1.4. Measures of Spread http://www.ck12.org
Standard Deviation
Thestandard deviationis an extremely important measure of spread that is based on the mean. Recall that the
mean is the numerical balancing point of the data. One way to measure how the data is spread is to look at how far
away the values are from the mean. The difference between the actual value and the mean is called thedeviation.
Written symbolically it would be:
Deviation=x−xLet’s take a simple data set of three randomly selected individuals’ shoe sizes:
912 , 1112 ,and 12
The mean of this data set is 11. The deviations then would be as follows:
TABLE1.6: Table of Deviations
x x−x ̄
9. 5 9. 5 − 11 =− 1. 5
11. 5 11. 5 − 11 = 0. 5
12 12 − 11 = 1Notice that the deviation of a point that is less than the mean is negative. Points that are above the mean have positive
deviations.
We need a statistic that can summarizeallof the deviations. The standard deviation is such a summary. It is a
measure of the “typical” or “average” deviation for all of the data points from the mean. However, the very property
that makes the mean so special also makes it tricky to calculate a standard deviation. Because the mean is the
balancing point of the data, when you add the deviations, they sum to 0, in effect canceling each other out.
TABLE1.7: Table of Deviations, Including the Sum.
Observed Data Deviations
9. 5 9. 5 − 11 =− 1. 5
11. 5 11. 5 − 11 = 0. 5
12 12 − 11 = 1
Sum of the deviations→ − 1. 5 + 0. 5 + 1 = 0So we need all the deviations to be positive before we add them up. One way to do this would be to simply make
them positive by taking their absolute values. This is a technique we use for a similar measure called themean
absolute deviation, but for the standard deviation, we square all the deviations. The square of any real number is
always positive.
TABLE1.8:
Observed Data Deviations (x−x ̄^2 )
9. 5 − 1. 5 − 1. 52 = 2. 25
11. 5 0. 5 0. 52 = 0. 25
12 1 12 = 1
Sum of the deviations= 0Now find the sum of the squared deviations: