Weighted Average vs Expected Value

I find my self inclined to compare these two formulas because they can become the same in one single case which I will explain below.

The first thing I would like to clear out is that the Expected Value formula has nothing to do with data distribution as the case with Weighted Average.

Weighted average basically calculates the average value for the contribution of all weights divided by the total weight where the contribution of each weight can be 0 , 1 or 2.

Lets say you have some data where you are only given percentages for each value contribution but not the real data and it is required to find the mean value.

In that case Weighted Average formula is the solution. The weight can be frequency or percentages. The two formulas are shown below for each case.

Note : P(x) is the probability which is also the percentage.

Here is an example:

On the right, it is required to calculate the expected earning after making 3 investments where every investment would return certain amount of profit if certain percentage of the capital is invested.

On the left 3 values have certain contribution to the data set, and it is given in the form of percentages. It is required to calculate the mean value of the data set.

As you can see by doing the math the two equations are similar to each other which is true only for that specific case where percentages are given instead of frequency for every value in the data set.

Notice that these are two different problems but they happen to lead to the same way of calculation.

In the first one an average is calculated which gives the same value as the Expected value because it is divided by 1 while in the other case the Expected value only is calculated without dividing by 1 which makes these two very similar.

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