Formulas for Confidence Limits in Large Samples

Most of the approximate methods for determining confidence limits are based on the assumption that your sample statistic has a sampling distribution that's (at least approximately) normally distributed. Fortunately, there are good theoretical and practical reasons to believe that almost every sample statistic you're likely to encounter in practical work will have a nearly normal sampling distribution, for large enough samples.

For any normally distributed sample statistic, the lower and upper confidence limits can be calculated very simply from the observed value (V) and standard error (SE) of the statistic:

CLL = Vk × SE
CLU = V + k × SE

Confidence limits computed this way are often referred to as normal-based, asymptotic, or central-limit-theorem (CLT) confidence limits. (The CLT provides good reason to believe that almost any sample statistic you're likely to encounter will be nearly normally distributed for large samples.)

The value of k in the formulas depends on the desired confidence level and can be obtained from a table of critical values for the normal distribution or from a web page such as StatPages.

Multipliers for Normal-Based Confidence Intervals
Confidence Level Tail Probability k Value
50% 0.50 0.67
80% 0.20 1.28
90% 0.10 1.64
95% 0.05 1.96
98% 0.02 2.33
99% 0.01 2.58

For the most commonly used confidence level, 95 percent, k is 1.96, or approximately 2. This leads to the very simple approximation that 95 percent confidence limits are about two standard errors above and below the observed value.

The distance of each confidence limit from the measured value, k × SE, is called the margin of error (ME). Because MEs are almost always calculated at the 95 percent confidence level, they're usually about twice as large as the corresponding SEs.

MEs are most commonly used to express the precision of the results of a survey, such as "These poll results have a margin of error of ± 5 percent." This usage can lead to some confusion because the SE is also usually expressed as a ± number.

For this reason, it's probably best to use the CI instead of the ME to express precision when reporting clinical research results. In any event, be sure to state which one you're using when you report your results.

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