Download CHAPTER FOUR Central Tendency and Variability NOTE TO

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 and X bar are
statistics because they refer to samples
whereas,  is a parameter and is used for a
population.
6. The full equation for the mean is M = X/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 41, 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
2.
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, .
> 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®.