In Reality: What is “The Average”?
Before we approach the subject of how to calculate weighted averages, let’s understand the basic averaging formula. When you have
several values (a1,a2,…, an), the average (or mean) value can be calculated as follows:
AVG = Σ(a1+a2+…+an) / n
The meaning is simple: you should sum up all your values and divide by total number of those. Remember, this is the general average formula, NOT the formula for calculating weighted averages!
John has five dollars, Sam has seven dollars, Mary has 12 dollars and Jane has four dollars. Calculate the average amount of money the four children have.
Summarizing 5+7+12+4, we get $28. There are four children, so we need to divide 28 by 4 to get the average 28/4 = $7.
Of course, this is a very basic and “slow" way – but mathematically it is the most correct one. Alternatively, we could immediately see that John and Sam have (5+7)/2 = $6 in average; and Mary and Jane have (4+12)/2 = $8 in average. So, we can now easily calculate overall average, getting (6+8)/2 = $7.
Let’s now assume that in addition to those four, we also have Tasha, and she also has 12 dollars. What will be the average in this case?
The straight-forward solution is as follows:
Let’s summarize all the money: (5+7+12*2(weighted!)+4) / 5 = 40/5 = $8.
But this is definitely not the fastest way. Someone said that mathematics is about trying to turn every problem into one that we have already solved.
So, here we are – we have four kids with seven dollars on average, and a Tasha, who has 12 dollars.Thus, the average would be (7+12)/2 = $9.5….
Stop! That’s not what we got in the “slow" solution. Of course! We had not taken into account that seven and 12 have different “weights," treating the four kids’ average the same way as Tasha’s money! So, here is the question: how do we calculate a weighted average?
Formula for Calculating Weighted Averages
The formula for calculating weighted averages for several values (a1,a2,…,an), each of them having a different “weight" (v1,v2,…,vn), would
AVGw = Σ(a1*v1+a2*v2+…+an*vn) / Σ(v1+v2+…+vn)
This means that in order to understand how to calculate weighted averages, you have to multiply each value by its weight, summarize those multiplications, and then divide the result by the sum of all weights.
Solving Problem Correctly and Quickly
Let’s get back to our problem with Tasha and other kids
We have four kids with an average of seven dollars and one girl, Tasha, with 12 dollars.
So, applying the formula for calculating weighted averages, we get 7*4+12*1 = 28+12 = 40. Now let’s divide it by the total number of children (4+1): 40/5 = $8. Now this corresponds with the answer we got in the first place!
Sources and Image Credits
Arithmetic Mean Tutorial, https://easycalculation.com/statistics/learn-mean.php
Weighted Average Calculation, https://www.buzzle.com/articles/weighted-average-calculation.html
Equation, by Author
Seesaw With Kids, Wikimedia Commons/The Clip Art Book, https://commons.wikimedia.org/wiki/File:SeesawWithKids_wb.png