Sample Size Calculator
Responses needed for a target margin and confidence.
What it does
Sample size answers a budget question: how many replies make the result worth trusting. Too few and the margin swallows the finding; too many and money burns for decimal dust. A short formula sets the count before invites go out.
The tool above applies the standard proportion formula with the z for 90, 95, or 99 % confidence, then corrects for small populations. Enter margin in points and the estimated share (50 % when unsure). It rounds every count up.
NIST statistics guidance and Penn State methods courses teach this same form: variance over squared margin, scaled by confidence, then the finite population correction. The 50 % default is deliberate, since even splits vary most and demand the largest sample.
Read the adjusted number as completes, not invites. Nonresponse, bad frames, and biased draws sit outside the math, so pad the send and protect randomness before quoting the margin.
How to use
- Pick 90, 95, or 99 % confidence.
- Enter the margin of error in points.
- Enter the estimated share, or keep 50 %.
- Enter population, or 0 for very large.
- Read the base and the population-adjusted count.
- Invite extra for nonresponse.
Formula
- z
- Normal value for the confidence level.
- E
- Margin of error as a decimal (5 % is 0.05).
- p
- Estimated share; 0.5 is the safe default.
- N
- Population size; 0 means very large.
z is 1.645, 1.96, or 2.576 for 90, 95, 99 % confidence. E is margin as a decimal, p the estimated share. The second term corrects for small populations N; with N at 0 the base stands.
Worked example
Example: 10,000 customers, 95 %, 5 %
Survey of 10,000 customers at 95 % confidence within 5 points.
- 95 % confidence (z 1.96), 5 % margin, 50 % proportion.
- Base: 1.96^2 x 0.25 / 0.05^2 = 384.2, round up to 385.
- Population is 10,000.
- Adjusted: 385 / (1 + 384 / 10000) = 370.8, round up to 371.
- Invite more than 371 to cover nonresponse.
Reference table
| Confidence | 3 % margin | 5 % margin |
|---|---|---|
| 90 % | 752 | 271 |
| 95 % | 1,068 | 385 |
| 99 % | 1,844 | 666 |
Limits
- Random sampling assumed. Bias survives any count.
- Proportion formula; means and rates need their own forms.
- Ignores cost, strata, and clustering effects.
- Nonresponse can dwarf the math.
Common mistakes
- Rounding down to save invites.
- Sizing for the full poll, then slicing tiny groups.
- Using 50 % when the split is known lopsided.
- Forgetting the population correction on small lists.
- Quoting margin on a convenience sample.
Try it yourself
Quick checks. Answers sit tucked under each question.
Sample for 99 %, 3 % margin, open population?
Base is 2.576^2 x 0.25 / 0.03^2 = 1843.3, so 1,844. No correction applies.
Sample for 90 %, 5 %, 30 % share, 2,000 people?
Base is 1.645^2 x 0.21 / 0.05^2 = 227.3, so 228. Adjusted: 228 / 1.1135 = 204.8, so 205.
Questions
Why round up instead of rounding normally?
The z for the confidence: 1.645 at 90 %, 1.96 at 95 %, 2.576 at 99 %. Higher confidence widens the net. The tool always rounds up, since 370.2 people cannot answer.
Why default to 50 % proportion?
Yes, near 50 %. A 50/50 split varies most, so it demands the biggest sample. Known lopsided splits (say 80/20) need fewer responses for the same margin.
When does population size matter?
The finite population correction. Below a few thousand people it trims the count; past 20,000 it barely matters. Enter 0 for open-ended or online audiences.
What does halving the margin cost?
Roughly four times. Margin sits under a square root, so halving it quadruples the sample. That trade drives most survey budgets.
Does a big sample fix a biased poll?
Only with a random draw. Convenience samples and self-selected polls keep their bias no matter the count. Method beats math past a point.
How do I handle nonresponse?
Add 20 to 40 % extra invites for the expected ignore rate, and track who answers versus who was asked. Weight or chase the gaps before trusting slices.