How QuickCalcs computes the confidence interval of a count.
One of GraphPad's free web QuickCalcs can compute the confidence interval of a count.
If you enter the number of objects observed in a certain volume or the number of events observed in a certain length of time, QuickCalc can compute the 95% confidence interval of the average number of objects in that volume overall or the average number of events that occur in that length of time. These calculations are based on the Poisson distribution. This assumes that the objects are well mixed and not clumped, or that the events were counted without error.
For example if you enter the number 25 into that calculator, the results look like this:
The 90% confidence interval extends from 17.38 to 34.92
The 95% confidence interval extends from 16.18 to 36.90
The 99% confidence interval extends from 14.00 to 41.00
There are many ways to compute the confidence itnerval of a value from a Poisson distribution. The method we use, called the Garwood method (1) and which is quite standard, is the first (of 19) methods listed in Patil and Kulkarni,
Define C to be the entered count (25 in this case).
Define P to be the desired confidence level. If you want 95% confidence intervals, set P to 95.
Calculate L to 50 + P/2. So if P is 95 (for the usual 95% confidence interval), L is set to 97.5.
Calculate the confidence limits using these Excel formulae:
Chisq.inv.rt is the Excel function that computes the chi square statistic such that the entered probability (the first parameter) is the fraction of the area uner the curve to the right of that chi-square statistic. (This form of the function was new to Excel 2010. Older version used quiinv) QuickCalc does not rely on Excel, but has its own version of this function built in.
Edited Dec. 22, 2013. The Excel equations were wrong. Until now, the value of L entered into Excel was 95 for 95% confidence levels, rather than the correct 97.5.
1. Garwood, F. (1936). Fiducial limits for the Poisson distribution, Biometrika,
28, 437–442.
2. Patil, V. V. & Kulkarni, H. V. Comparison of confidence intervals for the Poisson mean: some new aspects. REVSTAT–Statistical Journal 10, 211–227 (2012).