Mean deviation: Definition and examples
This article will be about The main concept of the mean deviation, with formula and examples.
Mean deviation definition
The mean deviation is a dispersion measure, what it does is express how much do the elements of a set vary from the arithmetic mean, the mean deviation does not make any difference between the deviation above the mean or deviation below the mean, the thing that matters here is the average deviation from the mean, this is why the mean deviation formula contains absolute values.
For example, if we have a set with the following numbers: 1, 3 and 5, in this case the arithmetic mean is 3, as we can see the mean is equals to the second number, the first number is 2 units lower than the mean and the third number is 2 units higher than the mean, but like we said, in the mean deviation is not important the direction of the deviation.
The mean deviation in statistics is used to know the composition of a set. We already know that the arithmetic mean is the average of the values of a data set, but the mean deviation is a complement for the arithmetic mean, because even when 2 sets have the same mean, when we decompose the sets, they could be much different than the expected, for example, the set A = {1,2,90,90} and the set B = {45,44,47,48} have the same mean (46), but when we see the elements of each data set we can see that the elements of A are much more dispersed, while the elements of B are much closer to the mean, and is in this case where the mean deviation is useful, to define how disperse are the elements of a set.
Mean deviation formula
As a reminder, an absolute value is the numeric value of the number, this means that we are not going to take into account the sign of the number, for example, the absolute value of -4 is 4. When we use the absolute value, it is expressed with the number with two lateral bars, like this: |-a| = a, it is necessary to remind this because the mean deviation formula uses absolute values.
Xi: is each value of a set, x is the mean of the set, so if we do not have the arithmetic mean we must find it, and N is the number of elements.
- Formula
- dm = Σ |xi - x|n
Examples of the mean deviation
Example 1: Find the mean of the following elements
{1, 3, 7, 12, 24}
- First we find the mean
- x = 1 + 3 + 7 + 12+ 245
- x = 475
- x = 9.4
- Now we find the mean deviation
- dm = Σ |xi - x|n
- dm = | 1 - 9.4 | + | 3 - 9.4 | + | 7 - 9.4| + | 12 - 9.4 | + | 24 - 9.4 |5
- dm = 8.4 + 6.4 + 2.4 + 2.6 + 14.65
- dm = 34.45
- dm = 6.88
Example 2: Find the mean of the 5 first even numbers
{2, 4, 6, 8, 10}
- Calculate the mean
- x = 2 + 4 + 6 + 8 + 105
- x = 125
- x = 6
- And now the mean deviation
- dm = Σ |xi - x|n
- dm = | 2 - 6 | + | 4 - 6 | + | 6 - 6 | + | 8 - 6 | + | 10 - 6 |5
- dm = 4 + 2 + 0 + 2 + 45
- dm = 125
- dm = 2.4
Example 3: The height of 4 basketball players was taken , calculate the mean deviation of the height of the players.
{1.67, 1.82, 1.75, 1.80}
- Calculate the mean
- x = 1.67 + 1.82 + 1.75 + 1.804
- x = 7.044
- x = 1.76
- And then we find the mean
- dm = Σ |xi - x|n
- dm = | 1.67 - 1.76 | + | 1.82 - 1.76 | + | 1.75 - 1.76 | + | 1.80 - 1.76 |4
- dm = 0.09 + 0.06 + 0.01 + 0.044
- dm = 0.24
- dm = 0.05cm