Weighted Mean

What does "Weighted" Mean?

When we looked at finding the mean before, we assumed that each number had the same "weight". What we mean by "weight" is that some data points contribute to the data set more than others.

What is a "Weighted Mean"?

How we perform mean (or average) at the moment is adding all the values in the data set together, and then dividing by the number of data points in the set. I will refer to this as the aritmetic mean.

By finding the arithmetic mean, we can assume that each number has the same weight, for example:

Say we have the data set $1,1,2,3,4$. If we want to find the mean of the set, we will do: $$\frac{1+1+2+3+4}{5}$$ Where $5$ is the total number of points in the dataset. Performing this calculation, we find that we get a mean of $\frac{11}{5}$, or $2.2$.

We can think that each of these numbers has a "weight" of $\frac{1}{5}$, since there are $5$ numbers. Therefore we could also do: $$\frac{1}{5} \times 1 + \frac{1}{5} \times 1 + \frac{1}{5} \times 2 + \frac{1}{5} \times 3 + \frac{1}{5} \times 4$$

How to Find a Weighted Mean

Let's have the same data set as above, except that the weight of $4$ is $0.6$, and the rest of the numbers have a weight of $0.1$. Therefore, we would calculate our mean by doing: $$0.1 \times 1 + 0.1 \times 1 +0.1 \times 2 + 0.1 \times 3 + 0.6 \times 4$$ This gives us a weighted mean of $3.1$. This is $0.9$ higher than the arithmetic mean due to the fact that it is being "pulled" to this higher number by the weight of the $4$.

Let's do another example.

Example of Weighted Means

Kevyn wants to buy a new drum set, and decides on the following rating system to justify his purchase.

Bass Quality: $50$% Cymbal Quality: $30$% Color: $20$%

The following scores are out of $10$

The Jersey drum set gets $8$ for bass quality, $6$ for cymbal quality and $7$ for color.

The Pitt drum set gets $9$ for bass quality, $4$ for cymbal quality and $6$ for color.

Which drum set is best?

Solution Jersey Drum Set: $0.5 \times 8 + 0.3 \times 6 + 0.2 \times 7 = 7.2$ Pitt Drum Set: $0.5 \times 9 + 0.3 \times 4 + 0.2 \times 6 = 6.9$

Kevyn should buy the Jersey drum set, since its weighted mean is higher than that of the Pitt drum set.

Weights that Don't Add to $1$

If the sum of the weights do not add up to $1$, you have to divide by the sum of weights. I'll show you an example to solidify this concept.

Example of Weights that Don't Add to $1$

Lisa usually works $7$ days a week, but sometimes just $1,2$, or $5$ days.

For $2$ weeks, she only worked $1$ day each week

For $10$ weeks, she only worked $2$ days each week

For $8$ weeks, she only worked $5$ days each week

For $32$ weeks, she worked all $7$ days

For this example, we will use the number of weeks as the weighting, therefore:

Weeks x Days: $(2 \times 1) + (10 \times 2) + (8 \times 5) + (32 \times 7) = 286$

Weeks: $2 + 10 + 8 + 32 = 52$

Mean: $\frac{286}{52} = 5.5$ days

On average, during the $52$ weeks she worked, she worked $5.5$ days per week.