Transcript No Slide Title

Chapter 5
Measures of
Variability
Fundamental Statistics for the
Behavioral Sciences, 5th edition
David C. Howell
©2003 Brooks/Cole Publishing Company/ITP
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Chapter 5 Measures of Variability
Major Points
• The general problem
• Range and related statistics
• Deviation scores
• The variance and standard deviation
• Boxplots
• Review questions
Chapter 5 Measures of Variability
The General Problem
• Central tendency only deals with the
center
• Dispersion
 Variability of the data around something
 The spread of the points
• Example: Mice and Music
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Chapter 5 Measures of Variability
Mice and Music
• Study by David Merrell
• Raised some mice in quiet environment
• Raised some mice listening to Mozart
• Raised other mice listening to Anthrax
• Dependent variable is the time to run a
straight alley maze after 4 weeks.
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5
Chapter 5 Measures of Variability
Results
• Anthrax mice took much longer to run
• Much greater variability in Anthrax group
 See following graphs for Anthrax and Mozart
 Both X axes are 500 units wide
• We often see greater variability with
larger mean
Mozart Group
12
10
8
6
4
2
Std. D ev = 36.10
Mean = 114.6
N = 24.00
0
27.8
83.3 138.9 194.4 250.0 305.6 361.1 416.7 472.2
WEEK4
Anthrax Group
10
8
6
4
2
Std. D ev = 103.14
Mean = 1825.9
N = 24.00
0
1600.0
1700.0
1650.0
WEEK4
1800.0
1750.0
1900.0
1850.0
2000.0
1950.0
2050.0
Chapter 5 Measures of Variability
Range and Related Statistics
• The range
 Distance from lowest to highest score
 Too heavily influenced by extremes
• The interquartile range (IQR)
 Delete lowest and highest 25% of scores
 IQR is range of what remains
 May be too little influenced by extremes
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Chapter 5 Measures of Variability
Trimmed Samples
• Delete a fixed (usually small) percentage
of extreme scores
• Trimmed statistics are statistics
computed on trimmed samples.
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Chapter 5 Measures of Variability
Deviation Scores
• Definition
 distance between a score and a measure of
central tendency
 usually deviation around the mean
(X  X )
• Importance
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Chapter 5 Measures of Variability
Variance
• Definitional formula
( X  X )
s 
N 1
2
• Example
 See next slide
2
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Chapter 5 Measures of Variability
Calculation

X
2 4 5 8 7 4
30
(X  X )
-3 -1 0 3 2 -1
0
(X  X )
9 1 0 9 4 1
24
2
( X  X )
24
s 

 4.80
N 1
5
2
2
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Chapter 5 Measures of Variability
Standard Deviation
• Definitional formula
 The square root of the variance
( X  X )
s s 
N 1
2
2
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Chapter 5 Measures of Variability
Computational Formula
(X )2 2 2 2 2 2 2 30 2
X 
2  4 5 8 7  4 
2
N
6
s 

N 1
5
2
 4.80
( X ) 2
X 
N  4.8  2.19
s
N 1
2
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Chapter 5 Measures of Variability
Estimators
• Mean
 Unbiased estimate of population mean ()
• Define unbiased
 Long range average of statistic is equal to the
parameter being estimated.
• Variance
( X  X ) 2
s 
N 1
2
 Unbiased estimate of 2
Cont.
Chapter 5 Measures of Variability
Estimators--cont.
 Using
•
2

(
X

X
)
s2 
N
gives biased estimate
 Standard deviation
• use square root of unbiased estimate.
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Chapter 5 Measures of Variability
Merrell’s Music Study SPSS
Printout
WEEK4
Treatment
Mean
N
Std. Deviation
Quiet
307.2319
23
71.8267
Mozart
114.5833
24
36.1017
Anthrax
1825.8889
24
103.1392
755.4601
71
777.9646
Total
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Chapter 5 Measures of Variability
Boxplots
• The general problem
 A display that shows dispersion for center and tails of
distribution
• Calculational steps (simple solution)
 Find median
 Find top and bottom 25% points (quartiles)
 eliminate top and bottom 2.5% (fences)
 Draw boxes to quartiles and whiskers to fences, with
remaining points as outliers
• Boxplots for comparing groups
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Chapter 5 Measures of Variability
Combined Merrell Data
3000
2000
1000
0
-1000
N=
71
W EEK4
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Chapter 5 Measures of Variability
Merrell Data by Group
3000
2000
1000
WEEK4
0
-1000
N=
23
24
24
Quiet
Mozart
Anthrax
Treatment Condition
Chapter 5 Measures of Variability
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Review Questions
• What do we look for in a measure of
dispersion?
• What role do outliers play?
• Why do we say that the variance is a
measure of average variability around the
mean?
• Why do we take the square root of the
variance to get the standard deviation?
Cont.
Chapter 5 Measures of Variability
Review Questions--cont.
• How does a boxplot reveal dispersion?
• What do David Merrell’s data tell us
about the effect of music on mice?
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