Margin of Error Calculator
This margin of error calculator shows how precise a sample is. Example: with 50 % agreement, 1,000 respondents and 95 % confidence, the margin of error is about ± 3.1 percentage points.
Source: Wikipedia – Margin of error. Updated: .
How it is calculated
The margin of error states how far the result of a random sample can be from the true value at a given confidence level. It is the number after the “±” quoted by pollsters and half the width of the confidence interval.
Formulas
The margin of error is E = z · SE. For a proportion p, SE = √(p(1 − p) / n); for a mean, SE = s / √n. The z values are 1.645 for 90 %, 1.960 for 95 % and 2.576 for 99 %. If you know the population size N, multiply by √((N − n) / (N − 1)); this matters when the sample is more than about 5 % of it. The calculator shows every intermediate step and the interval.
Examples
Proportion 50 %, n = 1,000, 95 %: SE = √(0.25 / 1,000) ≈ 0.0158 and E = 1.96 · 0.0158 ≈ 0.031, so ± 3.1 percentage points; the interval runs from 46.9 % to 53.1 %. Mean with s = 15, n = 100, 95 %: SE = 1.5 and E = 1.96 · 1.5 = 2.94. For a mean of 100 that gives 97.06 to 102.94.
- Quadrupling the sample size halves the margin of error.
- Proportions near 50 % have the largest margin of error, those near 0 % or 100 % the smallest.
Frequently asked questions
What does the margin of error mean in surveys?
It gives the range around the result in which the true value lies at the chosen confidence. With 52 % and ± 3 points the true value lies between 49 % and 55 %.
How many respondents do I need for ± 3 %?
At 95 % confidence and the worst-case proportion of 50 %, about 1,067 people. The sample size calculator is made for that.
When do I use the finite population correction?
When you know the population size and the sample is more than about 5 % of it, such as 200 of 1,000 employees. The margin of error then gets smaller.
Do small samples need a t value instead of z?
In mean mode the calculator uses z values. If s is estimated from a small sample (n < 30), the t value gives a larger margin; at n = 10 and 95% it is about 15% more.
Sources and legal basis
As of: · Editorial policy
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