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Chapter 18: A. Basic Statistics and Applications 209
Standard Deviation
Standard deviation is obtained by taking the positive square root (with positive
sign) of the variance. The population standard deviation s and the sample standard
deviation s are defined as follows:
1
( )
2 i
s =+
N
x x
i
N
2
(18.9)
s =+
( x )
2
1
2 i
x
i
n 1 n
(18.10)
Empirical Rule. Now we see how the standard deviation of a data set helps
to measure the variability of the data. If the data have a distribution that is approximately
bell-shaped, the following rule, known as the empirical rule, can be used
EXAMPLE 18.10
The following data give the length (in millimeters) of material chips removed during a
machining operation.
4, 2, 5, 1, 3, 6, 2, 4, 3, 5
Calculate the variance and the standard deviation for the data.
Solution:
There are three simple steps involved in calculating the variance of any data set:
Step 1. Calculate x i , the sum of all the data values. Thus we have
x i = 4 + 2 + 5 + 1 + 3 + 6 + 2 + 4 + 3 + 5 = 35.
Step 2. Calculate x i2 , the sum of squares of all the observations, that is,
x i2 = 4 2 + 2 2 + 5 2 + 1 2 + 3 2 + 6 2 + 2 2 + 4 2 + 3 2 + 5 2 = 145.
Step 3. Since the sample size is n = 10, by inserting the values x i and x i2 ,
calculated in step 1 and step 2 in formula 18.9, we have
Part IV.A.2
=
10 1 145 35
10
2
1
s 2
( )
1
145 122 5 2 5
= 9 ( . )= . .
The standard deviation is obtained by taking the square root of the variance, that is
s = 25 . = 158 . .
Note. It is important to remember that the value of s 2 , and therefore of s, is always
greater than zero, except where all the data values are equal, in which case it is zero.