Margin of error Totally Explained

Everything about Margin Of Error totally explained

The margin of error is a statistic expressing the amount of random sampling error in a survey's results. The larger the margin of error, the less confidence one should have that the poll's reported results are close to the "true" figures; that is, the figures for the whole population.

Explanation

The margin of error is usually defined as the radius of a confidence interval for a particular statistic from a survey. One example is the percent of people who prefer product A versus product B. When a single, global margin of error is reported for a survey, it refers to the maximum margin of error for all reported percentages using the full sample from the survey. If the statistic is a percentage, this maximum margin of error can be calculated as the radius of the confidence interval for a reported percentage of 50%.
   The margin of error has been described as an "absolute" quantity, equal to a confidence interval radius for the statistic. For example, if the true value is 50 percentage points, and the statistic has a confidence interval radius of 5 percentage points, then we say the margin of error is 5 percentage points. As another example, if the true value is 50 people, and the statistic has a confidence interval radius of 5 people, then we might say the margin of error is 5 people.
   In some cases, the margin of error isn't expressed as an "absolute" quantity; rather it's expressed as a "relative" quantity. For example, suppose the true value is 50 people, and the statistic has a confidence interval radius of 5 people. If we use the "absolute" definition, the margin of error would be 5 people. If we use the "relative" definition, then we express this absolute margin of error as a percent of the true value. So in this case, the absolute margin of error is 5 people, but the "percent relative" margin of error is 10% (because 5 people are ten percent of 50 people). Often, however, the distinction isn't explicitly made, yet usually is apparent from context.
   Like confidence intervals, the margin of error can be defined for any desired confidence level, but usually a level of 90%, 95% or 99% is chosen (typically 95%). This level is the probability that a margin of error around the reported percentage would include the "true" percentage. Along with the confidence level, the sample design for a survey, and in particular its sample size, determines the magnitude of the margin of error. A larger sample size produces a smaller margin of error, all else remaining equal.
   If the exact confidence intervals are used, then the margin of error takes into account both sampling error and non-sampling error. If an approximate confidence interval is used (for example, by assuming the distribution is normal and then modeling the confidence interval accordingly), then the margin of error may only take random sampling error into account. It doesn't represent other potential sources of error or bias such as a non-representative sample-design, poorly phrased questions, people lying or refusing to respond, the exclusion of people who couldn't be contacted, or miscounts and miscalculations.

Concept

Running example

A running example from the 2004 U.S. presidential campaign will be used to illustrate concepts throughout this article. According to an October 2, 2004 survey by Newsweek, 47% of registered voters would vote for John Kerry/John Edwards if the election were held on that day, 45% would vote for George W. Bush/Dick Cheney, and 2% would vote for Ralph Nader/Peter Camejo. The size of the sample was 1,013. Unless otherwise stated, the remainder of this article uses a 95% level of confidence.

Basic concept

Polls typically involve taking a sample from a certain population. In the case of the Newsweek poll, the population of interest is the population of people who will vote. Because it's impractical to poll everyone who will vote, pollsters take smaller samples that are intended to be representative, that is, a random sample of the population. It is possible that pollsters sample 1,013 voters who happen to vote for Bush when in fact the population is evenly split between Bush and Kerry, but this is extremely unlikely (p = 2^-1013 ≈ 1.13923782 × 10^-305) given that the sample is random. Sampling theory provides methods for calculating the probability that the poll results differ from reality by more than a certain amount, simply due to chance; for instance, that the poll reports 47% for Kerry but his support is actually as high as 50%, or is really as low as 44%. This theory and some Bayesian assumptions suggest that the "true" percentage will probably be fairly close to 47%. The more people that are sampled, the more confident pollsters can be that the "true" percentage is close to the observed percentage. The margin of error is a measure of how close the results are likely to be.
However, the margin of error only accounts for random sampling error, so it's blind to systematic errors that may be introduced by non-response or by interactions between the survey and subjects' memory, motivation, communication and knowledge.

Calculations assuming random sampling

This section will briefly discuss the standard error of a percentage, the corresponding confidence interval, and connect these two concepts to the margin of error. For simplicity, the calculations here assume the poll was based on a simple random sample from a large population.
The standard error of a reported proportion or percentage p measures its accuracy, and is the estimated standard deviation of that percentage. It can be estimated from just p and the sample size, n, if n is small relative to the population size, using the following formula:
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Given the observed percentage difference p − q (2% or 0.02) and the standard error of the difference calculated above (.03), any statistical calculator may be used to calculate the probability that a sample from a normal distribution with mean 0.02 and standard deviation 0.03 is greater than 0.
Applying these calculations to the Newsweek example results in a 75% probability that Kerry was "truly" leading.

Further Information

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