Download 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
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Discussion Question 4-1 (p. 32)
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Classroom Activity 4-1: Grade Expectancy and Study Habits (p. 33)
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Discussion Question 4-2 (p. 33)
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Classroom Activity 4-2: Working with Measure of Central Tendency
(p. 34)
III.
Measures of Variability
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Discussion Question 4-3 (p. 35)
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Classroom Activity 4-3: Creating Data to Calculate Central Tendency
and Variability (p. 35)
III.
Next Steps: The Interquartile Range
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Discussion Question 4-4 (p. 35)
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CENTRAL TENDENCY AND VARIABILITY
Handouts
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Handout 4-1: Survey (p. 36)
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Handout 4-2: Working with Measures of Central Tendency (p. 37)
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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 μ. M and X bar are statistics because they
refer to samples whereas, μ is a parameter and is used for a population.
6. The full equation for the mean is M = ΣX/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:
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The mean, the median, and the mode.
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To calculate the mean, or average, sum all the scores and divide by the
number of scores summed.
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CENTRAL TENDENCY AND VARIABILITY
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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.
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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?
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Run the analysis, and see how well the group estimated their results.
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Display the data graphically in a number of different ways to explore the
different options in SPSS.
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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.)
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CENTRAL TENDENCY AND VARIABILITY
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.)
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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.
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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.
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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 σ2.
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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, σ.
> Discussion Question 4-3
What are two measures of variability? How do you calculate them?
Your students’ answers should include:
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The two measures of variability are range and variance.
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To calculate range, subtract the lowest score from the highest score.
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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.
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As a class, use these data to calculate measures of central tendency.
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Also, have students calculate the range and standard deviation.
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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:
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The interquartile range is less susceptible to outliers.
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HANDOUT 4-1: SURVEY
What grade do you expect in this course? ________________________
Did you read Chapter 1 in your text? ____________________________
Did you read Chapter 2 in your text? ____________________________
How many hours of television did you watch last night? ____________
How many hours were you online last night?______________________
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HANDOUT 4-2: WORKING WITH MEASURES OF CENTRAL TENDENCY
Directions: Use Table II from Oshagbeni, T. (1997). Job satisfaction profiles of university teachers.
Journal of Managerial Psychology, 12, 27–39, and answer the following questions. For this
exercise, you will be using all of the medians and modes listed in the table as your data. (To view or
purchase this article, go to your local library or visit Emerald Publishing online at
http://www.emeraldinsight.com.)
1. Compare instances when the medians and means listed are not the same for each variable in the chart (e.g., for the teaching variable of group 1, the median is 5 and the
mean is 5.24). What do each of these instances tell you about the distribution of the
actual data?
2. Using all of the values for the median and mode as your data, calculate a new mean,
median, and mode for your scores.
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HANDOUT 4-3: CREATING DATA TO CALCULATE CENTRAL TENDENCY
AND VARIABILITY
Directions: For this exercise, first complete the Rosenberg (1965) Self-Esteem Scale that is provided. To do this, there is a list of statements dealing with your general feelings about yourself. If
you strongly agree, circle SA. If you agree with the statement, circle A. If you disagree, circle D. If
you strongly disagree, circle SD. Scoring instructions are at the bottom of this page.
Scoring Instructions: To score the scale, you will assign a value to each of the 10 items.
For items 1, 2, 4, 6, and 7:
Strongly Agree = 3
Agree = 2
Disagree = 1
Strongly Disagree = 0
For items 3, 5, 8, 9, and 10:
Strongly Agree = 0
Agree = 1
Disagree = 2
Strongly Disagree = 3
Once you have calculated your score, write it in the space below and give it to your
instructor. Continue the assignment on the following page once all of the scores have
been collected.
My score is: ___________
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HANDOUT 4-3 (continued)
Rosenberg (1965) Self-Esteem Scale
1
Strongly
Agree
2
3
Agree
Disagree
4
Strongly
Disagree
1. I feel that I’m a person of worth, at
least on an equal plane with others.
SA
A
D
SD
2. I feel that I have a number of good
qualities.
SA
A
D
SD
3. All in all, I am inclined to feel that I
am a failure.
SA
A
D
SD
4. I am able to do things as well as
most other people.
SA
A
D
SD
5. I feel I do not have much to be
proud of.
SA
A
D
SD
6. I take a positive attitude toward
myself.
SA
A
D
SD
7. On the whole, I am satisfied with
myself.
SA
A
D
SD
8. I wish I could have more respect for
myself.
SA
A
D
SD
SA
A
D
SD
SA
A
D
SD
9. I certainly feel useless at times.
10. At times I think I am no good at all.
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HANDOUT 4-3 (continued)
Directions: Using the scores from the class, answer the following questions.
1. Using the class data, calculate the scores for the three measures of tendency.
2. Calculate the range and the standard deviation.
3. Create a frequency table with this data.
4. Create a histogram with this data.