Margin of Error

In the popular media, the margin of error is the most frequently quoted measure of statistical accuracy for a sample estimate of a population parameter. Based on the conventional definition of the measure, the difference between the estimate and the targeted parameter should be bounded by the margin of error 95% of the time. Thus, only 1 in 20 surveys or studies should lead to a result in which the actual estimation error exceeds the margin of error.

Technically, the margin of error is defined as the radius or the half-width of a symmetric confidence interval. To formalize this definition, suppose that the targeted population parameter is denoted by θ. Let θ represent an estimator of θ based on the sample data. Let SDθ denote the standard deviation of θ (if known) or an estimator of the standard deviation (if unknown). SD is often referred to as the standard error.

Suppose that the sampling distribution of the standardized statistic

is symmetric about zero. Let Q(0.975) denote the 97.5th percentile of this distribution. (Note that the 2.5th percentile of the distribution would then be given by–Q(0.975).) A symmetric 95% confidence interval for θ is defined as θ ± Q(0.975) SDθ. The half-width or radius of such an interval, ME = Q(0.975)SDθ, defines the conventional margin of error, which is implicitly based on a 95% confidence level.

A more general definition of the margin of error is based on an arbitrary confidence level. Suppose 100(1–α)% represents the confidence level corresponding to a selected value of α between 0 and 1. Let θ denote the percentile of the sampling distribution of . A symmetric 100(1–α)% confidence interval is given by

leading to a margin of error of

The size of the margin of error is based on three factors: (1) the size of the sample, (2) the variability of the data being sampled from the population, and (3) the confidence level (assuming that the conventional 95% level is not employed). The sample size and the population variability are both reflected in the standard error of the estimator, SDθ, which decreases as the sample size increases and grows in accordance with the dispersion of the population data. The confidence level is represented by the percentile of the sampling distribution, θ. This percentile becomes larger as α is decreased and the corresponding confidence level 100(1–α)% is increased.

A common problem in research design is sample size determination. In estimating a parameter θ, an investigator often wishes to determine the sample size n required to ensure that the margin of error does not exceed some predetermined bound B; that is, to find the n that will ensure ME(α) ≤ B. Solving this problem requires specifying the confidence level as well as quantifying the population variability. The latter is often accomplished by relying on data from pilot or preliminary studies, or from prior studies that investigate similar phenomena. In some instances (such as when the parameter of interest is a proportion), an upper bound can be placed on the population variability. The use of such a bound results in a conservative sample size determination; that is, the resulting n is at least as large as (and possibly larger than) the sample size actually required to achieve the desired objective.

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