Frequency Distribution

A frequency distribution is a tabular representation of a survey data set used to organize and summarize the data. Specifically, it is a list of either qualitative or quantitative values that a variable takes in a data set and the associated number of times each value occurs (frequencies).

The frequency distribution is the basic building block of statistical analytical methods and the first step in analyzing survey data. It helps researchers (a) organize and summarize the survey data in a tabular [Page 293]format, (b) interpret the data, and (c) detect outliers (extreme values) in the survey data set.

For example, the following are the scores of a group of 25 students on a final math exam: 83, 75, 95, 100, 83, 85, 85, 83, 98, 89, 84, 65, 95, 98, 80, 95, 89, 75, 65, 80, 89, 80, 75, 98, and 69. The students' math scores are not listed in any order to make sense of the data or to help provide a descriptive and summary statement about the scores. From these data, as they are displayed here, it is difficult to answer questions such as the following: How many students had math scores between 80 and 90? What percentage of students had a score of 70 or less? To answer the first question, the researcher has to count up all the scores between 80 and 90. How difficult or easy this task is depends on how many math scores the researcher has. To answer the second question, the researcher needs to know how many math scores are 70 or less in the data compared to higher math scores.

To summarize this data set, the researcher needs to put the data in some sort of logical order and tally the number of times each value occurs. This simple frequency distribution is called raw (or ungrouped) frequency distribution. The necessary steps in creating the raw frequency distribution are as follows:

• Identify the lowest and highest variable values in the data set.
• List in ascending order all single values in the data set from the lowest to highest (e.g. see the column labeled “Score” in Table 1).
• Tally the number of the times the variable values occurred (e.g. see the column labeled “Tallies” in Table 1).
• Count the number of tallies for each variable value (e.g. see the column labeled “Frequency” in Table 1).

Thus, the simple frequency distribution of the listing of the 25 students' math exam scores will look like Table 1. (Of note, this table does not contain any percentages, which could be added to the table and are what is called relative frequency.)

In some situations, this simple frequency distribution tabulation is unpractical, even impossible, or simply not needed by the researcher, for instance, when the variable under consideration has continuous values with decimal points (e.g. 88.5, 75.6, 94.4) instead of discrete values (e.g. 88, 75) or when the number of possible data points (values) is too large to construct such simple frequency distribution.

In such situations, a different kind of tabulation, based on the range (interval) of values instead of a set of single values, is used. The data values are grouped into different intervals and the number of data values that belong to each interval is determined. Thus, instead of listing single variable values and tallying the frequencies for each listed value, as was done in creating the raw frequency distribution in Table 1, the researcher could use ranges (intervals) of variable values and count the frequencies for each interval. This tabulation scheme is called grouped frequency distribution.

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