Confidence Intervals for Percentages and
Counts
Introduction
This document describes the algorithms for computing confidence intervals for percentages
and counts for bar charts. The data are assumed to be from a simple random sample, and
each confidence interval is a separate or individual interval, based on a binomial proportion
of the total count. The computed binomial intervals are equal-tailed Jeffreys prior intervals
(see Brown, Cai, & DasGupta, 2001, 2002, 2003). Note that they are generally not symmetric
around the observed proportion. Therefore, the plotted interval bounds are generally not
symmetric around the observed percentage or count.
Notations
The following notation is used throughout this chapter unless otherwise noted:
Xi Distinct values of the category axis variable
i
w Rounded sum of weights for cases with value Xi
W = ∑
ii
w
Total sum of weights over values of X
pi Population proportion of cases at Xi
Specified error level for 100(1-FRQILGHQFHLQWHUYDOV
IDF.BETA(p,shape1,shape2) in COMPUTE gives the pth quantile of the beta distribution or
incomplete beta function with shape parameters shape1 and shape2. For a precise
mathematical definition, see page 2 of “Appendix 12: Cumulative Distribution, Percentile
Functions, and Random Numbers.”
Confidence Intervals for Counts (Wpi)
Lower bound for W pi = W [IDF.BETA( /2,wi +.5,W-wi +.5)].
Upper bound for W pi = W [IDF.BETA(1- /2,wi +.5,W-wi +.5)].
Confidence Intervals for Percentages (100pi)
Lower bound for 100 pi = 100 [IDF.BETA( /2,wi +.5,W-wi +.5)].
Upper bound for 100 pi = 100 [IDF.BETA(1- /2,wi +.5,W-wi +.5)].
