IV.

Tabulation and Presentation of Data

The collected data must be arranged, tabulated, and presented to permit ready and meaningful analysis and interpretation. To study and interpret the examination-grade distribution in a class of 30 pupils, for instance, the grades are arranged in ascending order: 30, 35, 43, 52, 61, 65, 65, 65, 68, 70, 72, 72, 73, 75, 75, 76, 77, 78, 78, 80, 83, 85, 88, 88, 90, 91, 96, 97, 100, 100. This progression shows at a glance that the maximum is 100, the minimum 30, and the range, or difference, between the maximum and minimum is 70.

In a cumulative-frequency graph, such as Fig. 1, the grades are marked on the horizontal axis and double marked on the vertical axis with the cumulative number of the grades on the left and the corresponding percentage of the total number on the right. Each dot represents the accumulated number of students who have attained a particular grade or less. For example, the dot A corresponds to the second 72; reading on the vertical axis, it is evident that there are 12, or 40 percent, of the grades equal to or less than 72.

In analyzing the grades received by 10 sections of 30 pupils each on four examinations, a total of 1200 grades, the amount of data is too large to be exhibited conveniently as in Fig. 1. The statistician separates the data into suitably chosen groups, or intervals. For example, ten intervals might be used to tabulate the 1200 grades, as in column (a) of the accompanying frequency-distribution table; the actual number in an interval, called the frequency of the interval, is entered in column (c). The numbers that define the interval range are called the interval boundaries. It is convenient to choose the interval boundaries so that the interval ranges are equal to each other; the interval midpoints, half the sum of the interval boundaries, are simple numbers, because they are used in many calculations. A grade such as 87 will be tallied in the 80-90 interval; a boundary grade such as 90 may be tallied uniformly throughout the groups in either the lower or upper intervals. The relative frequency, column (d), is the ratio of the frequency of an interval to the total count; the relative frequency is multiplied by 100 to obtain the percent relative frequency. The cumulative frequency, column (e), represents the number of students receiving grades equal to or less than the range in each succeeding interval; thus, the number of students with grades of 30 or less is obtained by adding the frequencies in column (c) for the first three intervals, which total 53. The cumulative relative frequency, column (f), is the ratio of the cumulative frequency to the total number of grades.

The data of a frequency-distribution table can be presented graphically in a frequency histogram, as in Fig. 2, or a cumulative-frequency polygon, as in Fig. 3. The histogram is a series of rectangles with bases equal to the interval ranges and areas proportional to the frequencies. The polygon in Fig. 3 is drawn by connecting with straight lines the interval midpoints of a cumulative frequency histogram.

Newspapers and other printed media frequently present statistical data pictorially by using different lengths or sizes of various symbols to indicate different values.