Margin of Error Calculator
Quantifying the Uncertainty Around an Estimate
Every sample-based estimate carries some amount of uncertainty, and margin of error puts an exact number on it — the "plus or minus" figure attached to poll results, survey averages, and experimental measurements. It marks the half-width of the confidence interval: the true population value is expected to fall within the sample estimate plus or minus this margin, at the stated confidence level.
The Formula
where Z is the critical value for the chosen confidence level, σ is the standard deviation, n is the sample size
How Margin of Error Shrinks With Sample Size
Holding the standard deviation fixed at 15 and confidence at 95% (Z = 1.96):
| Sample size (n) | Margin of Error |
|---|---|
| 25 | 5.880 |
| 50 | 4.158 |
| 100 | 2.940 |
| 200 | 2.079 |
| 400 | 1.470 |
| 1000 | 0.930 |
Values computed directly from the calculator's formula.
Quadrupling the sample from 25 to 100 only halves the margin of error — a direct consequence of the √n term in the denominator, and the reason very tight margins get expensive fast.
Where This Calculation Matters
- Poll and survey reporting — the familiar "±3 percentage points" line reported with any poll is a margin of error computed this way (using the proportion-based variant for percentages).
- Measurement uncertainty — laboratory and engineering measurements report a margin of error alongside the point estimate to communicate precision honestly.
- Deciding if a sample is big enough — checking the margin of error for a planned sample size before committing to a data collection effort.
How to Use This Calculator
- Select a confidence level: 90%, 95%, or 99%.
- Enter the standard deviation.
- Enter the sample size (n).
- Select Calculate to get the margin of error.
Related Calculations
To work backward from a target margin of error to a required sample size, use the Sample Size for Mean Calculator. For the full confidence interval built around this margin, see the Confidence Interval Calculator.