30 The accompanying sample consisting of n 5 20 observations on dielectric break-

Example 4.30 The accompanying sample consisting of n 5 20 observations on dielectric break-

  down voltage of a piece of epoxy resin appeared in the article “Maximum Likelihood Estimation in the 3-Parameter Weibull Distribution (IEEE Trans. on Dielectrics and Elec. Insul., 1996: 43–55). The values of (i 2 .5)n for which z percentiles are needed are (1 2 .5)20 5 .025, (2 2 .5)20 5 .075, c, and .975.

  Observation 24.46 25.61 26.25 26.42 26.66 27.15

  z percentile

  z percentile

  4.6 Probability Plots

  Figure 4.35 shows the resulting normal probability plot. The pattern in the plot is quite straight, indicating it is plausible that the population distribution of dielectric breakdown voltage is normal.

  x 31

  z percentile

  Figure 4.35 Normal probability plot for the dielectric breakdown voltage sample

  ■ There is an alternative version of a normal probability plot in which the z per-

  centile axis is replaced by a nonlinear probability axis. The scaling on this axis is constructed so that plotted points should again fall close to a line when the sampled distribution is normal. Figure 4.36 shows such a plot from Minitab for the break- down voltage data of Example 4.30.

  Figure 4.36 Normal probability plot of the breakdown voltage data from Minitab

  A nonnormal population distribution can often be placed in one of the follow- ing three categories:

  1. It is symmetric and has “lighter tails” than does a normal distribution; that is, the density curve declines more rapidly out in the tails than does a normal curve.

  2. It is symmetric and heavy-tailed compared to a normal distribution.

  3. It is skewed.

  CHAPTER 4 Continuous Random Variables and Probability Distributions

  A uniform distribution is light-tailed, since its density function drops to zero outside

  a finite interval. The density function f (x) 5 1[p(1 1 x 2 )] for 2` , x , ` 2 is heavy-tailed, since 1(1 1 x 2 ) declines much less rapidly than does e 2x .

  Lognormal and Weibull distributions are among those that are skewed. When the points in a normal probability plot do not adhere to a straight line, the pattern will frequently suggest that the population distribution is in a particular one of these three categories.

  When the distribution from which the sample is selected is light-tailed, the largest and smallest observations are usually not as extreme as would be expected from a normal random sample. Visualize a straight line drawn through the middle part of the plot; points on the far right tend to be below the line (observed value ,z percentile), whereas points on the left end of the plot tend to fall above the straight line (observed value .z percentile). The result is an S -shaped pattern of the type pictured in Figure 4.34.

  A sample from a heavy-tailed distribution also tends to produce an S -shaped plot. However, in contrast to the light-tailed case, the left end of the plot curves downward (observed ,z percentile), as shown in Figure 4.37(a). If the underlying distribution is positively skewed (a short left tail and a long right tail), the smallest sample observations will be larger than expected from a normal sample and so will the largest observations. In this case, points on both ends of the plot will fall above

  a straight line through the middle part, yielding a curved pattern, as illustrated in Figure 4.37(b). A sample from a lognormal distribution will usually produce such a pattern. A plot of (z percentile, ln(x)) pairs should then resemble a straight line.

  x

  x

  z percentile

  z percentile

  (a)

  (b)

  Figure 4.37 Probability plots that suggest a nonnormal distribution: (a) a plot consistent with a heavy-tailed distribution; (b) a plot consistent with a positively skewed distribution

  Even when the population distribution is normal, the sample percentiles will not coincide exactly with the theoretical percentiles because of sampling variability. How much can the points in the probability plot deviate from a straight-line pattern before the assumption of population normality is no longer plausible? This is not an easy question to answer. Generally speaking, a small sample from a normal distri- bution is more likely to yield a plot with a nonlinear pattern than is a large sample. The book Fitting Equations to Data (see the Chapter 13 bibliography) presents the results of a simulation study in which numerous samples of different sizes were selected from normal distributions. The authors concluded that there is typically

  4.6 Probability Plots

  greater variation in the appearance of the probability plot for sample sizes smaller than 30, and only for much larger sample sizes does a linear pattern generally predominate. When a plot is based on a small sample size, only a very substantial departure from linearity should be taken as conclusive evidence of nonnormality.

  A similar comment applies to probability plots for checking the plausibility of other types of distributions.