Cumulative Frequency Distribution

Cumulative Frequency Tables for Distributions with Ungrouped Scores

A cumulative frequency table for distributions with ungrouped scores typically includes the scores a variable takes in a particular sample, their frequencies, and the cumulative frequency. In addition, the table may include the cumulative relative frequency or proportion, and the cumulative percentage frequency. Table 1 illustrates the frequency, cumulative frequency, cumulative relative frequency, and cumulative percentage frequency for a set of data showing the number of credits a sample of students at a college have registered for in the autumn quarter.

The cumulative frequency is obtained by adding the frequency of each observation to the sum of the frequencies of all previous observations (which is, actually, the cumulative frequency on the previous row). For example, the cumulative frequency for the first row in Table 1 is 1 because there are no previous observations. The cumulative frequency for the second row is 1 + 0 = 1. The cumulative frequency for the third row is 1 + 2 = 3. The cumulative frequency for the fourth row is 3 + 1 = 4, and so on. This means that four students have registered for 13 credits or fewer in the autumn quarter. The cumulative frequency for the last observation must equal the number of observations included in the sample.

Cumulative relative frequencies or cumulative proportions are obtained by dividing each cumulative frequency by the number of observations. Cumulative proportions show the proportion of observations that fulfill a particular criterion or less. For example, the proportion of students who have registered for 14 credits or fewer in the autumn quarter is 0.60. The cumulative proportion for the last observation (last row) is always 1.

Cumulative percentages are obtained by multiplying the cumulative proportions by 100. Cumulative percentages show the percentage of observations that fulfill a certain criterion or less. For example, 40% of students have registered for 13 credits or fewer in the autumn quarter. The cumulative percentage of the last observation (the last row) is always 100.

Grouped Cumulative Frequency Distributions

For distributions with grouped scores, the cumulative frequency corresponding to each class equals the frequency of occurrence of scores in that particular class plus the sum of the frequencies of scores in all lower classes (which is, again, the cumulative frequency on the previous row). Grouped cumulative frequency distributions are calculated for continuous variables or for discrete variables that take too many values for the list of all possible values to be useful. The cumulative distribution table for this case is very similar to Table 1, and the entries in the table are computed in a similar way. The only difference is that instead of individual scores, the first column contains classes. Table 2 is an example of a cumulative frequency table for a sample of workers’ salaries at a factory.

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