Weighted Averages come up in cases surprisingly frequently. Today’s article will help you identify what weighted averages are, when to use them, how to calculate them, and what happens when the weights change.
What Are Weighted Averages and When Do We Use Them?
If you’re reading this site you probably know what an average (also called mean) is — it’s when you have a range of values for a specific quantity and you add them all up and divide by the number of values.
The average of a group of values is usually the most accurate estimation of that value, which is why we use it. (for example, if you measured your own height 100 times, added up the values, and divided by 100 you’d have a pretty good estimate of your own height).
A weighted average is slightly different, because each value doesn’t contribute equally to the final number.
Let’s take the example of grades, which most people are probably familiar with. Your grade for a class may be based on something like 20% quizzes, 20% homework, 60% tests.
In that case, you couldn’t just average your scores in each category because they don’t all contribute equally to the final grade.
That’s a weighted average, and you’ll see them anytime you have to calculate an average but the values you’re working with don’t contribute equally to your final number.
How Do We Calculate Them?
(credit: Ross 2011 Casebook Case # 1)
The easiest way to calculate weighted averages is to take each component of the weighted average, multiply it by its weight, and add these together.
Let’s use the above as an example. Just for some context, in this case we’re being asked to calculate the average annual profit per customer. Here’s how they did it:
What Happens When the Weights Change?
Now you’re asked to calculate the new average annual profit per customer with an additional $10 worth of fees every year (12 months x $0.85/month).
I’d break this calculation up into two parts:
1. What’s the New Profit Per Customer Segment?
Calculating the new profit per customer in each segment is pretty easy, just add $10 to the old profit.
Group 1: -5 + 10 = $5
Group 2: -25 + 10 = -$15
Group 3: -5 + 10 = $5
Group 4: 15 + 10 = $25
Group 5: 45 + 10 = $55
2. What’s the New Relative Size of Each Customer Segment?
The way I did it was I used a sample size of 100 customers to figure out the new relative size of each segment.
By the way, the % of Total = (# that Remain / Total) *100%, and I did a LOT of rounding just to show how I’d do it in an actual interview
3. What’s the Weighted Average?
At this point you can just take the weighted average of the size of each segment and the Profit / Segment. Note that I once again rounded and that the red numbers are negative numbers.
Now, I’m sure you’re thinking… this was actually a pretty difficult calculation. Isn’t there another way? In fact, there is (but it’s much more complicated). Here’s the answer from the casebook:
To sum it all up, use weighted averages to calculate an average when your values don’t contribute equally to the final number. In general, it’s much easier to multiply each value by its weight and add them all together at the end.
In the case where the numbers change, make sure to recalculate both the values and their weights.