What is Central Tendency?

Prerequisites
Distributions

Learning Objectives

  1. Identify situations in which knowing the center of a distribution would be valuable
  2. 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."