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The plain arithmetic mean is actually a special case of the weighted mean, except all the weights are equal to 1. The arithmetic mean is the sum of all the individual observations divided by the number of observations. With a weighted mean you multiply each observation by a weight, add those values together and then divide by the sum of the weights.
E.g. Let's say you have 3 observations: 4, 7, 12
The arithmetic mean is (4+7+12) / 3 = 23/3 = 7.67
Now let's assume that you want to weight the first observation by a factor of 10, the second observation by a factor of 5 and the third observation by a factor of 2:
The weighted mean is (4x10+7x5+12x3) / (10+5+2) = 111/17 = 6.53
You can see that if all the weights were 1 you would have the arithmetic mean shown above.
As it is mentioned above arithmetic mean is a special case of weighted mean. In the calculation of arithmetic mean all the observations are given an equal chance of occurance ie
the above mentioned problem can be written as
4*1/3+7*1/3+12*1/3=7.67
or inother words 7.67 is the number it takes if all are given equal chance
whereas in weighted mean the chance of occurance are not equal .This can be written as
4*10/17+7*5/17+12*2/17=6.53
in the above eg. 4 has given more weightage than 7 and 12 has the least weightage so the probability of 4 occurring is more when compared to 7 and 12 there fore the average obtained is seen to decrease as we have given more importance to 4 than others. It shows that the average is affected by the weightage given to the numbers
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