Mean Deviation

One of the most useful tools in the arsenal of an experimentalist are statistical tools. To be able to make sense of experimental data, one needs to understand many statistical concepts. Statistics is the prime branch of applied mathematics, which focuses on unearthing minute details from the mass of data generated through experiments. Of the many statistics concepts which you need to know about, one of the most important ones is mean deviation.

Mean or average value of any set of data points is obtained by summing all the values of a data set and dividing it with quantity or number of data points. To put it simply, to get mean, just add up all the values of data points and divide it by their total count.

What is Mean Deviation?

It is defined as the average of all the absolute deviations from the mean (of an entire distribution) for all data points. To get the mean deviation, you must subtract mean from each one of the data values in the distribution, take the absolute or mod value of each one, add them all together and divide it by the total number of data points. If you make a mean deviation vs standard deviation comparison, you'll see that the modulus part makes the calculation of the former, a bit more difficult compared to the latter.

Formula

Mean Deviation = [{Σ_n=1^N |x_n - X_Mean|}/{N}]

Here N is the total number of data points and X_Mean is the mean value of the distribution. When we take the absolute value of any number, remember that it's always positive. For example, |-3| equals to 3 and not -3.

Calculation

Let us consider a data set with the following five points - 2,5,10,6,7. The first step is to find the mean of this data set, which would be:

Mean = (2 + 5 + 10 + 6 + 7)/5 = 6

The next step is to calculate the absolute deviation of each data point from the mean. Here is a table depicting absolute deviation calculation.

Data(X)	|X - XMean|
2	|(2 - 6)| = 4
5	|(5 - 6)| = 1
10	|(10 - 6)| = 4
6	|(6 - 6)| = 0
7	|(7 - 6)| = 1

Therefore, the sum of deviations from the mean will be (4 + 1 + 4 + 0 + 1), which is equal to 10. Substituting N = 5 and sum of absolute deviations to be 10, we have:

Mean Deviation = 10/5 = 2

This calculation is quite simple, once you note down and arrange the data properly in an observation table. As explained before, it gives you an idea about how much is the average deviation among the data points from the mean. This makes the maximum amplitude of deviations from the average apparent.