194 CHAPTER 6 RANDOM SAMPLING AND DATA DESCRIPTIONthe squared deviations about the sample average must be used instead. However, the obser-
vations tend to be closer to their average, , than to the population mean, Therefore, to
compensate for this we use n1 as the divisor rather than n. If we used nas the divisor in the
sample variance, we would obtain a measure of variability that is, on the average, consistently
smaller than the true population variance.
Another way to think about this is to consider the sample variance as being based on
degrees of freedom.The term degrees of freedomresults from the fact that the ndevi-
ations always sum to zero, and so specifying the values of any
of these quantities automatically determines the remaining one. This was illustrated in
Table 6-1. Thus, only of the ndeviations, are freely determined.
In addition to the sample variance and sample standard deviation, the sample range,or
the difference between the largest and smallest observations, is a useful measure of variabil-
ity. The sample range is defined as follows.n 1 xix,n 1x 1 x, x 2 x, p , xnxn 1s^2^2xi x .xIf the nobservations in a sample are denoted by the sample rangeisrmax 1 xi 2 min 1 xi 2 (6-6)x 1 , x 2 , p , xn,DefinitionFor the pull-off force data, the sample range is Generally, as the vari-
ability in sample data increases, the sample range increases.
The sample range is easy to calculate, but it ignores all of the information in the sample
data between the largest and smallest values. For example, the two samples 1, 3, 5, 8, and 9
and 1, 5, 5, 5, and 9, both have the same range (r8). However, the standard deviation of the
first sample is while the standard deviation of the second sample is The
variability is actually less in the second sample.
Sometimes, when the sample size is small, say the information loss associ-
ated with the range is not too serious. For example, the range is used widely in statistical qual-
ity control where sample sizes of 4 or 5 are fairly common. We will discuss some of these
applications in Chapter 16.EXERCISES FOR SECTIONS 6-1 AND 6-2n8 or 10,s 1 3.35, s 2 2.83.r13.612.31.3.6-1. Eight measurements were made on the inside diameter
of forged piston rings used in an automobile engine. The data
(in millimeters) are 74.001, 74.003, 74.015, 74.000, 74.005,
74.002, 74.005, and 74.004. Calculate the sample mean and
sample standard deviation, construct a dot diagram, and com-
ment on the data.
6-2. In Applied Life Data Analysis(Wiley, 1982), Wayne
Nelson presents the breakdown time of an insulating fluid be-
tween electrodes at 34 kV. The times, in minutes, are as fol-
lows: 0.19, 0.78, 0.96, 1.31, 2.78, 3.16, 4.15, 4.67, 4.85, 6.50,
7.35, 8.01, 8.27, 12.06, 31.75, 32.52, 33.91, 36.71, and 72.89.
Calculate the sample mean and sample standard deviation.
6-3. The January 1990 issue of Arizona Trendcontains a
supplement describing the 12 “best” golf courses in the state.
The yardages (lengths) of these courses are as follows: 6981,
7099, 6930, 6992, 7518, 7100, 6935, 7518, 7013, 6800, 7041,and 6890. Calculate the sample mean and sample standard de-
viation. Construct a dot diagram of the data.
6-4. An article in the Journal of Structural Engineering
(Vol. 115, 1989) describes an experiment to test the yield
strength of circular tubes with caps welded to the ends. The
first yields (in kN) are 96, 96, 102, 102, 102, 104, 104, 108,
126, 126, 128, 128, 140, 156, 160, 160, 164, and 170.
Calculate the sample mean and sample standard deviation.
Construct a dot diagram of the data.
6-5. An article in Human Factors(June 1989) presented
data on visual accommodation (a function of eye movement)
when recognizing a speckle pattern on a high-resolution CRT
screen. The data are as follows: 36.45, 67.90, 38.77, 42.18,
26.72, 50.77, 39.30, and 49.71. Calculate the sample mean
and sample standard deviation. Construct a dot diagram of the
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