Calculate Average Deviation

You're in a small stats class of 10 students. A 100-question test is given, and the test scores are 70, 73, 74, 75, 76, 77, 78, 78, 79. The instructor says the class average is exactly 75.

A few weeks later, a second 100-question test is offered, and the scores this time are 47, 59, 66, 73, 78, 80, 84, 85, 89, 89. The class average is again 75.

Despite the class as a whole appearing to perform similarly on both exams, the second set of numbers are more spread out, showing a wider range of performances. This is reflected in a larger standard deviation and average deviation, two measures of variation in statistics. Sometimes you need to find average variation in addition to, or instead of, standard deviation.

When you add up a set of numbers and divide by the total in the set, you obtain what is called the average or mean, often represented by the Greek letter mu (μ). In the examples in the introduction, both sets of test scores add up to 750 total points, and with 10 students in the class, the average (mean) is thus 750/10 = 75 for both tests.

Variance, standard deviation and average deviation are terms that relate to the range of values in the set and are called measures of variation. A closely clustered set of data like the first exam above would have lower values for these metrics than the more widely spaced second data set from the other exam.

The standard deviation from the mean in statistics, usually just called "standard deviation," is the square root of a measure called variance. It is a way to get both a formal and a visual sense of the spread of a group of related numbers.

For example, if two exams both have averages of 75 questions out of 100, but the standard deviation is 5 in the first case but 15 in the second, then the spread was larger for the second exam, suggesting a greater range of preparation, student knowledge, etc.

The standard deviation formula is:

\(Standard\, deviation = \sqrt\frac{∑\limits_{i=1}^{n}(x−\mu)^2}{n – 1}\)

Here μ is the average, x is a given data point in the set and n is the number of points in the set. The funny sigma signal means "summation," and means that you perform the operation for all points in the set n=1, n=2 and so on, and then add up (sum) the results.

• The n

−

1 in the denominator becomes n when the equation is applied to whole populations instead of samples of the population.

The average deviation from the mean in statistics is similar to the standard deviation. It is not quite the same because it does not involve the squaring of the differences between individual data points and the mean. Instead, the absolute value of that difference is used (this is the reason for the vertical bars in the equation below, which translate to "change negative results to positive results and leave positive results alone."

The purpose of average deviation (also called mean deviation, just as "average" and "mean" are synonymous in statistics) is essentially the same as that of standard deviation. The mean deviation formula is:

\(Average\, deviation =∑\limits_{i=1}^{n}|x−\mu|\)

In a small set of numbers, you can find the mean and the associated values individually on a calculator and perform the required sums yourself, but it is easier to use your calculator to do this or an online tool. If you do not know how to access the relevant functions on your calculator, you should be able to Google the documentation for your particular machine.