# Calculating 95% CI of a proportion of 0%

I have a sample of 72 individuals. 0 of these had the trait I'm investigating. How do I calculate the 95% CI when the proportion is 0%?

• This is basically a duplicate of stats.stackexchange.com/questions/82720/… (that question has an additional assumption of a small sample size, but the answer is relevant no matter what sample size is used). One important point is that you shouldn’t use the normal approximation (the default in many software programs) even if the observed proportion was slightly higher than 0%; it has truly terrible properties for all probabilities close to 0% or 100%. – Karl Ove Hufthammer Mar 2 '14 at 7:21

The presumption is that the trait is possible, just not observed. So, sensible methods will give lower limits that are 0 and upper limits that are positive, depending on the exact assumptions.

Most good software will do this for you, but different methods will give different results.

With 0/72 observed, my favourite software gives upper 95% limits for the observed proportion that are variously .0499441, .0506512, .0341694, .0606849, depending on which method you use. This may seem surprising, but very competent statisticians disagree on how best to formulate the problem.

An excellent survey is

Brown, L. D., T. T. Cai, and A. DasGupta. 2001. Interval estimation for a binomial proportion. Statistical Science 16: 101-133.

If the trait is impossible, your confidence limits are identically zero.

• Excellent answer and a reference I have visited previously. Also worth noting that many report a 1-tailed confidence interval in the case of 0 estimate; just need to be very clear in reporting. – tristan Feb 26 '14 at 11:54
• It's not included in the above review but for the last decade or so, the particle physics world has been making heavy use of the Feldman-Cousins method for confidence intervals near logical bounds like 0 or 1 probabilities. Just looking over the review it seem to be in a similar vein. (As there is always the possibility that some physicists have re-invented something know to others, again.) – dmckee --- ex-moderator kitten Feb 26 '14 at 14:14
• Thanks in turn for that reference. I note that it was published on 1 April 1998. Sometimes a coincidence is just a coincidence. – Nick Cox Feb 26 '14 at 16:28
• The rule of three is easily calculated and gives a 95% interval of [0,3/72], or [0,0.0417]. Wikipedia has an article here: en.wikipedia.org/wiki/Rule_of_three_(statistics) – soakley Feb 26 '14 at 18:11
• Please vist [Link] lexjansen.com/nesug/nesug13/41_Final_Paper.pdf if you want to use SAS. – overwhelmed Aug 24 '16 at 4:17