Weighted averages

The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The notion of weighted mean plays a role in descriptive statistics and also occurs in a more general form in several other areas of mathematics.

If all the weights are equal, then the weighted mean is the same as the arithmetic mean. While weighted means generally behave in a similar fashion to arithmetic means, they do have a few counterintuitive properties, as captured for instance in Simpson's paradox.

Examples

Basic example

Given two school classes, one with 20 students, and one with 30 students, the grades in each class on a test were:

Morning class = 62, 67, 71, 74, 76, 77, 78, 79, 79, 80, 80, 81, 81, 82, 83, 84, 86, 89, 93, 98

Afternoon class = 81, 82, 83, 84, 85, 86, 87, 87, 88, 88, 89, 89, 89, 90, 90, 90, 90, 91, 91, 91, 92, 92, 93, 93, 94, 95, 96, 97, 98, 99

The mean for the morning class is 80 and the mean of the afternoon class is 90. The unweighted mean of the two means is 85. However, this does not account for the difference in number of students in each class (20 versus 30); hence the value of 85 does not reflect the average student grade (independent of class). The average student grade can be obtained by averaging all the grades, without regard to classes (add all the grades up and divide by the total number of students):

${\bar {x}}={\frac {4300}{50}}=86.$

Or, this can be accomplished by weighting the class means by the number of students in each class. The larger class is given more "weight":

${\bar {x}}={\frac {(20\times 80)+(30\times 90)}{20+30}}=86.$

Thus, the weighted mean makes it possible to find the mean average student grade without knowing each student's score. Only the class means and the number of students in each class are needed.

Convex combination example

Since only the relative weights are relevant, any weighted mean can be expressed using coefficients that sum to one. Such a linear combination is called a convex combination.

Using the previous example, we would get the following weights:

${\frac {20}{20+30}}=0.4$

${\frac {30}{20+30}}=0.6$

Then, apply the weights like this:

${\bar {x}}=(0.4\times 80)+(0.6\times 90)=86.$

Mathematical definition

Formally, the weighted mean of a non-empty finite multiset of data $\{x_{1},x_{2},\dots ,x_{n}\},$ with corresponding non-negative weights $\{w_{1},w_{2},\dots ,w_{n}\}$ is

${\bar {x}}={\frac {\sum \limits _{i=1}^{n}w_{i}x_{i}}{\sum \limits _{i=1}^{n}w_{i}}},$

which expands to:

${\bar {x}}={\frac {w_{1}x_{1}+w_{2}x_{2}+\cdots +w_{n}x_{n}}{w_{1}+w_{2}+\cdots +w_{n}}}.$

Therefore, data elements with a high weight contribute more to the weighted mean than do elements with a low weight. The weights cannot be negative. Some may be zero, but not all of them (since division by zero is not allowed).

The formulas are simplified when the weights are normalized such that they sum up to $1$, i.e.:

$\sum _{i=1}^{n}{w_{i}'}=1$.

For such normalized weights the weighted mean is then:

${\bar {x}}=\sum _{i=1}^{n}{w_{i}'x_{i}}$.

Note that one can always normalize the weights by making the following transformation on the original weights:

$w_{i}'={\frac {w_{i}}{\sum _{j=1}^{n}{w_{j}}}}$.

Using the normalized weight yields the same results as when using the original weights:

${\begin{aligned}{\bar {x}}&=\sum _{i=1}^{n}w'_{i}x_{i}=\sum _{i=1}^{n}{\frac {w_{i}}{\sum _{j=1}^{n}w_{j}}}x_{i}={\frac {\sum _{i=1}^{n}w_{i}x_{i}}{\sum _{j=1}^{n}w_{j}}}\\&={\frac {\sum _{i=1}^{n}w_{i}x_{i}}{\sum _{i=1}^{n}w_{i}}}.\end{aligned}}$

The ordinary mean ${\frac {1}{n}}\sum _{i=1}^{n}{x_{i}}$ is a special case of the weighted mean where all data have equal weights.

If the data elements are independent and identically distributed random variables with variance $\sigma$, the standard error of the weighted mean, $\sigma _{\bar {x}}$, can be shown via uncertainty propagation to be:

${\textstyle \sigma _{\bar {x}}=\sigma {\sqrt {\sum _{i=1}^{n}w_{i}'^{2}}}}$

Resources

• How To Find The Weighted Mean and Weighted Average In Statistics, The Organic Chemistry Tutor

Licensing

Content obtained and/or adapted from:

• Weight arithmetic mean, Wikipedia under a CC BY-SA license