Editing Weighted arithmetic mean (section)

== Examples ==
=== Basic example ===
Given two school {{nowrap|classes{{tsp}}{{mdash}}{{tsp}}one}} with 20 students, one with 30 {{nowrap|students{{tsp}}{{mdash}}{{tsp}}and}} test grades in each class as follows:

:Morning class = {62, 67, 71, 74, 76, 77, 78, 79, 79, 80, 80, 81, 81, 82, 83, 84, 86, 89, 93, 98}
{{line-height|2|}}
: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):
<math display="block">\bar{x} = \frac{4300}{50} = 86.</math>

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":

:<math>\bar{x} = \frac{(20\times80) + (30\times90)}{20 + 30} = 86.</math>

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:
:<math>\frac{20}{20 + 30} = 0.4</math>

:<math>\frac{30}{20 + 30} = 0.6</math>

Then, apply the weights like this:
:<math>\bar{x} = (0.4\times80) + (0.6\times90) = 86.</math>