Applied Statistics and Probability for Engineers

6-1 DATA SUMMARY AND DISPLAY 191

hypothetical population,because it does not physically exist. Sometimes there is an actual

physical population, such as a lot of silicon wafers produced in a semiconductor factory.

In previous chapters we have introduced the mean of a probability distribution, denoted

. If we think of a probability distribution as a modelfor the population, one way to think
of the mean is as the average of all the measurements in the population. For a finite popula-
tion with Nmeasurements, the mean is

(6-2)

The sample mean, , is a reasonable estimate of the population mean, . Therefore, the engi-

neer designing the connector using a 332-inch wall thickness would conclude, on the basis

of the data, that an estimate of the mean pull-off force is 13.0 pounds.

Although the sample mean is useful, it does not convey all of the information about a

sample of data. The variability or scatter in the data may be described by the sample variance

or the sample standard deviation.

x



a

N

i 1

xi

N



x = 13

12 14 15

Pull-off force

Figure 6-1 The

sample mean as a

balance point for a

system of weights.

If is a sample of nobservations, the sample varianceis

(6-3)

The sample standard deviation,s, is the positive square root of the sample variance.

s^2 

a

n

i 1

1 xix 22

n 1

x 1 , x 2 ,p, xn

Definition

The units of measurements for the sample variance are the square of the original units of

the variable. Thus, if xis measured in pounds, the units for the sample variance are (pounds)^2.

The standard deviation has the desirable property of measuring variability in the original units

of the variable of interest, x.

How Does the Sample Variance Measure Variability?

To see how the sample variance measures dispersion or variability, refer to Fig. 6-2, which

shows the deviations for the connector pull-off force data. The greater the amount of

variability in the pull-off force data, the larger in absolute magnitude some of the deviations

will be. Since the deviations always sum to zero, we must use a measure of vari-

ability that changes the negative deviations to nonnegative quantities. Squaring the deviations

is the approach used in the sample variance. Consequently, if is small, there is relatively

little variability in the data, but if s^2 is large, the variability is relatively large.

s^2

xix xix

xix

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