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194 CHAPTER 6 RANDOM SAMPLING AND DATA DESCRIPTION

the squared deviations about the sample average must be used instead. However, the obser-
x
vations tend to be closer to their average, , than to the population mean, .xx i Therefore, to
compensate for this we use n 1 as the divisor rather than n. If we used n as 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 2 .
Another way to think about this is to consider the sample variance s 2 as being based on
n 1 degrees of freedom. The term degrees of freedom results from the fact that the n devi-
ations x x, x x, p , x x always sum to zero, and so specifying the values of any
n
2
1
n 1 of these quantities automatically determines the remaining one. This was illustrated in
Table 6-1. Thus, only n 1 of the n deviations, x i x, 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.

Definition
If the n observations in a sample are denoted by x , x , p , x , the sample range is
2
n
1
2 min1x 2 (6-6)
r max1x i i

For the pull-off force data, the sample range is r 13.6 12.3 1.3. 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 (r 8). However, the standard deviation of the
first sample is s 3.35, while the standard deviation of the second sample is s 2.83. The
1
2
variability is actually less in the second sample.
Sometimes, when the sample size is small, say n 8 or 10, 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 6-2
6-1. Eight measurements were made on the inside diameter and 6890. Calculate the sample mean and sample standard de-
of forged piston rings used in an automobile engine. The data viation. Construct a dot diagram of the data.
(in millimeters) are 74.001, 74.003, 74.015, 74.000, 74.005, 6-4. An article in the Journal of Structural Engineering
74.002, 74.005, and 74.004. Calculate the sample mean and (Vol. 115, 1989) describes an experiment to test the yield
sample standard deviation, construct a dot diagram, and com- strength of circular tubes with caps welded to the ends. The
ment on the data. first yields (in kN) are 96, 96, 102, 102, 102, 104, 104, 108,
6-2. In Applied Life Data Analysis (Wiley, 1982), Wayne 126, 126, 128, 128, 140, 156, 160, 160, 164, and 170.
Nelson presents the breakdown time of an insulating fluid be- Calculate the sample mean and sample standard deviation.
tween electrodes at 34 kV. The times, in minutes, are as fol- Construct a dot diagram of the data.
lows: 0.19, 0.78, 0.96, 1.31, 2.78, 3.16, 4.15, 4.67, 4.85, 6.50, 6-5. An article in Human Factors (June 1989) presented
7.35, 8.01, 8.27, 12.06, 31.75, 32.52, 33.91, 36.71, and 72.89. data on visual accommodation (a function of eye movement)
Calculate the sample mean and sample standard deviation. when recognizing a speckle pattern on a high-resolution CRT
6-3. The January 1990 issue of Arizona Trend contains a screen. The data are as follows: 36.45, 67.90, 38.77, 42.18,
supplement describing the 12 “best” golf courses in the state. 26.72, 50.77, 39.30, and 49.71. Calculate the sample mean
The yardages (lengths) of these courses are as follows: 6981, and sample standard deviation. Construct a dot diagram of the
7099, 6930, 6992, 7518, 7100, 6935, 7518, 7013, 6800, 7041, data.