Suppose I have a discrete set of (finite) data (values are only positive integers) (which always remains discrete whenever the observation/survey is taken), divided into some identical categories . Now, I want to calculate the mean values of some random variable for each category, hence getting a sequence of mean values. Considering mean as a random variable (let's call it $M$), it's taking decimal values also (obviously possible). But I am not sure if $M$ is a discrete or a continuous random variable.
My intuition suggest me that it should be discrete. Since, my original data from which I calculated the sets of mean is discrete so the mean can take finitely many values which I suppose is not the case with a continuous random variable where the mean can take infinitely many values within an interval. So is my intuition right or wrong? Is it true that a discrete random variable takes finitely many values and continuous R.V. can take infinitely many possible values?
Best Answer
If you have a finite number of samples then your intuition is correct that the sample mean cannot take certain values. You mention that the discrete values are integers so clearly their mean cannot be irrational.
We know from the central limit theory that the sample mean tends to a continuous distribution as the sample size becomes large.
Even though a finite sample size doesn't technically create a continuous distribution, if your sample size is large or if the distance between discrete values is small for your purposes then you really should consider modelling it as a continuous variable.