ppa 415 – research methods in public administration lecture 4 – measures of dispersion

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 do you feel in your community?

10.09.08.07.06.05.04.03.02.0

How safe is your community?

Trafford3.5

3.0

2.5

2.0

1.5

1.0

.5

0.0

Std. Dev = 2.67

Mean = 6.8

N = 14.00

How safe do you feel in your community?

10.09.08.07.06.05.04.03.02.01.00.0

How safe is your community?

Red Hollow3.5

3.0

2.5

2.0

1.5

1.0

.5

0.0

Std. Dev = 3.96

Mean = 6.8

N = 7.00

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

sfrequencie squared theof sum thef

cases ofnumber N

categories ofnumber

:where

1

2

2

22

k

kN

fNkIQV

Index of Qualitative Variation

JCHA 1999: Ethnicity in Housing Communities Dixie Red Hickory Oak TerraceFultondale Brookside Warrior I Warrior II Bradford Manor Trafford Hollow Grove Ridge Manor

White f1 9 20 18 8 10 2 14 5 7 0 9Nonwhite f2 6 4 5 4 4 13 0 2 0 17 20

Total N 15 24 23 12 14 15 14 7 7 17 29Categories k 2 2 2 2 2 2 2 2 2 2 2

IQV 96.0% 55.6% 68.1% 88.9% 81.6% 46.2% 0.0% 81.6% 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.

)()( 251753 PQPQIQR

lowestHighestRange

Range and Interquartile Range: JCHA 1999 Example

StatisticsHow long have your lived at this address? N Valid 181

Missing 4Minimum 1Maximum 564Percentiles 25 24

75 108

Range = Maximum - Minimum 563IQR = P75-P25 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

N

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i

Deviation Average

N

XX

N

XX

i

i

2

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n)(populatioDeviation Standard

n)(populatio Variance

Sample Variance and Standard Deviation

1

deviation standard Sample1

varianceSample

2

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XXs

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i

Computational Variance and Standard Deviation - Sample

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2

2

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Deviation Standard Sample nalComputatio1

(Sample) Variance nalComputatio

ss

nn

xx

s

Examples – JCHA 1999

Safety (Xi) X2

10 1.9 1.9 3.61 1009 0.9 0.9 0.81 815 -3.1 3.1 9.61 255 -3.1 3.1 9.61 25

10 1.9 1.9 3.61 1007 -1.1 1.1 1.21 49

10 1.9 1.9 3.61 10010 1.9 1.9 3.61 10010 1.9 1.9 3.61 1005 -3.1 3.1 9.61 25

81 0.0 20.8 48.90 705N 10

8.1

X

)( XX i 2)( XX i XX i

Examples – Average and Standard Deviation

33.234.5

34.59

9.48

9

1.656705

91081

705

1

33.234.5

34.59

9.48

1

8.210

28

2

22

2

2

2

2

2

ss

nn

xx

s

ss

n

XXs

n

XXAD

i

i

Grouped Standard Deviation

1

2

2

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fxfx

s

mm

Grouped Standard Deviation Example

What is your monthly household income?

f xm x2m fxm fx2

m

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 14062500Total 38 26500 23875000

Missing Missing Values 5Total 43

Grouped Standard Deviation Example

84.381$6984.803,145

37

84.736,394,5

37

16.263,480,18000,875,23

3738500,26

000,875,23

1

22

2

s

s

nn

fxfx

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mm