Applied Statistics and Probability for Engineers

Figure 6-10 A

cumulative distribution

plot of the compressive

strength data from

Minitab.

6-4 FREQUENCY DISTRIBUTIOINS AND HISTOGRAMS 205

80

0

10

20

100 120 140 160 180 200 220 240

Strength

Frequency

Figure 6-9 A histogram of the compressive strength

data from Minitab with nine bins.

Figure 6-8 shows a histogram of the compressive strength data from Minitab. The “de-

fault” settings were used in this histogram, leading to 17 bins. We have noted that histograms

may be relatively sensitive to the number of bins and their width. For small data sets, his-

tograms may change dramatically in appearance if the number and/or width of the bins

changes. Histograms are more stable for larger data sets, preferably of size 75 to 100 or more.

Figure 6-9 shows the Minitab histogram for the compressive strength data with nine bins. This

is similar to the original histogram shown in Fig. 6-7. Since the number of observations is

moderately large (n80), the choice of the number of bins is not especially important, and

both Figs. 6-8 and 6-9 convey similar information.

Figure 6-10 shows a variation of the histogram available in Minitab, the cumulative fre-

quency plot.In this plot, the height of each bar is the total number of observations that are less

than or equal to the upper limit of the bin. Cumulative distributions are also useful in data in-

terpretation; for example, we can read directly from Fig. 6-10 that there are approximately 70

observations less than or equal to 200 psi.

When the sample size is large, the histogram can provide a reasonably reliable indicator of

the general shapeof the distribution or population of measurements from which the sample

was drawn. Figure 6-11 presents three cases. The median is denoted as. Generally, if the data

are symmetric, as in Fig. 6-11(b), the mean and median coincide. If, in addition, the data have

only one mode (we say the data are unimodal), the mean, median, and mode all coincide. If the

data are skewed(asymmetric, with a long tail to one side), as in Fig. 6-11(a) and (c), the mean,

median, and mode do not coincide. Usually, we find that modemedianmeanif the

x~

Figure 6-8 A histogram of the compressive strength

data from Minitab with 17 bins.

0

100 150 200 250

Strength

5

10

Frequency

0

100 150 200 250

Strength

40

10

20

30

50

60

70

80

Cumulative frequency

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