What is a weighted average?
A weighted average is an average where each data point has a different weight (think of weight as the amount of influence the data point has to the average). The weights usually correspond to the importance or significance the data point has. Some of the times, weights are calculated from 0 to 1 (percentages) and are multiplied to each data point value - sum them all up after multiplying and you get a weighted average.
A weighted average is an average where each data point has a different weight (think of weight as the amount of influence the data point has to the average). The weights usually correspond to the importance or significance the data point has. Some of the times, weights are calculated from 0 to 1 (percentages) and are multiplied to each data point value - sum them all up after multiplying and you get a weighted average.
How to use this concept?
We can use this concept in a lot of real-life scenarios, but first to be able to apply them, we need to go over how we can calculate it. Like we explained previously, the main idea is each datapoint has a weight assigned to it where each weight can range from 0 to 1 and the sum of all weights add up to 1. Each weight can be understood as a percentage of how much influence you would want the data point to have (except the percentage is divided by 100 to ensure it is between 0 and 1). Here is the formula of a weighted average applied to an example of finding the weight score/average of a student:
The weights in the case all added up to 1, however, sometimes this may not be the case. You may encounter some irregular weights not in decimals (between 0 and 1). In those situations, you can find out the weight of each category by just dividing the individual categories amount by the total sum amount of each category, giving you the weight. (Check out a solved example for a question of this type).
Sample Math Problems
Question 1:
A FedEx package deliverer wants to know the weighted average of the number of deliveries he does per day over the period of the past 2 weeks. Some days, he is busy delivering many packages, and on some days, he is quite free. The following is a breakdown for how many days he delivered a particular number of packages in the past two weeks:
For 1 day, he delivered 1 package
For 1 day, he delivered 3 packages
For 2 days, he delivered 5 packages
For 4 days, he delivered 6 packages
For 6 days, he delivered 8 packages
Answer:
For this question, the weights are not explicitly mentioned but they are actually the days divided by the total number of days. We know that the total number of days being observed are 14, and we have a breakdown of a set of days so we could calculate the numbers such that they would be between 0 and 1 like so:
weights: 1/14, 1/14, 2/14, 4/14, 6/14
Now that we have our weights, we can calculate the weighted average by multiplying the weights by the number of packages per day:
=(1/14 × 1)+(1/14 x 3)+(2/14 × 5)+(4/14 x 6)+(6/14 x 8)
=0.0714+0.214+0.7142+1.714+3.43
=6.14
Question 2:
Let’s say you are a college student worried about whether the teacher calculated your grade correctly. You know the grade breakdown, your grades and the fact that the final course grade uses a weighted average. Your teacher says you got a XX% in your course. Is this correct given the below information about your course grades?
Answer:
This question is quite similar to the one shown in the previous section. All we need to do is to multiply the weights to the grades, and sum them all up:
=0.3*96+0.2*63+0.2*77+0.3*89
=28.8+12.6+15.4+26.7
=83.5
Question 3:
If we are given that the data points, 77, 66, 67 and 78 are weighted 0.8, 0.06, 0.04, and 0.1, then what would the weight average be?
Answer:
Similar to the previous question, we will follow the weighted average formula that we have seen previously, we should get:
=0.8*77+0.06*66+0.04*67+0.1*78
=61.6+3.96+2.68+7.8
=76.04
Question 4:
If we are given that the data points, 10, 11, 15, 8, and 20 are weighted 0.1, 0.5, 0.05, 0.15, and 0.25, then what would the weight average be?
Answer:
Following the weighted average formula that we have seen previously, we should get:
=0.1*10+0.5*11+0.05*15+0.15*8+0.25*20
=1+5.5+0.75+1.2+5
=13.45
