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Weighted Average

It is observed that the average can be calculated only if the weights of all the factors, is same. So, the weighted average is a more generalized form of average. This can be further understood with the following illustration

Now, if we combine both these classes, then the average age of all the students = 2543.png years. This is one standard example of Average.

Let us see another example:
Now, if we combine these two classes, then the average can not be calculated by the above mentioned method, since the weights attached to different averages are different.

Some more cases of weighted average

  1. As we have observed above in the case of average, if per capita income of India is USD 500 and per capita income of US is USD 200, and if we merge India and US into one country, then the per capita income of this new country (India + US) cannot be found by just adding the per capita income of both the countries and dividing it by 2.
    The weights, i.e., the population attached to the different averages, i.e., the per capita income would also have to the considered.
  2. Average speed cannot be calculated by just adding the different speeds and then dividing it by 2. This can be understood by the following example:
    A person goes to A from B at a speed of 40 km/h and returns with a speed of 60 km/h, then the average speed for the whole journey can not be equal to 50 km/h.
    We know that average speed = 2582.png

Finding expression for weighted average

Let us assume there are N groups with the following structure:

If we combine all these groups, then the average age of all the members
= (N
1 × A1 + N2 × A2…+ NN × AN)/(N1 + N2 + N3 +…+ NN)
Considering that there are only two groups and both the groups are combined then the average age of all the members = (N1 × A1 + N2 × A2)/(N1 + N2) = Aw

Simplifying the above written expression, we get the conventional criss-cross method as given below
And we write this as: 2631.png
It is quite obvious that the ratio of the number of persons/items in different groups is proportionate to the deviations of their average from the average of all the people combined.
This average of all the members combined is known as weighted average, and is denoted by Aw. This process of mixing the two groups is also referred as alligation.

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