering the study for the effect you hope for, such as a 30% lift, because the resulting sample looks affordable.
Instead: plan for the smallest effect that would still change a decision. If only a 30% lift is worth shipping, fine; if 10% would also matter, plan for 10%.

Mistake: entering the margin of error as a proportion of the estimate (5% of 30% = 1.5 points) when the formula expects percentage points.
Instead: the survey mode takes absolute points: 5 means the interval runs from 25% to 35% around an estimate of 30%.

Mistake: reusing the survey figure of 385 for a comparison between two groups.
Instead: comparing two percentages is a different question and needs the A/B mode; detecting 50% against 55% takes 1,565 per group at 80% power.

Mistake: ignoring nonresponse and dropout.
Instead: 385 completed surveys at a 10% response rate means 3,850 invitations. Fill in the response rate or the dropout field.

Mistake: checking the results every day and stopping the first time p dips below 0.05.
Instead: fix n in advance with this calculator and analyze once, or use a sequential design built for repeated looks.

Sample size questions with numbers in the answers

Why is 385 the magic number for surveys?

It is z² × 0.5 × 0.5 / 0.05² with z = 1.95996, which gives 384.15, rounded up. That covers a ±5 point margin at 95% confidence for any share, because 50% is the worst case. With a finite population of a few thousand the number drops, for 2,000 people to 323.

How big should my sample be for a t-test?

For two groups, two-sided, alpha 0.05 and 80% power: 394 per group for a small effect (d = 0.2), 64 for a medium one (d = 0.5) and 26 for a large one (d = 0.8). The two-sample t-test calculator runs the test once the data are in.

Is a sample of 30 enough?

Thirty is a rule of thumb for when the sampling distribution of a mean looks roughly normal, not a sample size rule. Whether 30 is enough depends on the margin or the effect you need; for a medium effect between two groups it is about half of what is required.

Does a bigger population need a bigger sample?

Hardly. At ±5 points and 95% confidence a town of 50,000 needs 382 and a country of 50 million needs 385. The population only matters when the sample would be more than about 5% of it.

What should I enter as the standard deviation if I have no data?

Use a pilot of 10 to 20 measurements, a published study of the same outcome, or the expected range divided by 4. Overestimating the SD costs extra participants; underestimating it leaves the study underpowered, so round up when unsure.

How do I check the result after the test has run?

For rates, put the observed counts into the z-test calculator, which also shows the sample that would have been needed for the gap you saw. For a table with more than two groups, use the chi-square calculator.

Related tools

Z-Test Calculator

Tests an A/B result on two conversion rates, with the continuity-corrected p and the traffic that gap would need - See calculator

Chi-Square Calculator

Tests independence in a table of counts or the fit of one list to expected shares - See calculator

Two-Sample T-Test Calculator

Compares two group means with Welch, Student or a paired test - See calculator

One-Sample T-Test Calculator

Tests one mean against a target and reports the n needed for 80% power - See calculator

Coefficient of Variation Calculator

Shows spread relative to the mean, useful for guessing a standard deviation from earlier data - See calculator

What sample size do you need for a population of 4000?

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