Let's start by reviewing the definition for the standard deviation.Standard deviation is a measure of spread that represents the average distance between each data value and the mean. The formula for the standard deviation of a data set containing n elements { x_1, x_2, ..., x_n } with a mean value x is shown below.
Standard deviation [0.8em]
sqrt((x_1-x)^2+(x_2-x)^2+... +(x_n-x)^2/n)
The first step to calculate the standard deviation is to find the mean.
Finding the mean
The mean of a data set is defined as the sum of all the values in the data set divided by the total number of values.
Mean [0.8em]
x=x_1+x_2+x_3+ ... +x_n/n
In this case, the sum of the values is 68+ 71+ 75+ 74+ 75+ 71+ 73+ 71+ 72+ 74+ 79 = 803. As we have 11 data values, we will divide the total by 11 to find the mean.
x=803/11=73
Finding the squared deviations from the mean
Now we can find how much each value deviates from the mean by taking the difference between them. After that, we can square the deviations and add them together. This way we will find the numerator for the radicand in the standard deviation formula.
Data value, x
Deviation from the mean, x-x
Squared deviation, (x-x)^2
68
68-73 = - 5
(-5)^2=25
71
71-73=-2
(- 2)^2=4
75
75-73=2
(2)^2=4
74
74-73=1
(1)^2=1
75
75-73=2
(2)^2=4
71
71-73=-2
(- 2)^2=4
73
73-73=0
(- 0)^2=0
71
71-73=-2
(- 2)^2=4
72
72-73=-1
(- 1)^2=1
74
74-73=1
(1)^2=1
79
79-73=6
(6)^2=36
Total:
84
Since we have 11 values, we have to the divide the total of the squared deviations by 11. The squared root of this quotient will be the standard deviation.
