Identify situations in which knowing the center of a distribution
would be valuable

Give three different ways the center of a distribution can be defined

Describe how the balance is different for symmetric distributions than it is for asymmetric distributions.

What is "central tendency," and why do
we want to know the central tendency of a group of scores? Let
us first try to answer these questions intuitively. Then we will
proceed to a more formal discussion.

Imagine this situation: You are in a class with just
four other students, and the five of you took a 5-point pop quiz.
Today your instructor is walking around the room, handing back
the quizzes. She stops at your desk and hands you your paper.
Written in bold black ink on the front is "3/5." How
do you react? Are you happy with your score of 3 or disappointed?
How do you decide? You might calculate your percentage correct,
realize it is 60%, and be appalled. But it is more likely that
when deciding how to react to your performance, you will want
additional information. What additional information would you
like?

If you are like most students, you will immediately
ask your neighbors, "Whad'ja get?" and then ask the
instructor, "How did the class do?" In other words,
the additional information you want is how your quiz score
compares to other students' scores. You therefore understand
the importance of comparing your score to the class distribution
of scores. Should your score of 3 turn out to be among the
higher scores, then you'll be pleased after all. On the other
hand, if 3 is among the lower scores in the class, you won't
be quite so happy.

This idea of comparing individual scores to a distribution
of scores is fundamental to statistics. So let's explore it further,
using the same example (the pop quiz you took with your four classmates).
Three possible outcomes are shown in Table 1. They are labeled
"Dataset A," "Dataset B," and "Dataset
C." Which of the three datasets would make you happiest?
In other words, in comparing your score with your fellow students'
scores, in which dataset would your score of 3 be the most impressive?

In Dataset A, everyone's score is 3. This puts your
score at the exact center of the distribution. You can draw satisfaction
from the fact that you did as well as everyone else. But of course
it cuts both ways: everyone else did just as well as you.

Table 1. Three possible datasets for the 5-point make-up quiz.

Student

Dataset A

Dataset B

Dataset C

You

3

3

3

John's

3

4

2

Maria's

3

4

2

Shareecia's

3

4

2

Luther's

3

5

1

Now consider the possibility that the scores are described
as in Dataset B. This is a depressing outcome even though your
score is no different than the one in Dataset A. The problem is
that the other four students had higher grades, putting yours
below the center of the distribution.

Finally, let's look at Dataset C. This is more like
it! All of your classmates scorelower
than you so your score is above the center of the distribution.

Now let's change the example in order to develop more
insight into the center of a distribution. Figure 1 shows the
results of an experiment on memory for chess positions. Subjects
were shown a chess position and then asked to reconstruct it on
an empty chess board. The number of pieces correctly placed was
recorded. This was repeated for two more chess positions. The
scores represent the total number of chess pieces correctly placed
for the three chess positions. The maximum possible score was
89.

Two groups are compared. On the left are people who
don't play chess. On the right are people who play a great deal
(tournament players). It is clear that the location of the center
of the distribution for the non-players is much lower than the
center of the distribution for the tournament players.

Figure 1. Back-to-back stem and leaf display. The left side shows the memory scores of the non-players.
The right side shows the scores of the tournament players.

We're sure you get the idea now about the center
of a distribution. It is time to move beyond intuition. We need
a formal definition of the center of a distribution. In fact,
we'll offer you three definitions! This is not just generosity
on our part. There turn out to be (at least) three different ways
of thinking about the center of a distribution, all of them useful
in various contexts. In the remainder of this section we attempt
to communicate the idea behind each concept. In the succeeding
sections we will give statistical measures for these concepts
of central tendency.

Definitions of Center

Now we explain the three different ways of defining
the center of a distribution. All three are called measures of
central
tendency.

Balance Scale

One definition of central tendency is the point
at which the distribution is in balance. Figure 2 shows the
distribution of the five numbers 2, 3, 4, 9, 16 placed upon
a balance scale. If each number weighs one pound, and is placed
at its position along the number line, then it would be possible
to balance them by placing a fulcrum at 6.8.

Figure 2. A balance scale.

For another example, consider the distribution
shown in Figure 3. It is balanced by placing the fulcrum in the
geometric middle.

Figure 3. A distribution balanced on the tip of a triangle.

Figure 4 illustrates that the same distribution
can't be balanced by placing the fulcrum to the left of center.

Figure 4. The distribution is not balanced.

Figure 5 shows an asymmetric distribution. To
balance it, we cannot put the fulcrum halfway between the lowest
and highest values (as we did in Figure 3). Placing the fulcrum
at the "half way" point would cause it to tip towards
the left.

Figure 5. An asymmetric distribution balanced on the tip of a triangle.

The balance point defines one sense of a distribution's
center. The simulation in the next section "Balance
Scale Simulation" shows how to find the point at which the
distribution balances.

Smallest Absolute Deviation

Another way to define the center of a distribution
is based on the concept of the sum of the absolute deviations (differences).
Consider the distribution made up of the five numbers 2, 3, 4,
9, 16. Let's see how far the distribution is from 10 (picking
a number arbitrarily). Table 2 shows the sum of the absolute deviations
of these numbers from the number 10.

Table 2. An example of the sum of absolute deviations

Values

Absolute Deviations from 10

2
3
4
9
16

8
7
6
1
6

Sum

28

The first row of the table shows that the absolute
value of the difference between 2 and 10 is 8; the second row
shows that the absolute difference between 3 and 10 is 7, and similarly
for the other rows. When we add up the five absolute deviations,
we get 28. So, the sum of the absolute deviations from 10 is
28. Likewise, the sum of the absolute deviations from 5 equals
3 + 2 + 1 + 4 + 11 = 21. So, the sum of the absolute deviations
from 5 is smaller than the sum of the absolute deviations from
10. In this sense, 5 is closer, overall, to the other numbers
than is 10.

We are now in a position to define a second measure
of central tendency, this time in terms of absolute deviations.
Specifically, according to our second definition, the center of
a distribution is the number for which the sum of the absolute
deviations is smallest. As we just saw, the sum of the absolute
deviations from 10 is 28 and the sum of the absolute deviations
from 5 is 21. Is there a value for which the sum of the absolute
deviations is even smaller than 21? Yes. For these data, there
is a value for which the sum of absolute deviations is only 20.
See if you can find it. A general method for finding the center
of a distribution in the sense of absolute deviations is provided
in the simulation "Absolute Differences
Simulation."

Smallest Squared Deviation

We shall discuss one more way to define the
center of a distribution. It is based on the concept of the
sum of squared deviations (differences). Again, consider the distribution
of the five numbers 2, 3, 4, 9, 16. Table 3 shows the sum of
the squared deviations of these numbers from the number 10.

Table 3. An example of the sum of squared deviations.

Values

Squared Deviations
from 10

2
3
4
9
16

64
49
36
1
36

Sum

186

The first row in the table shows that the squared
value of the difference between 2 and 10 is 64; the second row
shows that the squared difference between 3 and 10 is 49, and so forth.
When we add up all these squared deviations, we get 186. Changing the
target from 10 to 5, we calculate the sum of the squared deviations
from 5 as 9 + 4 + 1 + 16 + 121 = 151. So, the sum of the squared
deviations from 5 is smaller than the sum of the squared deviations
from 10. Is there a value for which the sum of the squared deviations
is even smaller than 151? Yes, it is possible to reach 134.8.
Can you find the target number for which the sum of squared
deviations is 134.8?

The target that minimizes the sum of squared deviations
provides another useful definition of central tendency (the
last one to be discussed in this section). It can be challenging
to find the value that minimizes this sum. You will see how you
do it in the upcoming section "Squared
Differences Simulation."