Engineering Fundamentals: An Introduction to Engineering, 4th ed.c

640 Chapter 19 Probability and Statistics in Engineering

(19.3)

(19.4)

Therefore, the average of the deviations from the mean of the data set cannot be used to mea-

sure the spread of a given data set. What if one considers the absolute value of each deviation

from the mean? We can then calculate the average of the absolute values of deviations. The

result of this approach is shown in the third column of Table 19.4. For group A, the mean devi-

ation is 29, whereas for group B the mean deviation is 82. It is clear that the result provided by

group B is more scattered than the group A data. Another common way of measuring the dis-

persion of data is by calculating thevariance. Instead of taking the absolute values of each

deviation, one may simply square the deviations and compute their averages:

(19.5)

Notice, however, for the given example the variance yields units that are (kg /m

3

)

2

. To remedy
this problem, we can take the square root of the variance, which results in a number that is
calledstandard deviation.

(19.6)

This may be an appropriate place to say a few words about why we usen1 rather

thennto obtain the standard deviation. This is done to obtain conservative values because

(as we have mentioned) generally the number of experimental trials are few and limited.

Let us turn our attention to the standard deviations computed for each group of densities in

Table 19.5. Group A has a standard deviation that is smaller than group B’s. This shows the den-

sities reported by group A are bunched near the mean (r1000 kg /m

3

), whereas the results

reported by group B are more spread out. The standard deviation can also provide information

about the frequency of a given data set. For normal distribution (discussed in Section 19.5) of

a data set, we will show that approximately 68% of the data will fall in the interval

of ( means) to ( means), about 95% of the data should fall between ( mean 2 s)to

( mean 2 s), and almost all data points must lie between ( mean  3 s) to ( mean  3 s).

In Section 19.3, we discussed grouped frequency distribution. The mean for a grouped dis-

tribution is calculated from

(19.7)

where

xmidpoints of a given range

ffrequency of occurrence of data in the range

ngftotal number of data points

x

g 1 x f 2

n

s

R

a

n

i 1

1 xix 2

2

n 1

n

a

n

i 1

1 xix 2

2

n 1

a

n

i 1

dinxnx 0

a

n

i 1

di 

a

n

i 1

xi 

a

n

i 1

x

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