Download PPA 415 – Research Methods in Public Administration

PPA 415 – Research Methods
in Public Administration
Lecture 4 – Measures of
Dispersion
Introduction



By themselves, measures of central tendency
cannot summarize data completely.
For a full description of a distribution of scores,
measures of central tendency must be paired
with measures of dispersion.
Measures of dispersion assess the variability of
the data. This is true even if the distributions
being compared have the same measures of
central tendency.
Introduction – Example, JCHA
1999
How safe is your community?
How safe is your community?
Trafford
Red Hollow
3.5
3.5
3.0
3.0
2.5
2.5
2.0
2.0
1.5
1.5
1.0
1.0
Std. Dev = 2.67
.5
Mean = 6.8
N = 14.00
0.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
How safe do you feel in your community?
9.0
10.0
Std. Dev = 3.96
.5
Mean = 6.8
N = 7.00
0.0
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
How safe do you feel in your community?
9.0
10.0
Introduction

Measures of dispersion discussed.



Index of qualitative variation (IQV).
The range and interquartile range.
Standard deviation and variance.
Index of Qualitative Variation


Used primarily for nominal variables, but
can be used with any variable with a
frequency distribution.
Ratio of amount of variation actually
observed in a distribution of scores to the
maximum variation that could exist in that
distribution.
Index of Qualitative Variation


Maximum variation in a frequency
distribution occurs when all cases are
evenly distributed across all categories.
The measure gives you information on
how homogeneous or heterogeneous a
distribution is.
Index of Qualitative Variation
IQV 

k N2  f 2
N 2 k  1

where :
k  number of categories
N  number of cases
2
f
  the sum of the squared frequencie s
Index of Qualitative Variation
JCHA 1999: Ethnicity in Housing Communities
Dixie
Red
Hickory Oak Terrace
Fultondale Brookside Warrior I Warrior II Bradford Manor Trafford Hollow Grove Ridge Manor
White
f1
9
20
18
8
10
2
14
5
7
0
9
Nonwhite f2
6
4
5
4
4
13
0
2
0
17
20
Total
N
Categories k
IQV
15
2
24
2
23
2
12
2
96.0%
55.6%
68.1%
88.9%
14
2
15
2
81.6% 46.2%
14
2
7
2
0.0% 81.6%
7
2
17
2
29
2
0.0% 0.0%
85.6%
Range and Interquartile Range

Range: the distance between the highest and
lowest scores.



Only uses two scores.
Can be misleading if there are extreme values.
Interquartile range: Only examines the middle
50% of the distribution. Formally, it is the
difference between the value at the 75%
percentile minus the value at the 25th percentile.
Range and Interquartile Range

Problems: only based on two scores.
Ignores remaining cases in the
distribution.
Range  Highest  lowest
IQR  Q3 ( P75 )  Q1 ( P25 )
Range and Interquartile Range:
JCHA 1999 Example
Statistics
How long have your lived at
N
Valid
Missing
Minimum
Maximum
Percentiles
25
75
this address?
181
4
1
564
24
108
Range = Maximum - Minimum
IQR = P75-P25
563
84
The Standard Deviation


The basic limitation of both the range and the
IQR is their failure to use all the scores in the
distribution
A good measure of dispersion should



Use all the scores in the distribution.
Describe the average or typical deviation of the
scores.
Increase in value as the distribution of scores
becomes more heterogeneous.
The Standard Deviation



One way to do this is to start with the
distances between every point and some
central value like the mean.
The distances between the scores are the
mean (Xi-Mean X) are called deviation
scores.
The greater the variability, the greater the
deviation score.
The Standard Deviation


One course of action is to sum the
deviations and divide by the number of
cases, but the sum of the deviations is
always equal to zero.
The next solution is to make all deviations
positive.


Absolute value – average deviation.
Squared deviations – standard deviation.
Average and Population Standard
Deviation
Average Deviation
X

AD 
i
X
N
Variance (populatio n)
2

X


i  X

2
N
Standard Deviation (populatio n)

 X
i
X
N

2
Sample Variance and Standard
Deviation
Sample variance
s
2

X


X
i

2
n 1
Sample standard deviation
s
 X
i
X
n 1

2
Computational Variance and
Standard Deviation - Sample
Computatio nal Variance (Sample)

x

x  n
2
2
s2 
n 1
Computatio nal Sample Standard Deviation
s  s2
Examples – JCHA 1999

N
X
Safety (Xi )
10
9
5
5
10
7
10
10
10
5
81
10
8.1
(X
i
 X )
1.9
0.9
-3.1
-3.1
1.9
-1.1
1.9
1.9
1.9
-3.1
0.0
X
i

X
1.9
0.9
3.1
3.1
1.9
1.1
1.9
1.9
1.9
3.1
20.8
( X i  X )2
3.61
0.81
9.61
9.61
3.61
1.21
3.61
3.61
3.61
9.61
48.90
X2
100
81
25
25
100
49
100
100
100
25
705
Examples – Average and
Standard Deviation
AD 
s
2
Xi  X

X

n
X
n 1
i

2

28
 2.8
10
48.9

 5.43
9
s  s 2  5.43  2.33

x

x  n
2
2
s2 
n 1
812
705 
705  656.1 48.9
10



 5.43
9
9
9
s  s 2  5.43  2.33
Grouped Standard Deviation
s
 fx
m

fx 


2
2
m
n 1
n
Grouped Standard Deviation
Example
What is your monthly household income?
f
xm
x 2m
fx m
fx 2m
Valid
$500 or less
13 250
62500 3250
812500
$1,000 or less
16 750 562500 12000 9000000
$1,500 or less
9 1250 1562500 11250 14062500
Total
38
26500 23875000
Missing Missing Values
5
Total
43
Grouped Standard Deviation
Example
s
 fx
m

fx 


2
2
m
n 1
n
26,500
23,875,000 
38

37
2
23,875,000  18,480,263.16
5,394,736.84
s

37
37
s  145,803.6984  $381.84