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

Definitions of Center

Here we present three 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 1 shows the distribution
of the five numbers 2, 3, 4, 9, 15 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 1. A balance scale.

Figure 2 shows an asymmetric distribution. To
balance it, we cannot put the fulcrum halfway between the lowest
and highest values (as we did in Figure 2). Placing the fulcrum
at the "half way" point would cause it to tip towards
the left.

Figure 2. 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 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 differences
of these numbers from the number 10.

Table 2. An example of the sum of
absolute deviations

Values

Absolute differences 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 difference between 3 and 10 is 7, and similarly
for the other rows. When we add up the five absolute differences,
we get 28. So, the sum of the absolute differences from 10 is
28. Likewise, the sum of the absolute differences from 5 equals
3 + 2 + 1 + 4 + 11 = 21. So, the sum of the absolute differences
from 5 is smaller than the sum of the absolute differences from
10. In this sense, 5 is closer, overall, to the other numbers
than is 10.

We are now in position to define a second measure
of central tendency, this time in terms of absolute differences.
Specifically, according to our second definition, the center of
a distribution is the number for which the sum of the absolute
differences is smallest. As we just saw, the sum of the absolute
differences from 10 is 28 and the sum of the absolute differences
from 5 is 21. Is there a value for which the sum of the absolute
difference is even smaller than 21? Yes. For these data, there
is a value for which the sum of absolute deviation is only 20.
See if you can find it. A general method for finding the center
of a distribution in the sense of absolute difference 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 is based on the concept of the
sum of squared differences. Again, consider the distribution
of the five numbers 2, 3, 4, 9, 16. Table 3 shows the sum of
the squared differences of these numbers from the number 10.

Table 3. An example of the sum of
squared deviations

Values

Squared differences 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 difference between 3 and 10 is 49, and so forth.
When we add up all these differences, we get 186. Changing the
target from 10 to 5, we calculate the sum of the squared differences
from 5 as 9 + 4 + 1 + 16 + 121 = 151. So, the sum of the squared
differences from 5 is smaller than the sum of the absolute differences
from 10. Is there a value for which the sum of the squared difference
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 differences
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. We'll how you how to do
it in the upcoming section "Squared
Differences Simulation."