Applied Statistics and Probability for Engineers

204 CHAPTER 6 RANDOM SAMPLING AND DATA DESCRIPTION

(1) Label the bin (class interval) boundaries on a horizontal scale.

(2) Mark and label the vertical scale with the frequencies or the relative

frequencies.

(3) Above each bin, draw a rectangle where height is equal to the frequency (or rel-

ative frequency) corresponding to that bin.

Constructing a

Histogram (Equal

Bin Widths)

observations. The last row of Table 6-4 expresses the relative frequencies on a cumulative ba-

sis. Frequency distributions are often easier to interpret than tables of data. For example, from

Table 6-4 it is very easy to see that most of the specimens have compressive strengths between

130 and 190 psi and that 97.5 percent of the specimens fail below 230 psi.

The histogram is a visual display of the frequency distribution. The stages for construct-

ing a histogram follow.

70

0 0

90 110 130 150 170 190 210 230 250

Compressive strength (psi)

5

10

15

20

25

0.0625

0.1250

0.1895

0.2500

0.3125

Relative frequency

Frequency

Figure 6-7 Histogram

of compressive strength

for 80 aluminum-

lithium alloy

specimens.

Figure 6-7 is the histogram for the compression strength data. The histogram, like the stem-

and-leaf diagram,provides a visual impression of the shape of the distribution of the meas-

urements and information about the central tendency and scatter or dispersion in the data.

Notice the symmetric, bell-shaped distribution of the strength measurements in Fig. 6-7. This

display often gives insight about possible choices of probability distribution to use as a model

for the population. For example, here we would likely conclude that the normal distribution

is a reasonable model for the population of compression strength measurements.

Sometimes a histogram with unequal bin widthswill be employed. For example, if the

data have several extreme observations or outliers, using a few equal-width bins will result

in nearly all observations falling in just of few of the bins. Using many equal-width bins will

result in many bins with zero frequency. A better choice is to use shorter intervals in the region

where most of the data falls and a few wide intervals near the extreme observations. When the

bins are of unequal width, the rectangle’s area(not its height) should be proportional to the

bin frequency. This implies that the rectangle height should be

In passing from either the original data or stem-and-leaf diagram to a frequency distribu-

tion or histogram, we have lost some information because we no longer have the individual

observations. However, this information loss is often small compared with the conciseness

and ease of interpretation gained in using the frequency distribution and histogram.

Rectangle height

bin frequency

bin width

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