Statistical Methods for Psychology

A term that we will need shortly is the median location.The median location of N

numbers is defined as follows:

Median location

Thus, for five numbers the median location 5 (5 1 1)/2 5 3, which simply means that the

median is the third number in an ordered series. For 12 numbers, the median location 5

(12 1 1)/2 5 6.5; the median falls between, and is the average of, the sixth and seventh numbers.

For the data on reaction times in Table 2.2, the median location 5 (300 1 1)/2 5 150.5.

When the data are arranged in order, the 150th time is 59 and the 151st time is 60; thus the

median is (59 1 60)/2 5 59.5 hundredths of a second. You can calculate this for yourself

from Table 2.2. For the electronic games data there are 100 scores, and the median location

is 50.5. We can tell from the stem-and-leaf display in Figure 2.4 that the 50th score is 44 and

the 51st score is 46. The median would be 45, which is the average of these two values.

The Mean

The most common measure of central tendency, and one that really needs little explana-

tion, is the mean, or what people generally have in mind when they use the word average.

The mean( ) is the sum of the scores divided by the number of scores and is usually des-

ignated (read “Xbar”).^8 It is defined (using the summation notation given on page 30) as

follows:

where is the sum of all values of X, and Nis the number of Xvalues. As an illustration,

the mean of the numbers 3, 5, 12, and 5 is

For the reaction time data in Table 2.2, the sum of the observations is 18,078. When we di-

vide that number by N 5 300, we get 18,078/300 5 60.26. Notice that this answer agrees

well with the median, which we found to be 59.5. The mean and the median will be close

whenever the distribution is nearly symmetric (as defined on page 27). It also agrees well

with the modal interval (60–64).

Relative Advantages and Disadvantages of the Mode,

the Median, and the Mean

Only when the distribution is symmetric will the mean and the median be equal, and

only when the distribution is symmetric and unimodal will all three measures be the

same. In all other cases—including almost all situations with which we will deal—some

measure of central tendency must be chosen. There are no good rules for selecting a

measure of central tendency, but it is possible to make intelligent choices among the

three measures.

31511215

4

=

25

4

=6.25

gX

X= a

X

N

X

X

=

N 11

2

Section 2.7 Measures of Central Tendency 33

(^8) The American Psychological Association would like us to use Mfor the mean instead of , but I have used for
so many years that it would offend my delicate sensibilities to give it up. The rest of the statistical world generally
agrees with me on this, so we will use throughout.X
X X
median location
mean