In this exercise we consider how to deal with class lengths that are unequal (and with open-ended classes) when setting up histograms. Often data are published in this form and we wish to construct a histogram. An example is provided by data concerning the benefits of ISO 9000 registration published by CEEM Information Services. According to CEEM:'°
ISO 9000 is a series of international standards for quality assurance management systems. It establishes the organizational structure and processes for assuring that the production of goods or services meet a consistent and agreed-upon level of quality for a company's customers.
CEEM presents the results of a Quality Systems Update/Deloitte & Touche survey of ISO 9000-registered companies conducted in July 1993. Included in the results is a summary of the total annual savings associated with ISO 9000 implementation for surveyed companies. The findings (in the form of a frequency distribution of ISO 9000 savings) are given on the page margin. Notice that the classes in this distribution have unequal lengths and that there is an open-ended class (>$500K).
To construct a histogram for these data, we select one of the classes as a base. It is often convenient to choose the shortest class as the base (although it is not necessary to do so). Using this choice, the 0 to $10K class is the base. This means that we will draw a rectangle over the 0 to $1()K class having a height equal to 162 (the frequency given for this class in the published data). Because the other classes are longer than the base, the heights of the rectangles above these classes will be adjusted. Remembering that the area of a rectangle positioned over a particular class should represent the relative proportion of measurements in the class, we proceed as follows. The length of the $10K to 25K class differs from the base class by a factor of (25 - 10)/( 10 - 0) = 3/2, and. therefore, we make the height of the rectangle over the $1()K to 25K class equal to (2/3)(62) = 41.333. Similarly, the length of the $25K to 50K class differs from the length of the base class by a factor of (50 - 25)/(10 - 0) = 5/2, and, therefore, we make the height of the rectangle over the $25K to 50K class equal to (2/5X53) = 21.2.
a. Use the procedure just outlined to find the heights of the rectangles drawn over all the other classes (with the exception of the open-ended class, > $500K).
b. Draw the appropriate rectangles over the classes (except for > $500 K). Note that the $250K to 500K class is a lot longer than the others. There is nothing wrong with this as long as we adjust its rectangle's height.
c. We complete the histogram by placing a star (*) to the right of > $500 K on the scale of measurements and by noting "37" next to the * to indicate 37 companies saved more than $500K. Complete the histogram by doing this.