Lecture Data ans statistics Applications in Business and Economics

Ogive
A graph of a cumulative distribution, called an ogive, shows data values on the horizontal
axis and either the cumulative frequencies, the cumulative relative frequencies, or the cumulative
percent frequencies on the vertical axis. Figure 2.6 illustrates an ogive for the cumulative
frequencies of the audit time data in Table 2.7.
The ogive is constructed by plotting a point corresponding to the cumulative frequency
of each class. Because the classes for the audit time data are 10–14, 15–19, 20–24, and so
on, one-unit gaps appear from 14 to 15, 19 to 20, and so on. These gaps are eliminated by
plotting points halfway between the class limits. Thus, 14.5 is used for the 10 –14 class, 19.5
is used for the 15–19 class, and so on. The “less than or equal to 14” class with a cumulative
frequency of 4 is shown on the ogive in Figure 2.6 by the point located at 14.5 on the
horizontal axis and 4 on the vertical axis. The “less than or equal to 19” class with a cumulative
frequency of 12 is shown by the point located at 19.5 on the horizontal axis and 12
on the vertical axis. Note that one additional point is plotted at the left end of the ogive. This
point starts the ogive by showing that no data values fall below the 10 –14 class. It is plotted
at 9.5 on the horizontal axis and 0 on the vertical axis. The plotted points are connected
by straight lines to complete the ogive.


Exploratory Data Analysis:
The Stem-and-Leaf Display
The techniques of exploratory data analysis consist of simple arithmetic and easy-to-draw
graphs that can be used to summarize data quickly.Onetechnique—referred to as a stem-andleaf
display—can be used to show both the rank order and shape of a data set simultaneously.
To illustrate the use of a stem-and-leaf display, consider the data in Table 2.8. These
data result from a 150-question aptitude test given to 50 individuals recently interviewed
for a position at Haskens Manufacturing. The data indicate the number of questions answered
correctly.
To develop a stem-and-leaf display, we first arrange the leading digits of each data value
to the left of a vertical line. To the right of the vertical line, we record the last digit for each
data value. Based on the top row of data in Table 2.8 (112, 72, 69, 97, and 107), the first
five entries in constructing a stem-and-leaf display would be as follows:

For example, the data value 112 shows the leading digits 11 to the left of the line and the

last digit 2 to the right of the line. Similarly, the data value 72 shows the leading digit 7
to the left of the line and last digit 2 to the right of the line. Continuing to place the last
digit of each data value on the line corresponding to its leading digit(s) provides the
following: