Significance of continuity of normal vs. Poisson distribution in central limit theorem

Question

Say I have an experiment where we have a radioactive source, and we measure the particles detected. At the start, we use a thin metal sheet to cover the detector such that the mean number of counts detected per unit time is quite small. I understand that this set of results, if used to produce a histogram for example, could look like a Poisson distribution. By the central limit theorem, when the mean is large we would be able to approximate the results to a normal distribution.

When scaling the data, what is the significance of both distributions? What are the implications of Poisson being discrete and Gaussian continuous in this situation?

Answer 1

If the average value $\bar x$ becomes "large" the Poissonian distribution is "well" approximated by the Gaussian distribution. The error between these two distributions is of the order of $1/\bar x$. Therefore, you should really decide for yourself if the difference is of any significance to your questions. E.g. if you are interested in extremely rare events, the tails of these distributions are important, and the factor $1/\bar x$ has relevance. In contrast, often we only know the sample average value $\bar x$, but the formulas use the population mean value $\mu$. By replacing $\mu \to \bar x$ we omit the associated uncertainty. As a result, the calculations obtained by using the Normal distribution often out-perform the calculations, which uses the Poisson distribution.

So, let's calculate the difference of the two densities for $\lambda = 100$. This is what the following picture shows:

On the y axis I plot the difference of the two densities, and on the x axis I first remove the average value $\lambda$ -- so that the average value is shifted to zero --, and then I divide by the standard deviation $\sqrt{\lambda}$ -- so that we see that the x values span from $\bar x \pm 5 \cdot SD$. We see that the absolute error is smallest in the tails of the distribution, and largest around the average value. However, in the tails both distributions are extremely small. Therefore, it is informative to plot the relative error as well. This is done in the following graph:

The shifting and scaling is done analogously. Now we see that the relative error is large in the tails of the distribution and it is smallest around the average value. This is why we can not answer your question about the severity of the approximation in general, but the answer must be: The severity depends on the details of your question.

Minor point:

Your sentence is misleading:

By the central limit theorem, when the mean is large we would be able to approximate the results to a normal distribution.

The central limit theorem applies to the average value $\bar x$. Thus, if you measure the distribution of how many particles you count per second for 1min, you will obtain an average count $\bar x$. If you repeat this experiment $n$ times you get a distribution of this average counts, $\{\bar x_1, \ldots, \bar x_n\}$. The central limit theorem states that $\bar x$ is normally distributed, if certain conditions apply.

Answer 2

Your question is not specific to radioactivity, but as your title correctly states, is quite general. Another example would be scooping sand up from a beach and counting the grains you got, for ever larger scoops.

The correct distribution in these cases will be the Poisson distribution. Because you have a discrete variable (radioactive counts, grains of sand, etc), the Gaussian distribution is simply never completely correct. In the limit where you don't care about that difference, approximating the true Poisson distribution as a Gaussian is fine. But of course as soon as you care, e.g. by zooming in on how often you counted 75000 times versus 75001 times, the distinction between a continuous and a discrete distribution becomes visible and relevant again