Binomial || Poisson ||
Set Conf Levels
This page computes exact confidence intervals for samples from the Binomial
and Poisson distributions.
By default, it calculates symmetrical 95% confidence intervals, but you can
change the "tail areas" to anything you'd like.
The formulas used in this web page are also available as Excel macros, which
you can download in the file: confint.xls (85k
long). This spreadsheet now includes an extra page that can generate a customized
table of confidence limits around the observed numerator (x) and around the
observed proportion (x/N), for any value of the denominator (N). My
thanks to Prof. Patrick J. Laycock (University of Manchester) for that
If you would like to have the six functions (BinomLow, BinomHigh, BinomP,
PoisLow, PoisHigh, and PoisP) which appear in this spreadsheet available
all the time (just as if they were built-in Excel functions), you can download
and install the confint.xla Excel "add-in". Save
the downloaded file in some reasonably "permanent" location on your computer's
hard disk. Then install it, using the appropriate steps for your version
of Excel (see Excel's Help section for "add-ins").
Note: Before using this page for the first time,
make sure you read the JavaStat user interface
guidelines for important information about interacting with JavaStat
Enter the observed numerator and denominator counts, then click the
Exact Confidence Interval around Proportion:
Enter the number of observed number of events, then click the Compute
Exact Confidence Interval around Mean Event Rate:
Normally you will not need to change anything in this section. People usually
use symmetrical 95% confidence intervals, which correspond to a 2.5% probability
in each tail.
If you want a different confidence level, you can replace the 95 with your
preferred level, then click the Compute button. The program will split the
tail area evenly between the Lower and Upper tails.
If you want asymmetrical limits, you can change the % Area numbers in the
Lower Tail and Upper Tail cells shown below, then click the Compute button.
The program will adjust the Confidence Level level accordingly.
% Area in Upper Tail:
% Area in Lower Tail:
Note: Over the years, I have grappled with
the issue of whether or not any special action has to be taken , in computing
the classic Clopper-Pearson binomial confidence intervals, when the
observed count falls at one or the other end of the range of possible
values (such as when the observed "Numerator" is equal to zero, or equal
to the "Denominator" for the binomial case, or when the "Observed Events"
is zero for the Poisson case). Specifically, the question arises as to whether,
in such a situation, the confidence interval should be made one-sided; that
is, should all of the 5% tail probability (for 95% CI's) be put onto one
side, instead of being split half-and-half between the left and right side.
The rationale for taking such action is that, in these situations, one side
of the CI will be equal to the observed value (that is, there will be no
"confidence region" on that side), so it would seem to make sense that all
of the tail area should be re-allocated over to the other side, giving a
slightly narrower confidence interval.
In the original version of this page (and the corresponding
Excel spreadsheet), I took no special action. So, for example, an observed
event count of zero would result in a 95% Poisson CI of 0 to 3.689 . But
over the years, people pointed out that the one-sided 95% Poisson CI for
an observed count of 0 was 0 to 2.996, so on June 19, 2004, I revised
this web page to apply this one-sided adjustment automatically whenever the
observed Poisson count was zero, or whenever the observed binomial numerator
was zero or equal to the denominator. I also revised the Excel spreadsheet
(and the included macros) to do the same thing.
Then in 2007, in a series of e-mail communications
with Karl Schlag (of the Economics Department, European University Institute,
in Florence, Italy), I came to realize that this special action was
not justified -- It is not valid for the CI algorithm to turn a 2-sided
CI into a 1-sided CI "on the fly" for certain values of the observed value.
Taking this special action produces a CI that, for certain ranges of the
true parameter, fails to produce at least a 95% coverage probability, thereby
violating the strict requirement for an exact CI. The decision to use a 1-sided
or a 2-sided CI has to be made beforehand, and applied consistently no matter
what the observed value turns out to be. So, on August 11, 2007, I changed
the web page's algorithm back to the way it was prior to June 19, 2004.
I also made the same changes to the Excel spreadsheet and its macros.
Reference: CJ Clopper and ES Pearson, "The use of confidence or fiducial
limits illustrated in the case of the binomial." Biometrika
Reference: F Garwood, "Fiducial Limits for the Poisson Distribution"
Biometrica 28:437-442, 1936.