2.7 Measures of Central Tendency
We have seen how to display data in ways that allow us to begin to draw some conclusions
about what the data have to say. Plotting data shows the general shape of the distribution and
gives a visual sense of the general magnitude of the numbers involved. In this section you
will see several statistics that can be used to represent the “center” of the distribution. These
statistics are called measures of central tendency. In the next section we will go a step fur-
ther and look at measures that deal with how the observations are dispersed around that cen-
tral tendency, but first we must address how we identify the center of the distribution.
The phrase measures of central tendency,or sometimes measures of location,refers
to the set of measures that reflect where on the scale the distribution is centered. These
measures differ in how much use they make of the data, particularly of extreme values, but
they are all trying to tell us something about where the center of the distribution lies. The
three major measures of central tendency are the mode, which is based on only a few data
points; the median, which ignores most of the data; and the mean, which is calculated from
all of the data. We will discuss these in turn, beginning with the mode, which is the least
used (and often the least useful) measure.The Mode
The mode (Mo)can be defined simply as the most common score, that is, the score ob-
tained from the largest number of subjects. Thus, the mode is that value of Xthat corre-
sponds to the highest point on the distribution. If two adjacent times occur with equal (and
greatest) frequency, a common convention is to take an average of the two values and call
that the mode. If, on the other hand, two nonadjacent reaction times occur with equal (or
nearly equal) frequency, we say that the distribution is bimodal and would most likely re-
port both modes. For example, the distribution of time spent playing electronic games is
roughly bimodal (see Figure 2.7), with peaks at the intervals of 0–9 minutes and 40–49
minutes. (You might argue that it is trimodal, with another peak at 120 1 minutes, but that
is a catchall interval for “all other values,” so it does not make much sense to think of it as
a modal value.)The Median
The median (Mdn)is the score that corresponds to the point at or below which 50% of the
scores fall when the data are arranged in numerical order. By this definition, the median is
also called the 50th percentile.^6 For example, consider the numbers (5, 8, 3, 7, 15). If the
numbers are arranged in numerical order (3, 5, 7, 8, 15), the middle score would be 7, and
it would be called the median. Suppose, however, that there were an even number of scores,
for example (5, 11, 3, 7, 15, 14). Rearranging, we get (3, 5, 7, 11, 14, 15), and no score has
50% of the values below it. That point actually falls between the 7 and the 11. In such a
case the average (9) of the two middle scores (7 and 11) is commonly taken as the median.^732 Chapter 2 Describing and Exploring Data
(^6) A specific percentile is defined as the point on a scale at or below which a specified percentage of scores fall.
(^7) The definition of the median is another one of those things about which statisticians love to argue. The definition
given here, in which the median is defined as a point on a distribution of numbers, is the one most critics prefer. It
is also in line with the statement that the median is the 50th percentile. On the other hand, there are many who are
perfectly happy to say that the median is either the middle number in an ordered series (if Nis odd) or the average
of the two middle numbers (if Nis even). Reading these arguments is a bit like going to a faculty meeting when
there is nothing terribly important on the agenda. The less important the issue, the more there is to say about it.
measures of
central tendency
measures of
location
mode (Mo)
median (Mdn)