Hong Life CalculatorsUnderstanding the margin of error calculator — concept and principle
The margin of error is the "±" figure reported with poll and survey results, such as "42% support, margin of error ±3.1 percentage points". It tells you how far the sample percentage could be from the true population percentage purely because of random sampling. Use it to judge whether a poll lead, a month-to-month change or a survey difference is meaningful.
How it is calculated
MOE = z × √[p(1 − p) ÷ n] × FPC
- z: critical value for the confidence level (1.96 for 95%, 2.576 for 99%).
- p: the observed or assumed proportion, as a decimal. 50% gives the largest, most conservative margin.
- n: sample size (number of completed responses).
- FPC (finite population correction) = √[(N − n) ÷ (N − 1)], used only when you enter a population size N; with 0 it equals 1.
The result is in percentage points of the reported proportion.
The principle
A sample proportion is an average of yes/no answers, so by the central limit theorem it is approximately normally distributed around the true proportion, with standard error √[p(1 − p)/n]. Multiplying the standard error by z gives the half-width of the confidence interval. The square root explains why precision grows slowly: quadrupling the sample only halves the margin. When the sample is a large share of the population, there is less left to be uncertain about, and the FPC shrinks the margin.
Worked example
- Inputs: n = 1,000, p = 50%, confidence 95%, population 0 (infinite).
- z = 1.959964.
- Standard error = √(0.5 × 0.5 ÷ 1,000) = √0.00025 = 0.0158114.
- MOE = 1.959964 × 0.0158114 = 0.030990.
- As a percentage: 3.1% (±3.1 percentage points).
| Sample size | Margin of error |
|---|---|
| 250 | 6.2% |
| 500 | 4.38% |
| 750 | 3.58% |
| 1,000 | 3.1% |
| 1,250 | 2.77% |
| 1,500 | 2.53% |
| 2,000 | 2.19% |
How to read the result
If a candidate polls 52% with a ±3.1-point margin, the 95% interval is roughly 48.9% to 55.1%. The margin for the difference between two candidates in the same poll is larger, roughly up to twice the stated figure, so a 52–48 race is not a clear lead. Typical values at 95% and p = 50%: about ±4.9 points for n = 400, ±3.1 for 1,000 and ±2.2 for 2,000. At 99% confidence n = 1,000 gives about ±4.1 points.
Common mistakes
- Applying the overall margin to subgroups; a subgroup of 250 respondents has a margin near ±6.2 points.
- Treating the margin as covering all error. It does not include non-response, question wording or coverage bias.
- Confusing percentage points with percent: ±3 points around 40% is 37–43%, not 38.8–41.2%.
Frequently asked questions
Why do pollsters quote the margin at 50%?
Because p(1 − p) is largest at 50%, the margin quoted there is valid as an upper bound for every percentage in the poll.
Does the population size matter for a national poll?
Hardly. With N in the millions and n = 1,000, the FPC is essentially 1.
How do I get a ±2-point margin?
At 95% and p = 50% you need about 2,401 completed responses.
Sources: American Association for Public Opinion Research (AAPOR), guidance on margin of sampling error. · Cochran, W. G. (1977). Sampling Techniques (3rd ed.). Wiley.