The Mode The Median

C. Analysis of Individual Observations

In the analysis of individual observations, or ungrouped data, consideration will be given to all levels of measurement to determine which descriptive measures can be used, and under what conditions each is appropriate. One of the most widely used descriptive measures is the ‘average’. One speaks of the ‘average age’, average response time’, or ‘average score’ often without being very specific as to precisely what this means. The use of the average is an attempt to find a single figure to describe or represent a set of data. Since there are several kinds of average, or measures of central tendency, used in statistics, the use of precise terminology is importantμ each ‘average’ must be clearly defined and labelled to avoid confusion and ambiguity. At least three kinds of common uses of the ‘average’ can be describedμ 1. An average provides a summary of the data. It represents an attempt to find one figure that tells more about the characteristics of the distribution of data than any other. For example, in a survey of several hundred undergraduates the average intelligence quotient was 105: this one figure summarizes the characteristic of intelligence. 2. The average provides a common denominator for comparing sets of data. For example, the average score on the Job Descriptive Index for British managers was found to be 144, this score provides a quick and easy comparison of levels of felt job satisfaction with other occupational groups. 3. The average can provide a measure of typical size. For example, the scores derived for a range of dimensions of personality can be compared to the norms for the group the sample was taken from; thus, one can determine the extent to which the score for each dimension is above, or below, that to be expected.

1. The Mode

The mode can be defined as the most frequently occurring value in a set of data; it may be viewed as a single value that is most representative of all the values or observation in the distribution of the variable under study. It is the only measure of central tendency that can be appropriately used to describe nominal data. However, a mode may not exist, and even if it does, it may not be unique: 1 2 3 4 5 6 7 8 9 10 . . . . . . . . . . . . No mode Y Y N Y N N N N Y . . . . . . . . . . . . Unimodal N 1 2 2 3 4 4 4 4 5 5 . . . . . . . . . . . . Unimodal 4 1 2 2 2 3 4 5 5 5 6 . . . . . . . . . . . . Bimodal 2, 5 1 2 2 3 4 4 5 6 6 7 . . . . . . . . . . . . Multimodal 2, 4, 6 With relatively few observations, the mode can be determined by assembling the set of data into an array. Large numbers of observations can be arrayed by means of Microsoft EXCEL, or other statistical software programs: Subject Reaction Time Array in mseconds 000123 625 460 000125 500 480 000126 480 500 000128 500 500 000129 460 500 000131 500 500 Mode 000134 575 510 000137 530 525 000142 525 530 000144 500 575 000145 510 625

2. The Median

When a measurement of a set of observation is at least ordinal in nature, the observations can be ranked, or sorted, into an array whereby the values are arranged in order of magnitude with each value retaining its original identity. The median can be defined as the value of the middle item of a set of items that form an array in ascending or descending order of rank: the [N+1}2 position. In simple terms, the median splits the data into two equal parts, allowing us to state that half of the subjects scored below the median value and half the subjects scored above the median value. If an observed value occurs more than once, it is listed separately each time it occurs: Subject Reaction Time in msecs Reaction Time shown in array: shortest to longest Reaction Time shown in array: longest to shortest 000123 625 460 625 000125 500 480 575 000126 480 500 530 000128 500 500 525 000129 460 500 510 000131 500 500 500 Median 000134 575 510 500 000137 530 525 500 000142 525 530 500 000144 500 575 480 000145 510 625 460

3. The Arithmetic Mean