I ran a simple Monte Carlo based test to compute the expected bias between a Poisson sample and the “equivalent” Gaussian sample. The result is shown in the plot below.

The jagged red line is the fractional expected bias relative to the true intensity. The typical recommendation in high-energy astronomy is to bin up events until there are about 25 or so counts per bin. This leads to an average bias of about 2% in the estimate of the true intensity. The bias drops below 1% for counts >50. The smooth blue line is the reciprocal of the square-root of the intensity, reflecting the width of the Poisson distribution relative to the true intensity, and is given here only for illustrative purposes.

Poisson-Gaussian bias

Exemplar IDL code that can be used to generate this kind of plot is appended below:

nlam=100L & nsim=20000L
lam=indgen(nlam)+1 & sct=intarr(nlam,nsim) & scg=sct & dct=fltarr(nlam)
for i=0L,nlam-1L do sct[i,*]=randomu(seed,nsim,poisson=lam[i])
for i=0L,nlam-1L do scg[i,*]=randomn(seed,nsim)*sqrt(lam[i])+lam[i]
for i=0,nlam-1L do dct[i]=mean(sct[i,*]-scg[i,*])/(lam[i])
plot,lam,dct,/yl,yticklen=1,ygrid=1
oplot,lam,1./sqrt(lam)