# What is the arithmetic mean of no numbers?

I have two programs that both behave nearly identically: they both take in any numbers you give them and can tell you the arithmetic mean and how many numbers were given. However, when you don’t give them any numbers, one says the arithmetic mean is 0.0, and the other says it’s NaN (“Not a Number”). Which of these answers, if any, is more correct, and why?

Note: Although I use “programs” as a metaphor here, this isn’t a programming question; I could’ve just as easily said “computers”, “machines”, “wise men”, etc. and my question would be the same

More mathematically, taking the average is a linear operation, which means if you add a constant $c$ to each observation, then the average $a$ becomes $a+c$. Now if you add $c$ to each observation in the empty set, you get the empty set again, and thus the average will have to satisfy $a+c=a$ for all $c$, clearly nonsense.