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| Second, a reformulation for the necessary sample size to achieve a specific margin of error. | Second, a reformulation for the necessary sample size to achieve a specific margin of error. The final formula again assumes a 95% confidence level. |
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| As an example, again taking a 95% confidence level, this can be used to calculate necessary sample size for a 5% margin of error: | As an example, the necessary sample size for a 5% margin of error is given by: |
Margin of Error
The margin of error describes the confidence interval about a point estimate due to sampling error.
Contents
Description
Simply put, the margin of error is the radius of the confidence interval. The confidence interval is formed by the point estimate plus and minus the margin of error.
Therefore, the margin of error is calculated in terms of standard errors.
This can lead to a misunderstanding that margins of error are a duplicative concept. The important detail is that margins of error are reported for non-overlapping domains, not for variables/estimates. A sample design may be selected to ensure margins of error of 5% for all domains, i.e. each U.S. state.
Formulation
For a given confidence level, the z-score is calculated and plugged into:
In several fields (survey statistics included), the metrics of interest are Bernoulli-distributed variables. Note that variance is maximized at a proportion of 0.5, giving σ2 = 0.25. Margins of error are usually reported assuming this maximum.
Given the popularity of the 95% confidence level, the z-score is often implicitly substituted with 1.96.
As such, there are two prevalent expression of the formula. First, a shorthand for calculating the margin of error with a 95% confidence level:
Second, a reformulation for the necessary sample size to achieve a specific margin of error. The final formula again assumes a 95% confidence level.
As an example, the necessary sample size for a 5% margin of error is given by:
