Transcript less - Cengage Learning

CHAPTER 4
Measures of Dispersion
Chapter Outline
 Introduction
 The Index of Qualitative Variation
(IQV)
 The Range (R) and Interquartile
Range (Q)
 Computing the Range and
Interquartile Range
 The Standard Deviation
Chapter Outline
 Computing the Standard Deviation:
An Additional Example
 Computing the Standard Deviation for
Grouped Data
 Interpreting the Standard Deviation
 Interpreting Statistics: The Central
Tendency and Dispersion of Income in
the United States
In This Presentation
 Measures of dispersion.
 You will learn
 Basic Concepts
 How to compute and interpret the Range
(R) and the standard deviation (s)
 These other measures of dispersion are
covered in the text:
 IQV
 Q
 s2
The Concept of Dispersion
 Dispersion = variety, diversity,
amount of variation between scores.
 The greater the dispersion of a
variable, the greater the range of
scores and the greater the differences
between scores.
The Concept of Dispersion:
Examples
 Typically, a large city will have more
diversity than a small town.
 Some states (California, New York)
are more racially diverse than others
(Maine, Iowa).
 Some students are more consistent
than others.
The Concept of Dispersion
 The taller curve has less dispersion.
 The flatter curve has more dispersion.
The Range
 Range (R) = High Score – Low Score
 Quick and easy indication of
variability.
 Can be used with ordinal or intervalratio variables.
 Why can’t the range be used with
variables measured at the nominal
level?
Standard Deviation
 The most important and widely used
measure of dispersion.
 Should be used with interval-ratio
variables but is often used with
ordinal-level variables.
Standard Deviation
 Formulas for variance and standard
deviation:
Standard Deviation
 To solve:
Subtract mean from each score.
Square the deviations.
Sum the squared deviations.
Divide the sum of the squared deviations
by N.
 Find the square root of the result.




Interpreting Dispersion
 Low score=0, Mode=12, High score=20
 Measures of dispersion: R=20–0=20, s=2.9
Years of Education (Full Sample)
900
800
700
600
500
400
300
200
100
0
Interpreting Dispersion
 What would happen to the dispersion
of this variable if we focused only on
people with college-educated
parents?
 We would expect people with highly
educated parents to average more
education and show less dispersion.
Interpreting Dispersion
 Low score=10, Mode=16, High Score=20
 Measures of dispersion: R=20-10=10, s=2.2
Years of Education (Both Parents w BA)
45
40
35
30
25
20
15
10
5
0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Interpreting Dispersion
 Entire sample:
 Mean = 13.3
 Range = 20
 s = 2.9
 Respondents with college-educated
parents:
 Mean = 16.0
 R = 10
 s =2.2
Interpreting Dispersion
 As expected, the smaller, more
homogeneous and privileged group:
 Averaged more years of education
 (16.0 vs. 13.3)
 And was less variable
 (s = 2.2 vs. 2.9; R = 10 vs. 20)
Measures of Dispersion
 Higher for more diverse groups (e.g.,
large samples, populations).
 Decrease as diversity or variety
decreases (are lower for more
homogeneous groups and smaller
samples).
 The lowest value possible for R and s
is 0 (no dispersion).