Central Tendency

… a core concept used in Quantitative Methods and Atlas104

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Concept description

OnlineStatBook (link on right) describes central tendency as a loosely defined concept that has to do with the location of the center of a distribution.

OnlineStatBook describes three different ways of defining the centre of a distribution:

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 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.

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.”

Smallest square 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.”

Atlas topic, subject, and course

Describing Distributions (core topic) in Quantitative Methods and Atlas104 Quantitative Methods.

Sources

David M. Lane and Heidi Ziemer, OnlineStatBook, at http://onlinestatbook.com/2/summarizing_distributions/what_is_ct.html, accessed 10 June 2017.

Page created by: Ian Clark, last modified 10 June 2017.

Image: OnlineStatBook, at http://onlinestatbook.com/2/summarizing_distributions/what_is_ctM.html, accessed 10 June 2017.