Normal distribution is better than Poisson for large sample set?

Question

I'm at a beginner level, so please bear with me.

This is a call center use case. For every week, certain number of calls are received. The average is about 20. This seemed like a Poisson distribution to me (rate per interval). I took about 100 data points. I tried to answer the question "what is the probability of receiving more than 5,10,20,30 etc. calls per week". I computed the corresponding Poisson results. I created another column and answered the same question using Normal distribution. I then generated the probability by directly querying from the database (ie # of weeks where there were more than x calls/total number of weeks). I found something interesting, the actual data followed Normal distribution and not Poisson distribution. Why is that ? I assumed that "number of calls" is discrete & so it should be Poisson.

The question I'm getting at is

1. When is it appropriate to use normal distribution when you have discrete variables ?
2. Where is Poisson distribution appropriate ?
3. What is the distribution to use to model discrete variables ?

I use Libre Office and SQL Server.

Answer 1

When the expectation of the poisson distribution becomes large, the normal distribution becomes a good approximation. Also note that the normal has two free parameters while the poisson has only one, making for greater freedom when fitting the normal to the data. That is related to the next point:

The poisson distribution is a natural point of departure when modeling count data, but it arises under quite restrictive assuptions. The phone calls arrives at the call center as a stochastic (point) process. If the rate is constant is time, and number of points in disjoint intervals are independent, the process will be poisson. That could well not be the case for your call center. One incident could generate multiple calls. Some weeks the rate could be lower. That could well lead to overdispersion relative to the poisson distribution. You could try some other count distribution, like the negative binomial. Or the normal distribution could be good enough as approximation, it does not impose the poisson restriction that mean and variance are equal.