You are perhaps familiar with the concept of an average, and recognize that the purpose of this statistic is to convey a sense of "normal" or "expected."

If you know, for example, that the average height of a newly discovered creature from a distant planet is 100 cm (about 3 feet, 4 inches), you would probably not be surprised to see any individual figures in the range of 80 to 120 cm. But you might be suspicious of numbers like 20 cm or 500 cm despite knowing nothing else about this creature.

When calculating a mathematical *average*, every data point is usually considered to have equal weight. That is, even data points that clearly represent rare findings and are thus *statistical outliers* for some reason (such as a person who can run a mile in under four minutes or can speak 20 languages fluently).

In some situations, however, it is desirable to treat certain kinds of data as being more or less important in order to generate an accurate picture of what is happening; this is where the **weighted average** comes in.

## What Is an Average?

A basic average, or mean, is just the **sum of all the observations in a sample divided by the number of observations in the sample**. If someone has five children, and their weights are 20, 35, 80, 100 and 145 pounds, their average weight is (20 +35 + 80 +100 + 145)/5 = 280/5 = 56 pounds.

Notice that in this simple calculation, all of the data points are treated as having the same importance in the calculation of average. This is evident from the fact that none of the points is manipulated in any way (e.g., multiplied or divided by another number) before the division step occurs. If this sounds weird, keep reading.

## Why Use Averages?

Averages, usually simple averages with no weighting, paint a statistical picture of what people have good reason to expect. If you take a quiz and are told that the average score among the 25 students in the class is 40 percent, and your score is 45, you know that in spite of getting fewer than half of the questions right, you did a little better than a "typical" student.

Averages offer solid information for planning and other civic purposes. If the average level of air pollution in a given city is higher than the national average, then the leaders of that city should perhaps consider environmental measures a top priority.

## Weighted Average Formula

There is no fixed formula to determine a weighted average because the weights assigned to each variable can change from situation to situation. In general, the equation would be of the form:

**(Ax _{1} + Bx_{2} + Cx_{3}... + Zx_{n})/n**

Where the capital letters are coefficients corresponding to the weighing factors and n is the total number of data points in the set.

**Example:** A quiz includes 10 questions: five about science and five about history. The science questions are given twice the "weight" of the history questions.

If students get an average of four science and three history questions correct, what is the simple class average?

- The answer in this case is just (4 + 3)/10 = 7/10 = 7.

What is the *weighted* class average?

- The answer this time is [(2)(4) + (1)(3)/10] = (8 + 1)/10 = 11.

What would the weighted average be if the average scores on each part of the test were reversed, with the average science score being 3 and the average history score being 4?

- This would change the above equation to [(2)(3) + (1)(4)/10] = (8 + 1)/10 = 10.

You can see from this example that the teacher intends to reward science knowledge more that history knowledge with this quiz.

## Weighted Average Calculator

See the Resources for an example of a site that allows you to input any number of weighting coefficients and data points to find weighted averages.

References

About the Author

Kevin Beck holds a bachelor's degree in physics with minors in math and chemistry from the University of Vermont. Formerly with ScienceBlogs.com and the editor of "Run Strong," he has written for Runner's World, Men's Fitness, Competitor, and a variety of other publications. More about Kevin and links to his professional work can be found at www.kemibe.com.