What is a Weighted Average?
A weighted average is a type of mean that gives differing importance to the values in a dataset. In contrast, the regular average, or arithmetic mean, gives equal weight to all observations. The weighted average is also known as the weighted mean, and I’ll use those terms interchangeably.
Use a weighted mean when you must consider the relative significance of values in a dataset. In other words, you’re placing different weights on the values in the calculations.
For example, use a weighted average in the following situations:
- A professor weights projects, exams, and quizzes to reflect varying difficulty.
- An investor weights the share price by the number of stocks they purchase to reflect the changing prices.
In these examples, a weighted average gives differing importance to each value according to relevant criteria.
In this post, learn how to calculate a weighted average and go through two worked examples.
Weighted Average Formula
Calculating the mean is a simple process of summing all your values and dividing them by the number of values. That process gives each value an equal weight.
Now let’s see how that procedure contrasts with the weighted average calculation.
The weighted average formula is the following:
Where:
- w = the weight for each data point.
- x = the value of each data point.
Calculating the weighted average involves multiplying each data point by its weight and summing those products. Then sum the weights for all data points. Finally, divide the weight*value products by the sum of the weights. Voila, you’ve calculated the weighted mean!
Two broad calculation cases exist when using the weighted average formula:
- The weights sum to 1.
- They don’t sum to 1.
Notice how you divide the products by the sum of the weights in the denominator. Consequently, when the weights sum to one, the weighted average simply equals the sum of the products in the numerator. However, you’ll need to perform the division when the denominator does not equal one.
Let’s use the weighted mean formula to work through two examples.
Example Weighted Average Calculations
We’ll start with an example where the weights sum to one. This situation frequently occurs when someone intentionally builds the weighted average into a process. For example, a teacher might devise a grading system using weights and, for simplicity, has the weights sum to one.
The teacher has given weights ranging from 0.05 for quizzes to 0.4 for the group project. Because the teacher devised the weights to equal one, it’s easy to understand the importance of each observation. For instance, the group project accounts for 40% of the grade!
Let’s calculate the weighted mean for one student’s grade! In the column headers, I use notation that matches the weighted average formula above.
The student did well on the quizzes and exams but not so well on the group project. The resulting weighted average is 79.70.
The regular mean is 84.5, but the considerable importance of the group project brought their weighted average grade down to 79.7. Ouch!
Example Weighted Mean Calculations
Next, let’s work through an example where the weights don’t sum to one.
An investor is building up a particular stock in his portfolio. He purchases the same stock at different prices over time. He can calculate the weighted mean for the average share price. In this example, the prices are the values, and the numbers of stocks are the weights.
Here, the weights sum to 125. Consequently, we need to divide the sum of the products (2,985) by 125. The weighted average price per stock is $23.88.
In closing, I’d like to point out that the statistical concept of the expected value is a specialized weighted average that uses the probabilities of each outcome for the weights. To learn more, read my post Expected Values: Definition, Formula & Finding.
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