Fundamentals of Materials Science and Engineering: An Integrated Approach, 3e

GTBL042-07 GTBL042-Callister-v2 August 9, 2007 13:52

230 • Chapter 7 / Mechanical Properties

determined. For example, instead of asking the question, “What is the fracture

strength of this alloy?” the engineer should become accustomed to asking the ques-

tion, “What is the probability of failure of this alloy under these given circumstances?”

It is often desirable to specify a typical value and degree of dispersion (or scatter)

for some measured property; such is commonly accomplished by taking the average

and the standard deviation, respectively.

Computation of Average and Standard Deviation Values

An average value is obtained by dividing the sum of all measured values by the num-

ber of measurements taken. In mathematical terms, the averagexof some parameter

xis

x=

∑n

i= 1

xi

n

(7.26)

Computation of

average value

wherenis the number of observations or measurements andxiis the value of a

discrete measurement.

Furthermore, the standard deviationsis determined using the following expres-

sion:

Computation of

standard deviation s=

⎡

⎢

⎣

∑n

i= 1

(xi−x)^2

n− 1

⎤

⎥

⎦

1 / 2

(7.27)

wherexi,x, andnare defined above. A large value of the standard deviation corre-

sponds to a high degree of scatter.

EXAMPLE PROBLEM 7.6

Average and Standard Deviation Computations

The following tensile strengths were measured for four specimens of the same

steel alloy:

Sample Number Tensile Strength(MPa)

1 520

2 512

3 515

4 522

(a)Compute the average tensile strength.

(b)Determine the standard deviation.