The Confidence Interval around a Proportion
If you were to survey 100 typical children and find that 70 of them like chocolate, you'd estimate that 70 percent of children like chocolate. What is the 95 percent confidence interval (CI) around that 70 percent estimate?
There are many approximate formulas for confidence intervals around an observed proportion (also called binomial confidence intervals). The simplest method is based on approximating the binomial distribution by a normal distribution. It should be used only when N (the denominator of the proportion) is large (at least 50), and the proportion is not too close to 0 or 1 (say, between 0.2 and 0.8). You first calculate the SE of the proportion:
And then you use the normal-based confidence interval formulas: CI = p ± k×SE, as described below.
Using the numbers from the preceding example, you have p = 0.7 and N = 100, so the SE for the proportion is
k is 1.96 for normal-based 95 percent confidence limits. So the lower and upper confidence limits (CLL and CLU) are given by: CLL = 0.7 – 1.96 x 0.046 and CLU = 0.7+ 1.96 x 0.046, which works out to a 95 percent CI of 0.61 to 0.79. To express these fractions as percentages, you report your result this way: "The percentage of children in the sample who liked chocolate was 70 percent, 95%CI = 61–79%."
Many other approximate formulas for CIs around observed proportions exist, most of which are more reliable when N is small. There are also several exact methods, the first and most famous of which is called the Clopper-Pearson method, named after the authors of a classic 1934 article. The Clopper-Pearson calculations are too complicated to attempt by hand, but fortunately, many statistical packages can do them for you.
You can also go to the "Binomial Confidence Intervals" section of the online web calculator at StatPages.info. Enter the numerator (70) and denominator (100) of the fraction, and press the Compute button. The page calculates the observed proportion (0.7) and the exact confidence limits (0.600 and 0.788), which you can convert to percentages and express as 95%CI = 60–79%.
For this example, the normal-based approximate CI (61–79%) is very close to the exact CI, mainly because the sample size was quite large. For small samples, you should report exact confidence limits, not normal-based approximations.