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Figure 6-1 The sample mean as a balance point for a system of weights.
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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 model for the population, one way to think of the mean is as the average of all the measurements in the population. For a nite population with N measurements, the mean is
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The sample mean, x, is a reasonable estimate of the population mean, . Therefore, the engineer designing the connector using a 3 32-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. De nition If x1, x2, p , xn is a sample of n observations, the sample variance is a 1xi
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The sample standard deviation, s, is the positive square root of the sample variance.
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The units of measurements for the sample variance are the square of the original units of the variable. Thus, if x is 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 xi x 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 xi x will be. Since the deviations xi x always sum to zero, we must use a measure of variability that changes the negative deviations to nonnegative quantities. Squaring the deviations is the approach used in the sample variance. Consequently, if s2 is small, there is relatively little variability in the data, but if s2 is large, the variability is relatively large.
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Figure 6-2 How the sample variance measures variability through the deviations xi x.
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Table 6-1 displays the quantities needed for calculating the sample variance and sample standard deviation for the pull-off force data. These data are plotted in Fig. 6-2. The numerator of s2 is a 1xi
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so the sample variance is s2 1.60 8 1 1.60 7 0.2286 1pounds2 2
and the sample standard deviation is s
0.48 pounds
Computation of s2 The computation of s2 requires calculation of x, n subtractions, and n squaring and adding operations. If the original observations or the deviations xi x are not integers, the deviations xi x may be tedious to work with, and several decimals may have to be carried to ensure
Table 6-1 Calculation of Terms for the Sample Variance and Sample Standard Deviation i 1 2 3 4 5 6 7 8
12.6 12.9 13.4 12.3 13.6 13.5 12.6 13.1 104.0
0.4 0.1 0.4 0.7 0.6 0.5 0.4 0.1 0.0
x2 2
0.16 0.01 0.16 0.49 0.36 0.25 0.16 0.01 1.60
numerical accuracy. A more ef cient computational formula for the sample variance is obtained as follows: a 1xi
i 1 n
x2 2 1
s2 and since x
a 1xi
x2 n 1
2xxi 2
a xi
nx2 n 1
2x a xi
n 11 n2 g i
xi, this last equation reduces to
a xi s2
a a xi b
n n 1 (6-4)
Note that Equation 6-4 requires squaring each individual xi, then squaring the sum of the xi, subtracting 1 g xi 2 2 n from g x2, and nally dividing by n 1. Sometimes this is called the i shortcut method for calculating s2 (or s). EXAMPLE 6-3 We will calculate the sample variance and standard deviation using the shortcut method, Equation 6-4. The formula gives
n 2 a xi
a a xi b
s2 and
n n 1
1353.6 10.2286 7
11042 2 8
1.60 7