# Weighted average

A weighted average (or weighted arithmetic mean) is an average of a data set in which certain data points carry more weight; in other words, certain data points contribute more to the final average of the data set. Normally, when computing an average, each data point carries the same weight.

For example, the average of the set {1, 3, 7, 10, 15} is:

In this case, each data point has an equal weight of 1/5. This would not be the case with a weighted average. For example, weighted averages are commonly used when determining a student's final grade in a class. Exam, quiz, and homework scores typically carry a different weight. A student may achieve an average score of 90 for their homework, 85 for their quizzes, and 80 for their exams. If each of these categories had the same weight, their final grade would be:

(90 + 85 + 80)/3 = 85

However, this is typically not the case. Instead, exams may be worth 50% of the final grade, quizzes may be worth 35%, and homework may be worth 15% (totaling 100%). To calculate the student's final grade, these weights would be applied to their average scores in each category as follows:

90(0.15) + 85(0.35) + 80(0.5) = 83.25

Thus, once the weight of each score is taken into account, the student's final grade is actually lower than if each score were weighted equally.

## Weighted average formula

For a set of data containing the elements {x_{1}, x_{2}, x_{3}, ..., x_{n}} with corresponding weights {w_{1}, w_{2}, w_{3}, ..., w_{n}}, the weighted average, x is:

The weight that each element carries can be shown by the frequency with which it occurs in the set. The frequency of an element can also be defined as a fraction or percentage of the set.

Example

John commutes to his classes using a bicycle. He rides his bike for a certain number of minutes each day. The table below shows the frequency with which he rides a certain number of minutes in a given day over the course of 28 days.

# of days | Minutes ridden |
---|---|

8 | 55 |

10 | 40 |

6 | 30 |

4 | 0 |

Thus, for 8/28 days, John rode 55 minutes per day; for 10/28 days, John rode 40 minutes per day; for 6/28 days, John rode 30 minutes per day; for 4/28 days, John did not ride at all. What is the average time, in minutes, that John rode his bike each day over the 4 week period?

Thus, John rode his bike an average of 36.4 minutes per day over the 4 week period.

If the weights for the elements in the set are defined by a fraction or a percentage, the sum of the weights must equal 1 or 100%, respectively, in order for all elements in the set to be accounted for. Thus, when computing a weighted average where the weights are defined in terms of a fraction or percentage, the formula for the weighted average is:

Example

A machine at a manufacturing plant that crushes rock runs at a certain capacity for given periods of each day. For 40% of the day, the machine runs at 75% capacity; for 35% of the day, it runs at 100% capacity; for the rest of the day, the machine does not run at all. What is the average capacity that the machine runs at each day?

This problem calls for the use of the following formula of the weighted average:

= 0.65

Thus, the machine runs at an average of 65% capacity over the course of a day.