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In statistics, including opinion polls and similar surveys, a margin of error is the radius of a confidence interval -- often a 90% confidence interval -- for a proportion of a population.

## Example

For example, suppose the quantity of interest is the proportion of voters who will vote "yes" in a referendum. A random sample of the population of voters is taken, and it is found that 60% of voters in the sample will vote "yes". Then the estimated proportion of the whole population who will vote "yes" may be taken to be 60%. If a 3% margin of error is reported, that means a procedure was used that will be within 3% of the proportion to be estimated, 90% of the time. Consequently the interval from 57% to 63% is a 90% confidence interval for the proportion of voters in the whole population who will vote "yes". The radius of that interval is 3%; that is the margin of error.

## How to compute a margin of error

is approximately normally distributed with expected value 0 and variance 1. Consulting tabulated percentage points of the normal distribution reveals that P(-1.645 < Z < 1.645) = 0.9, or, in other words, there is a 90% chance of this event. We have

This is equivalent to

Replacing p in the first and third members of this inequality by the estimated value X/n seldom results in large errors if n is big enough. This operation yields

The first and third members of this inequality depend on the observable X/n and not on the unobservable p, and are the endpoints of the confidence interval. In other words, the margin of error is  