Lecture Data ans statistics Applications in Business and Economics

Cumulative Distributions
A variation of the frequency distribution that provides another tabular summary of quantitative
data is the cumulative frequency distribution. The cumulative frequency distribution
uses the number of classes, class widths, and class limits developed for the frequency
distribution. However, rather than showing the frequency of each class, the cumulative frequency
distribution shows the number of data items with values less than or equal to the
upper class limit of each class. The first two columns of Table 2.7 provide the cumulative
frequency distribution for the audit time data.
To understand how the cumulative frequencies are determined, consider the class with
the description “less than or equal to 24.” The cumulative frequency for this class is simply
the sum of the frequencies for all classes with data values less than or equal to 24. For the
frequency distribution in Table 2.5, the sum of the frequencies for classes 10–14, 15–19, and
20–24 indicates that 4 _ 8 _ 5 _ 17 data values are less than or equal to 24. Hence, the cumulative
frequency for this class is 17. In addition, the cumulative frequency distribution in
Table 2.7 shows that four audits were completed in 14 days or less and 19 audits were completed
in 29 days or less.
As a final point, we note that a cumulative relative frequency distribution shows the
proportion of data items, and a cumulative percent frequency distribution shows the percentage
of data items with values less than or equal to the upper limit of each class. The cumulative
relative frequency distribution can be computed either by summing the relative
frequencies in the relative frequency distribution or by dividing the cumulative frequencies
by the total number of items. Using the latter approach, we found the cumulative relative
frequencies in column 3 of Table 2.7 by dividing the cumulative frequencies in column 2
by the total number of items (n _ 20). The cumulative percent frequencies were again computed
by multiplying the relative frequencies by 100. The cumulative relative and percent
frequency distributions show that .85 of the audits, or 85%, were completed in 24 days or
less, .95 of the audits, or 95%, were completed in 29 days or less, and so on.