If you were interested in considering the proportion of women in several departments of a specific company it is tempting to set confidence intervals on the proportion of 0.5 for departments of various sizes and then consider the observed proportions department by department. This will result in increasing the risk of false positive results. The procedures discussed above are, strictly speaking, applicable in the central range of proportions, say, 0.2 to 0.8 or perhaps 0.1 to 0.9. With values outside the central region one had best consult a professional. There are times when the use of confidence limits is inadvisable.
Most committees are small.
If the numbers are small then the confidence limits we have considered will be very wide, perhaps ridiculously wide, i.e. extending to negative values or beyond 1.0. Even with appropriate statistical procedures the probabilities of 'unusual' results will tend to be large and decision making in this manner fuzzy at best. For such problems one should turn attention to the ways in which committee members are recruited or the hiring practices of an organization rather than to statistical analysis. In this case it is important to evaluate whether equality of opportunity is present and operable. We have shown how to determine the statistical uncertainty in a calculated proportion from data and how to compare a proportion with a standard or expected proportion as well as comparing two different proportions We have pointed out aspects of data collection that may introduce considerable additional uncertainty We have shown how to determine whether or not to attribute an observed difference to statistical uncertainty or to a genuine effect.