Why is my Weighted Average = Simple Average Problem is as follows:
I have a dataset as follows:
#. Category Value

*

*Item A,   1

*Item B,   2

*Item B,   2

If I do a simple average of value, it is 1.67.
If I do a weighted Average as follows, then the answer is the same.
(1/3) x 1 + (2/3) x  2 = 1.67.
So would you call this a simple or a weighted average?

Break down the arithmetic.
$$
1(1/3) +2(2/3) = (1+4)/3=(1+2+2)/3
$$
$(1+2+2)/3$ is exactly the usual mean.
There are two ways to look at the calculation.

*

*Add up the values, and then divide by however many values you added up (the usual way of thinking about the mean).


*Weight the unique values according to how often they occur, and then add up those products of values and weights (a weighted average of the unique values).
You see an example where those coincide, and that applies in general.
