Central Tendency and Variability

CHAPTER FOUR

Central Tendency and Variability

NOTE TO INSTRUCTORS

In this chapter, instructors should emphasize the measures of central tendency and variability because these will be used extensively in the coming chapters. Instructors should spend time explaining why there are three types of central tendency and the use of each one. Furthermore, use examples from the chapter, the discussion questions, and the classroom exercises to demonstrate how and why these three measures differ.

For the measures of variability, provide specific examples of how and why the range and standard deviation can differ. Because many students have difficulty calculating the standard deviation, which will be used frequently in future chapters, it is important to continue to demonstrate how it is calculated until students are comfortable with it. In addition, since students have a tendency to be intimidated by the term variability, show them how the words “variability” and “variable” are used in everyday language and how this could be applied to working with numbers. For example, draw an analogy by discussing what it means to say, “My mood has been variable,” and apply their responses to working with numbers.

OUTLINE OF RESOURCES

III. Central Tendency

 Discussion Question 4-1 (p. 32)

 Classroom Activity 4-1: Grade Expectancy and Study Habits (p. 33)

 Discussion Question 4-2 (p. 33)

 Classroom Activity 4-2: Working with Measure of Central Tendency (p. 34)

III. Measures of Variability

 Discussion Question 4-3 (p. 35)

 Classroom Activity 4-3: Creating Data to Calculate Central Tendency and Variability (p. 35)

III. Next Steps: The Interquartile Range

 Discussion Question 4-4 (p. 35)

IV. Handouts

 Handout 4-1: Survey (p. 36)

 Handout 4-2: Working with Measures of Central Tendency (p. 37)

 Handout 4-3: Creating Data to Calculate Central Tendency and Variability (p. 38)

Chapter Guide

I. Central Tendency

1.The central tendency is a descriptive statistic that best represents the center of a data set. In other words, it is the particular value that all the other data seem to be gathering around. If we represent our distribution visually, the central tendency is usually at or near the highest point of the histogram or polygon.

2.There are three different kinds of central tendency: mean, median, and mode.

3.The mean is the arithmetic average of our scores. In other words, we add up all of our scores and divide the sum by the number of scores in our distribution. The mean is used to represent the “typical” score in a distribution.

4.Visually, the mean is the point that perfectly balances both sides of the distribution.

5.The mean can be symbolized in a number of ways. The current text will use M. Other texts will use X bar, or m. M and X bar are statistics because they refer to samples whereas, m is a parameter and is used for a population.

6.The full equation for the mean is M = SX/N.

7.The median is the middle score of all the scores in a sample when scores are arranged in ascending order.

8.The median is at the 50th percentile and can be abbreviated to mdn.

9.To find the median, line up the data in ascending order. If there is an odd number of scores, find the middle score (there should be equal amounts of data on both sides). If there is an even number, take the mean of the two middle scores. Alternatively, you can divide the number of scores by 2 and add ½ to find the middle score. With the numbers in order, count that many places to find the median.

10.The mode is the most common score in the sample.

Discussion Question 4-1

What are the three measures of central tendency? How do you calculate each one?

Your students’ answers should include:

 The mean, the median, and the mode.

 To calculate the mean, or average, sum all the scores and divide by the number of scores summed.

 To find the median, or 50th percentile, line up the scores in ascending order. If the total number of scores is an odd number, the median is the middle score. If the total number of scores is an even number, the median is the mean of the two middle scores.

 To find the mode, or most frequently occurring score, search a list of scores to find the score that occurs most frequently, or construct a frequency table to find the most frequently occurring score.

Classroom Activity 4-1
Grade Expectancy and Study Habits

Have your students complete anonymously Handout 4-1, a survey found at the end of this chapter. (Data are always more meaningful when they are relevant to the students. Using information taken from your students as data for this exercise will engage the class and help prompt students to participate.) Once you collect the data from your students, enter the data into SPSS.

 Have your students eyeball the data file and estimate what the mean, median, and mode are for each variable. Have them estimate the variability of the data. Which variable is most variable?

 Run the analysis, and see how well the group estimated their results.

 Display the data graphically in a number of different ways to explore the different options in SPSS.

 Discuss issues of distribution (normalcy and skewness with the graphs options).

This is also a good exercise for discussing grade expectations and study habits!

11.A distribution can be unimodal, or have one mode; bimodal with two modes; or multimodal with more than two modes.

12.The mean is most often identified as the central tendency. However, the median or mode can be used when the data are skewed or lopsided, which, when it occurs, is frequently due to a statistical outlier. An outlier is an extreme score, either very high or very low in comparison with the rest of the scores in the sample. When the data are skewed, the median is most often used. The mode can be used if a particular score dominates or in bimodal and multimodal distributions.

Discussion Question 4-2

Although the mean is most often used as the measure of central tendency, when would you want to avoid using the mean? Why?

Your students’ answers may include:

 Use the mode, not the mean, when reporting nominal values, such as the percentage of females in a population. (The mode, not the mean, accurately represents percentages.)

 Use the median or mode, not the mean, when data are lopsided, or skewed. (The mean will not accurately represent the average score when data are skewed.)

13.Statistics can often be used to provide false information about the distribution. As a result, it is usually best to use and report multiple measures of central tendency rather than rely on only one.

Classroom Activity 4-2
Working with Measure of Central Tendency

Use Table II from Oshagbeni (1997). Job satisfaction profiles of university teachers. Journal of Managerial Psychology, 12, 27–39. (To view or purchase this article, go to your local library or visit Emerald Publishing online at http://www.emeraldinsight.com.)

 Look at the individual values in the table and have students think about how to interpret the fact that the mean is different from the median.

 Next, have students use the medians and modes listed in the table as raw scores and calculate the new mean, median, and mode for this set of data.

 Finally, have them draw a histogram and a frequency polygon using these data.

See Handout 4-2 at the end of this chapter.

II. Measures of Variability

1.Variability is a numerical way of describing how much spread there is in a distribution.

2.The range is a measure of variability calculated by subtracting the lowest score from the highest score. It is the easiest measure of variability to calculate.

3.Variance is the average of the squared deviations from the mean. It basically refers to variability.

4.In order to calculate the standard deviation, we need to calculate the deviations from the mean, the term for the amount that a score differs from the mean of the sample. We calculate this simply by subtracting each of our individual scores from the mean.

5.The next step in calculating standard deviation is to square all of the deviations from the mean. If we take the average of these deviations squared, we will have the variance.

6.When we have taken the sum of all of our squared deviations, we have calculated the sum of squares. This is abbreviated as SS.

7.Then, we divide the sum of squares by the total number of the sample (N).

8.The variance can be symbolized by SD2, s2, or MS when calculated from a sample. When estimating a sample, the symbol for variance is s2.

9.The standard deviation is the typical amount that scores in a sample vary from the mean. It is calculated by taking the square root of the average of the squared deviations from the mean. It is the most commonly used measure of variability.

10.Our final step in calculating the standard deviation is to take the square root of the variance, which is symbolized by SD, s, or the parameter, s.

Discussion Question 4-3

What are two measures of variability? How do you calculate them?

Your students’ answers should include:

 The two measures of variability are range and variance.

 To calculate range, subtract the lowest score from the highest score.

 To calculate the variance, or standard deviation:

a. subtract the mean from every score to get the deviations;

b. square all the deviations; and

c. find the mean of the squared deviations by summing them and dividing by N.

Classroom Activity 4-3
Creating Data to Calculate Central Tendency and Variability

Have students complete the Rosenberg Self-Esteem Scale. See Handout
4-3 at the end of this chapter. They should then score their scales and turn in their scores.

 As a class, use these data to calculate measures of central tendency.

 Also, have students calculate the range and standard deviation.

 Next, arrange the data as a frequency table and construct a histogram.

III. Next Steps: The Interquartile Range

1.The interquartile range is a measure of the distance between the first and third quartiles.

2.The first quartile marks the 25th percentile of a data set whereas the third quartile marks the 75th.

3.To calculate the interquartile range, we first calculate the median. Then we look at the scores below the median and find the median of these scores. The lower half of these scores is known as the first quartile, or Q1. The third quartile, or Q3, is calculated by finding the median of the top half of the scores. Next, subtract Q1 from Q3.

4.The interquartile range is often abbreviated as IQR.

5.The advantage of the IQR over the range is that it is less susceptible to outliers.

Discussion Question 4-4

What is an advantage of using the interquartile range instead of the range?

Your students’ answers should include:

 The interquartile range is less susceptible to outliers.

PLEASE NOTE: Due to formatting, the Handouts are only available in Adobe PDF®.