Improved approximations of Poissonian errors for high confidence levels

We present improved numerical approximations to the exact Poissonian confidence limits for small numbers n of observed events following the approach of Gehrels (1986). Analytic descriptions of all parameters used in the approximations are provided to allow their straightforward inclusion in computer algorithms for processing of large data sets. Our estimates of the upper (lower) Poisson confidence limits are accurate to better than 1% for n ≤ 100 and values of S, the derived significance in units of Gaussian standard deviations, of up to 7 (5). In view of the slow convergence of the commonly used Gaussian approximations toward the correct Poissonian values, in particular for higher values of S, we argue that, for n ≤ 40, Poissonian statistics should be used in most applications, unless errors of the order of, or exceeding, 10% are acceptable.