Margin of Error Calculator
Poll error band and interval from percent and sample size.
What it does
Poll leads mean little without their wobble: 52 to 48 at plus or minus 3 is a tie wearing a headline. Margin of error hangs the honest band around every sample percent.
The tool above takes sample percent, respondent count, and confidence. It returns plus or minus points with the interval and the sample size.
The default 50% with 1,068 interviews at 95% gives the classic plus or minus 3 points, or 47 to 53. The tighter case of 400 interviews reads 4.9 points.
NIST statistical handbook and Penn State survey courses teach the same z times root formula with 1.96 anchoring 95%. The 1,068 respondent standard exists precisely because it rounds to 3 at 50%.
Error shrinks with the square root of n, so halving the band costs four times the interviews. Subgroup slices rerun the math on smaller n, which is why crosstabs wobble wider than toplines.
How to use
- Type the sample percent.
- Type the respondent count.
- Pick 90, 95, or 99%.
- Read plus or minus first.
- Read the interval beside it.
- Compare leads against the band.
Formula
- p
- Sample share as percent, 50% most conservative.
- n
- Respondent count, 30 plus.
- z
- 1.645, 1.96, or 2.576.
NIST/SEMATECH e-Handbook of Statistical Methods carries confidence intervals for proportions with the same z root formula. Penn State STAT 500 mirrors it for survey proportions. Normal approximation wants n of 30 plus with middling p; tiny samples need exact methods instead.
Worked example
Example: classic 1,068 poll
Defaults: 50%, n = 1,068, 95% confidence.
- Poll: 50% with 1,068 interviews at 95%.
- Spread: 0.5 x 0.5 / 1068 = 0.000234.
- Root: 0.0153, times z 1.96 = 0.030.
- Points: x 100 = 3 points.
- Interval: 50 minus 3 to 50 plus 3 = 47 to 53.
- Tool shows 3 points, 47 to 53, 1,068 interviews.
Reference table
| Interviews | MOE | Use |
|---|---|---|
| 1,068 | 3 points | National polls |
| 400 | 4.9 points | City polls |
| 100 | 9.8 points | Focus panels |
| 2,500 | 2 points | High precision |
Limits
- Sampling error only, not bias.
- Normal approximation needs n 30 plus.
- Simple random sample assumed.
- Weights and clusters widen real MOE.
Common mistakes
- Calling inside MOE leads.
- Quoting topline MOE for subgroups.
- Using 50% math on skewed splits.
- Ignoring nonresponse bias.
- Rounding n before math.
Try it yourself
Quick checks. Answers sit tucked under each question.
50% with 400 at 95%?
4.9 points, interval 45.1 to 54.9, 400 interviews.
30% with 100 at 90%?
7.5 points, interval 22.5 to 37.5, 100 interviews.
n = 10?
Under 30 breaks the normal approximation: gather more interviews.
Questions
The formula?
z times root(p(1-p)/n): 1.645 at 90%, 1.96 at 95%, 2.576 at 99%. Times 100 for points.
Why 1,068?
1,068 at 50% gives the famous plus or minus 3 at 95%. Quadruple n to halve it.
Why 50% default?
50% maximizes p(1-p) and the error. Reported MOE usually assumes it.
Total error?
No, it covers sampling wobble only. Bias, wording, and nonresponse sit outside.
Lead smaller than MOE?
Near ties within one MOE are statistical dead heats, not leads.
Subgroup margins?
Subgroups shrink n and widen MOE fast. Always recompute per slice.