No.45

Mean Deviation

The mean of the distances of each value from their mean.

Yes, we use "mean" twice: Find the mean ... use it to work out distances ... then find the mean of those!

Three steps:

• 1. Find the mean of all values
• 2. Find the distance of each value from that mean (subtract the mean from each value, ignore minus signs)
• 3. Then find the mean of those distances

Like this:

Example: the Mean Deviation of 3, 6, 6, 7, 8, 11, 15, 16

Step 1: Find the mean:

Mean =	3 + 6 + 6 + 7 + 8 + 11 + 15 + 16	=	72	= 9
	8		8

Step 2: Find the distance of each value from that mean:

Value	Distance from 9
3	6
6	3
6	3
7	2
8	1
11	2
15	6
16	7

Which looks like this:

Step 3. Find the mean of those distances:

Mean Deviation =	6 + 3 + 3 + 2 + 1 + 2 + 6 + 7	=	30	= 3.75
	8		8

So, the mean = 9, and the mean deviation = 3.75

It tells us how far, on average, all values are from the middle.

In that example the values are, on average, 3.75 away from the middle.

For deviation just think distance

Formula

The formula is:

Mean Deviation =	Σ|x - μ|
	N

Let's learn more about those symbols!

Firstly:

• μ is the mean (in our example μ = 9)
• x is each value (such as 3 or 16)
• N is the number of values (in our example N = 8)

Absolute Deviation

Each distance we calculated is called an Absolute Deviation, because it is the Absolute Value of the deviation (how far from the mean).

To show "Absolute Value" we put "|" marks either side like this: |-3| = 3 For any value x: Absolute Deviation = |x - μ|

From our example, the value 16 has Absolute Deviation = |x - μ| = |16 - 9| = |7| = 7

And now let's add them all up ...