Chapter 3: Central Tendency

Chapter Outline

3.1 Overview (The Reason for Measuring Central Tendency)

3.2 The Mean

Alternative Definitions for the Mean

The Weighted Mean

Computing the Mean from a Frequency Distribution Table

Characteristics of the Mean

3.3 The Median

Finding the Median for Most Distributions

Finding the Precise Median for a Continuous Variable

The Median, the Mean, and the Middle

3.4 The Mode

3.5 Selecting a Measure of Central Tendency

When to Use the Median

When to Use the Mode

In The Literature: Reporting Measures of Central Tendency

Presenting Means and Medians in Graphs

3.6 Central Tendency and the Shape of the Distribution

Symmetrical Distributions

Skewed Distributions

Inherit

Learning Objectives and Chapter Summary

1. Students should understand the purpose of measuring central tendency.

Because central tendency serves two purposes (identifying the center of the distribution

and identifying the best representative score), no single measure is always best for both

purposes. Therefore, there are three methods for measuring central tendency. At times,

one of the three measures is sufficient but occasionally two are three are used together to

obtain a complete and accurate description of the distribution (See Figure 3.1).

Instructor Notes – Chapter 3 – page 32

2. Students should be able to define and compute each of the three measures of central tendency.

It may seem obvious, but remind students that two values are needed to compute the

mean: the number of scores and the sum of the scores. They need to obtain both of these

values before the mean can be computed. Most students find that the median and the

mode are easier to compute if a distribution is displayed in a frequency distribution

histogram.

3. Students should understand how the mean is affected when a set of scores is modified (a new

score is added, a score is removed, or a score is changed).

Before attempting to compute the mean, students should be able to visualize what is

going to happen. Remind them that the mean is the balance point and they can visualize

the original distribution as a set of blocks on a see-saw that is balanced at the mean. The

modification involves adding a new block, taking away a block, or moving a block. In

any case, they should be able to see how the modification affects the balance.

Again, the key to calculating the mean is to find the number of scores (n) and the sum of

the scores (ΣX). Students first need to determine exactly how the modification affects the

number of scores. Then, they must determine how the modification affects the sum of

the scores. Once these two values are determined, the mean is easy.

4. Students should understand the circumstances in which each of the three measures of central

tendency is appropriate.

Most of the time, the mean is the preferred measure, especially if the scores are numerical

values. The median is most commonly used in situations where the mean can be

calculated but the calculated value is not a good representative score. Typically, this

occurs with skewed distributions or when a distribution has a few extreme scores (often

called outliers). The mode is the only measure available for non-numerical (nominal)

data. It also is used as a supplemental measure (along with the mean) to help describe the

shape of a distribution.

5. Students should understand how the three measures of central tendency are related to each

other in symmetrical and skewed distributions.

Of the three measures, the mean is the one that is most affected by extreme scores. Thus,

the mean is displaced toward the extreme scores in the tail of a skewed distribution. If

you begin with a skewed distribution and change a score in the tail so that it becomes

even more extreme, only the mean will be affected by the change.

Instructor Notes – Chapter 3 – page 33

6. Students should be able to draw and to understand figures/graphs that display several different

means (or medians) representing different treatment conditions or different groups.

Line graphs are typically used when the treatment conditions or groups correspond to

numerical values (interval or ratio scales). Otherwise, bar graphs are used to display the

means.

Other Lecture Suggestions

1. The mean can be introduced as the “average” that students already know how to calculate.

For example, ask students to find the average telephone bill if one month is $20 and the next

month is $30. Everyone gets this right. Now, add a third month with a bill of $100. Many

students will tell you that the mean is now $75. This is a good chance to introduce the formula

for the mean, showing that the total cost (ΣX = $150) is divided over three (n = 3) months.

2. Sketch a simple histogram and label the values along the X axis using 1, 2, 3, 4, and 5. Ask

the students what will happen to the distribution if you add 10 points to every score. Answer:

The whole distribution moves 10 points to the right. (You can keep the same sketch, simply re-

label the values on the X axis to 11, 12, 13, 14, and 15.) Note that the mean (middle) has shifted

10 points to the right.

Go back to the original distribution and ask what would happen if every score were

multiplied by 10. This time the 1s become 10s, the 2s become 20s, and so on. Again, you can

keep the same sketch, simply re-label the values on the X axis to 10, 20, 30, 40, and 50. After

multiplying, the mean (that used to be at 3) is located at 30 (10 times bigger).

3. The idea that the mean is not always a central, representative value can be demonstrated by

starting with a simple, symmetrical distribution consisting of scores 1, 2, 3, 4, and 5 with

frequencies of 1, 2, 4, 2, and 1, respectively. Just looking at the frequency distribution

histogram, it is easy to see that the mean is 3, but you can also demonstrate that the n = 10 scores

add up to ΣX = 30.

Now, move the score at X = 5 to a new location at X = 55 and have students find the new

mean (adding 50 points to one score adds 50 points to the total, so ΣX is now 80 and the mean is

8). Note that the new mean, 8, is not a representative value. In fact, none of the scores is located

around X = 8. Finally, you can introduce the median using the original distribution (Median = 3)

and then see what happens to this measure of central tendency when X = 5 is moved to X = 55

(the median is still 3). The median is relatively unaffected by extreme scores.

Instructor Notes – Chapter 3 – page 34

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