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Review
Mean—arithmetic average, sum of all
scores divided by the number of scores
Median—balance point of the data, exact
middle of the distribution, 50th percentile
Mode—highest frequency, can be more
than one
1
Review
X
5
4
3
2
1
f
2
5
3
2
2
Find the mean,
median, mode
2
Review
X
5
4
3
2
1
f fX
2
5
3
2
2
N ∑fX
Find the mean,
median, mode
Mean=sum of all
scores(∑fX) /number
of scores(N)
3
Review
X
5
4
3
2
1
f fX
2
5
3
2
2
Find the mean,
median, mode
Mean=sum of all
scores(∑fX) /number
of scores(N)
Median=middle point
(N-1/2)th position
4
Review
X
5
4
3
2
1
f fX
2
5
3
2
2
Find the mean,
median, mode
Mean=sum of all
scores(∑fX) /number
of scores(N)
Median=middle point
(N-1/2)th position
Mode=greatest f
5
Measures of
Variability
Major Points
The general problem
Range and related statistics
Deviation scores
The variance and standard deviation
Boxplots
Review questions
7
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
8
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.
Borrowed from David Howell, 2000
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Results
Anthrax mice took much longer to run
Much greater variability in Anthrax group

See following graphs for Anthrax and Mozart
We often see greater variability with larger
mean
10
Mozart Group
12
10
8
6
4
2
Std. Dev = 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. Dev = 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
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
13
Trimmed Samples
Delete a fixed (usually small) percentage
of extreme scores
Trimmed statistics are statistics computed
on trimmed samples.
14
Deviation Scores
Definition


distance between a score and a measure of
central tendency
usually deviation around the mean
(X  X )
Importance
15
Variance
Definitional formula
( X  X )
s 
N 1
2
2
Example

See next slide
16
Computing the Variance
X
2
4
5
8
7
4
30
X-X
¯
(X - X)
¯2
Definitional formula
2

(
X

X
)
s2 
N 1
Find the mean
N=6
∑X=30
30/6=5
17
Computing the Variance
X
X-X
¯
2
-3
4
-1
5
0
8
3
7
2
4
-1
30
0
(X - ¯
X)2
2

(
X

X
)
s2 
N 1
Calculate the
difference between
each score and the
mean and sum
18
Computing the Variance
X
X-X
¯
(X - ¯
X)2
2
-3
9
4
-1
1
5
0
0
8
3
9
7
2
4
4
-1
1
30
0
24
( X  X ) 2 24
s 

 4.80
N 1
5
2
Calculate the square
of the difference
between each score
and the mean and
sum
Standard Deviation is
the square root
19
Standard Deviation
Definitional formula

The square root of the variance
( X  X )
s s 
N 1
2
2
Computational formula based on algebraic
manipulation

Makes it easier to calculate
20
Computational Formula
( X )
30
2
2
2
2
2
2
X 
2  4 5 8 7  4 
2
N
6
s 

N 1
5
2
2
2
 4.80
( X )
X 
N  4.8  2.19
s
N 1
2
2
21
Try one
X
5
4
3
2
1
f
2
5
3
2
2
N
(X )
X 
N
s
N 1
2
2
22
Try one
X
5
4
3
2
1
f fX
2
5
3
2
2
N ∑fX
(X )
X 
N
s
N 1
2
2
23
Try one
X
5
4
3
2
1
f
2
5
3
2
2
14
fX
10
20
9
4
2
45
(X )
X 
N
s
N 1
2
2
24
Try one
X
5
4
3
2
1
f
2
5
3
2
2
14
fX
10
20
9
4
2
45
X2
fX2
(X )
X 
N
s
N 1
2
2
∑fX2
25
Try one
X
5
4
3
2
1
f
2
5
3
2
2
14
fX
10
20
9
4
2
45
X2
25
16
9
4
1
fX2
50
80
27
8
2
167
(X )
X 
N
s
N 1
2
2
26
X
5
4
3
2
1
f
2
5
3
2
2
14
fX
10
20
9
4
2
45
X2
25
16
9
4
1
fX2
50
80
27
8
2
167
(X )
X 
N
s
N 1
2
2
(45)
167 
14
s
14  1
2
2025
167 
14  167  144.64  22.36  1.72  1.31
s
13
13
13
27
Estimators
Mean

Unbiased estimate of population mean ()
Define unbiased

Long range average of statistic is equal to the parameter
being estimated.
Variance

2

(
X

X
)
s2 
N 1
Unbiased estimate of 2
28
Cont.
Estimators--cont.

Using
2

(
X

X
)
s2 
N
gives biased estimate

Standard deviation
use square root of unbiased estimate.
29
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
Total
755.4601 71
777.9646
30
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
31
Combined Merrell Data
3000
2000
1000
0
-1000
N =
71
WEEK4
32
Merrell Data by Group
3000
2000
1000
WEEK4
0
-1000
N =
23
24
24
Quiet
Mozart
Anthrax
Treatment Condition
33
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?
34
Cont.
Review Questions--cont.
How does a boxplot reveal dispersion?
What do David Merrell’s data tell us about
the effect of music on mice?
35