Average of a list of numbers, read one at a time I have a dilemma
Lets take the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$
If we wanted to calculate the average of this set we would add up all the numbers in the set $(45)$ and then divide by the total number of items in the set $(9)$ and arrive at the correct average of $5$
What happens if we are not given the full set? What if we are only given one number at a time and have to calculate the correct average?
For instance if we are given $1$ we know the average is $1$, then if we are given $2$ then we add $1+2$ and get $3$, then divide $3$ by $2$ and arrive at the correct average of $1.5$.
Then what if we are given the next number $3$ we would add $1.5$ and $3$ giving $4.5$ and divide that by $2$ arriving at $2.25$ which is incorrect since $1+2+3=6$  and $6/3=2$.
Is there a way of calculating the correct total average like this?

You can:


*

*Keep track of the total number of values you have seen so far, call it $n$, and the average of the numbers so far, $A$.
Then when you see a new number $x$, map
\begin{align*}
n &\mapsto n+1 \\
A &\mapsto \frac{nA + x}{n+1}
\end{align*}


Or:


*

*Keep track of the total number of values you have seen so far, call it $n$, and the sum of the numbers so far, $S$.
Then when you see a new number $x$, map
\begin{align*}
n &\mapsto n+1 \\
S &\mapsto S+x
\end{align*}
at the end, compute the average $\frac{S}{n}$.

