Measures of Central Tendency

Measures of central tendency are ways of summarizing a distribution of values with reference to its central point. They consist, primarily, of the MEAN and the MEDIAN. The MODE, although not strictly a measure of central tendency, also provides information about the distribution by giving the most common value in the distribution. It therefore tells us something about what is typical, an idea that we usually associate with a central point. Table 1 shows the number of dependent children in seven notional families. We can summarize the midpoint of this distribution with reference to the arithmetic mean (the sum of the values divided by the number of the observations), the median (or midpoint of the distribution), or the mode (the most common value).

Table 2 identifies the values of each of these three measures of central tendency as applied to the family sizes given in Table 1. It shows that each measure produces a different figure for summarizing the center of the distribution. In this case, the limited range of the variable and the observations means that the differences are not substantial, but when the range of values is much wider and the distribution is not symmetrical, there may be substantial differences between the values provided by the different measures.

Although the mean can only be used with interval data, the median may be used for ordinal data, and the mode can be applied to nominal or ordinal categorical data. On the other hand, when data are collected in continuous or near-continuous form, the mode may provide little valuable information: Knowing that, in a sample of 100 people, two people earn $173.56, whereas all the other sample members have unique earnings, tells us little about the distribution.

An additional, though rarely used, measure of central tendency is provided by the midrange, that is, the halfway point between the two ends of distribution. In the example in Table 1, the midrange would have the value 2.5. A modification of this is the midpoint between the 25th and 75th percentile of a distribution (i.e., the midpoint of the INTERQUARTILE RANGE).

Means can be related to any fixed point in the distribution, although they are usually assumed to be related to 0. For example, IQ tests set the standard for intelligence at 100. An average score for a class of students could be taken from the amount they differ from 100. The final average difference would then be added to (or subtracted from) 100. Such nonzero reference points are often taken as those in which the frequency of [Page 629]observations is greatest, as this can simplify the amount of calculation involved.