Download distances to center

Last Time
• Notions of Center
– Expected Value (probability distributions)
• Interpretation
• Binomial Expected Value
• Properties
– Median
• Comparison to mean
Administrative Matters
Midterm I Results:
• Check Scores
– Points taken off in circles
– Total points for each problem in [purple]
– Total for each page, lower right corner
– Total those to score on front
– Check Blackboard entry
Administrative Matters
Midterm I Results: Interpretation of scores:
Administrative Matters
Midterm I Results: Interpretation of scores:
84 – 100
Very Pleased
Administrative Matters
Midterm I Results: Interpretation of scores:
84 – 100
Very Pleased
(No letter grade yet, since to early, expect
many to change a lot)
Administrative Matters
Midterm I Results: Interpretation of scores:
84 – 100
Very Pleased
(Good to see many here,
including #(100s) = 12)
Administrative Matters
Midterm I Results: Interpretation of scores:
84 – 100
Very Pleased
65 – 83
OK
Administrative Matters
Midterm I Results: Interpretation of scores:
84 – 100
Very Pleased
65 – 83
OK
Administrative Matters
Midterm I Results: Interpretation of scores:
84 – 100
Very Pleased
65 – 83
OK
0 – 64
Reco: Drop Course
Administrative Matters
Midterm I Results: Interpretation of scores:
84 – 100
Very Pleased
65 – 83
OK
0 – 64
Reco: Drop Course
Don’t want to? Let’s talk…
Reading In Textbook
Approximate Reading for Today’s Material:
Pages 55-68, 319-326
Approximate Reading for Next Class:
Pages 59-62, 279-285, 62-64, 337-344
Big Picture
•
Margin of Error
•
Choose Sample Size
Need better prob tools
Start with visualizing probability distributions
Big Picture
•
Margin of Error
•
Choose Sample Size
Need better prob tools
Start with visualizing probability distributions,
Next exploit constant shape property of Bi
Big Picture
Start with visualizing probability distributions,
Next exploit constant shape property of Binom’l
Big Picture
Start with visualizing probability distributions,
Next exploit constant shape property of Binom’l
Centerpoint feels p
Big Picture
Start with visualizing probability distributions,
Next exploit constant shape property of Binom’l
Centerpoint feels p
Spread feels n
Big Picture
Start with visualizing probability distributions,
Next exploit constant shape property of Binom’l
Centerpoint feels p
Spread feels n
Now quantify these ideas, to put them to work
Notions of Center
Will later study “notions of spread”
Notions of Center
Textbook: Sections 4.4 and 1.2
Recall parallel development:
(a)
Probability Distributions
(b)
Lists of Numbers
Study 1st, since easier
Notions of Center
Caution about mean:
Works well for ~symmetric distributions
E.g. Buffalo Snowfalls
Manual bins Small Binwidth
8
7
6
3
2
1
130
125
120
115
110
105
100
95
90
85
80
75
70
65
60
55
50
45
40
35
0
30
Using AVERAGE)
4
25
(from Excel,
5
20
Mean = 80.3
Notions of Center
Caution about mean:
Works well for ~symmetric distributions
E.g. Buffalo Snowfalls
Manual bins Small Binwidth
8
7
6
3
2
1
130
125
120
115
110
105
100
95
90
85
80
75
70
65
60
55
50
45
40
35
0
30
Notion of “Center”
4
25
Visually sensible
5
20
Mean = 80.3
Notions of Center
Caution about mean:
But poorly for asymmetric distributions
Notions of Center
Caution about mean:
But poorly for asymmetric distributions
E.g. British Suicides Data
•
Time (in days) to suicide attempt
•
Of Suicide Patients
•
After Initial Treatment
Notions of Center
Caution about mean:
But poorly for asymmetric distributions
E.g. British Suicides Data
Analyzed in:
http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg6.xls
Notions of Center
Caution about mean:
But poorly for asymmetric distributions
E.g. British Suicides Data
British Suicides
Clearly not mound shaped
Very asymmetric
20
18
16
14
12
8
6
4
2
0
0
40
80
120
160
200
240
280
320
360
400
440
480
520
560
600
640
680
720
Called “right skewed”
10
Notions of Center
Caution about mean:
But poorly for asymmetric distributions
E.g. British Suicides Data
British Suicides
20
Mean = 122.3
18
Sensible as “center”??
14
16
12
10
Too Small…
8
6
4
2
0
0
40
80
120
160
200
240
280
320
360
400
440
480
520
560
600
640
680
720
%(data ≥) = 30.2%
Notions of Center
Perhaps better notion of “center”:
•
Take center to be point in middle
•
I.e. have 50% of data smaller
•
And 50% of data larger
This is called the “median”
Notions of Center
Median: = Value in middle (of sorted list)
Unsorted E.g:
Sorted E.g:
3
0
1
1
27
2
2
3
0
27
Notions of Center
Median: = Value in middle (of sorted list)
Unsorted E.g:
Sorted E.g:
3
0
1
1
27
2
2
3
0
27
One in middle???
NO, must sort
Notions of Center
Median: = Value in middle (of sorted list)
Unsorted E.g:
Sorted E.g:
3
0
1
1
27
2
2
3
0
27
Sensible version of “middle”
Notions of Center
What about ties?
Tie for point in
middle
Sorted E.g:
0
1
2
3
Notions of Center
What about ties?
Sorted E.g:
0
Tie for point in
1
middle
2
3
Break by taking average (of two tied values):
e.g. Median = 1.5
Notions of Center
Median: = Value in middle (of sorted list)
Unsorted E.g:
Sorted E.g:
3
0
1
1
27
2
2
3
0
27
EXCEL:
use function “MEDIAN”
Notions of Center
EXCEL:
use function “MEDIAN”
Very similar to other functions
E.g. see:
http://www.stat-or.unc.edu/webspace/courses/marron/UNCstor155-2009/ClassNotes/Stor155Eg6.xls
Notions of Center
E.g. Buffalo Snowfalls
Mean = 80.3
Manual bins Small Binwidth
8
Median = 79.6
7
6
3
2
1
130
125
120
115
110
105
100
95
90
85
80
75
70
65
60
55
50
45
40
35
0
30
symmetry)
4
25
(expected from
5
20
Very similar
Notions of Center
E.g. British Suicides Data
Mean = 122.3
Median = 77.5
British Suicides
20
Substantially different
But which is better?
18
16
14
12
10
8
6
4
2
0
0
40
80
120
160
200
240
280
320
360
400
440
480
520
560
600
640
680
720
Goal 1: ½ - ½ middle
Notions of Center
E.g. British Suicides Data
Mean = 122.3
Median = 77.5
British Suicides
20
Substantially different
But which is better?
18
16
14
12
10
Goal 2: long run average
8
6
4
2
0
0
40
80
120
160
200
240
280
320
360
400
440
480
520
560
600
640
680
720
Goal 1: ½ - ½ middle
Notions of Center
HW:
1.63 a (median only), c
1.65 (hint: use histogram)
Research Corner
Recall Hidalgo
StampData &
Movie over binwidth
Main point:
Binwidth drives
histogram performance
Research Corner
Less known fact:
Bin location also has
Serious effect
(even for fixed width)
Research Corner
How many bumps?
~2?
Research Corner
How many bumps?
~3?
Research Corner
How many bumps?
~7?
Research Corner
Compare with “smoothed
version” called
“Kernel Density Estimate”
Peaks appear:
when entirely in a bin
Research Corner
Compare with “smoothed
version” called
“Kernel Density Estimate”
Peaks disappear:
when split between
two bins bin
Research Corner
Question: What is a Kernel Density Estimate?
E.g. Chondrite Data
•
Type of Meteor (never part of planet)
Research Corner
Question: What is a Kernel Density Estimate?
E.g. Chondrite Data
•
Type of Meteor (never part of planet)
•
From how many sources do they come?
Research Corner
Question: What is a Kernel Density Estimate?
E.g. Chondrite Data
•
Type of Meteor (never part of planet)
•
From how many sources do they come?
(Current Issue: Meteors from Mars show life?)
Research Corner
Question: What is a Kernel Density Estimate?
E.g. Chondrite Data
•
Type of Meteor (never part of planet)
•
From how many sources do they come?
Research Corner
Question: What is a Kernel Density Estimate?
E.g. Chondrite Data
•
Type of Meteor (never part of planet)
•
From how many sources do they come?
•
Data Set
22 Chondrites
Research Corner
Question: What is a Kernel Density Estimate?
E.g. Chondrite Data
•
Type of Meteor (never part of planet)
•
From how many sources do they come?
•
Data Set
22 Chondrites
(Histograms as slippery)
Research Corner
Question: What is a Kernel Density Estimate?
E.g. Chondrite Data
•
Type of Meteor (never part of planet)
•
From how many sources do they come?
•
Data Set
•
Interesting measurement:
22 Chondrites
% Silica
Research Corner
Question: What is a Kernel Density Estimate?
Chondrite Data
% Silica
Research Corner
Question: What is a Kernel Density Estimate?
Chondrite Data
% Silica
Research Corner
Question: What is a Kernel Density Estimate?
Chondrite Data
% Silica
Area = 1/n for
each data point
Research Corner
Question: What is a Kernel Density Estimate?
Chondrite Data
% Silica
Area = 1/n for
each data point
Sum for Total
Area = 1
Research Corner
Question: What is a Kernel Density Estimate?
Smooth Histogram
Research Corner
Question: What is a Kernel Density Estimate?
Smooth Histogram
(Areas correspond
to frequencies)
Research Corner
Question: What is a Kernel Density Estimate?
Smooth Histogram
Shows 3 clear
bumps
Research Corner
Question: What is a Kernel Density Estimate?
Smooth Histogram
Shows 3 clear
bumps
Suggests 3 sources
of chondrites
Research Corner
Kernel Density Estimate For Hidalgo Stamps
Research Corner
Kernel Density Estimate For Hidalgo Stamps
Kernel width is crucial
Research Corner
Kernel Density Estimate For Hidalgo Stamps
Kernel width is crucial
Deep question:
How to choose?
Notions of Center
Another view of mean vs. median:
Notions of Center
Another view of mean vs. median:
Use e-textbook Applet: “Mean and Median”
Notions of Center
Another view of mean vs. median:
Use e-textbook Applet: “Mean and Median”
Interactive Construction of Toy Examples
Mean and Median Applet
On Stats Portal:
http://courses.bfwpub.com/ips6e
• Login
• Resources
• Student Resources – Statistical Applets
• Mean and Median
Mean and Median Applet
Toy Example 1: Understand Applet
• Click below line
to add data pt.
Mean and Median Applet
Toy Example 1: Understand Applet
• Click below line
to add data pt.
• Add another
Mean and Median Applet
Toy Example 1: Understand Applet
• Click below line
to add data pt.
• Add another
• And one more
Mean and Median Applet
Toy Example 1: Understand Applet
•
Show mean
(recall balance
point) in green
Mean and Median Applet
Toy Example 1: Understand Applet
•
Show median
(recall ½ -way
point) in red
Mean and Median Applet
Toy Example 1: Understand Applet
•
Big difference,
since skewed
data set
Mean and Median Applet
Toy Example 2: Effect of Single Outlier
Mean and Median Applet
Toy Example 2: Effect of Single Outlier
• Note mean
outside range of
other data points
Mean and Median Applet
Toy Example 2: Effect of Single Outlier
• Note mean
outside range of
other data points
• While median
is inside
Mean and Median Applet
Toy Example 2: Effect of Single Outlier
• Fun to “grab”
outlier and
move around
Mean and Median Applet
Toy Example 2: Effect of Single Outlier
• Fun to “grab”
outlier and
move around
Mean and Median Applet
Toy Example 2: Effect of Single Outlier
• Fun to “grab”
outlier and
move around
• Median stable
• Mean “feels”
outlier
Mean and Median Applet
Robust Statistics:
•
Mean is sensitive to outliers
•
Median is not sensitive
Called a “robust notion of center”
•
Studied as part of statistical research
•
There are many others
(beyond scope of this course)
Mean and Median Applet
Toy Example 3: Bimodal distribution
• Two lumps of data
Mean and Median Applet
Toy Example 3: Bimodal distribution
• Two lumps of data
• Where is center
of population???
Mean and Median Applet
Toy Example 3: Bimodal distribution
• Two lumps of data
• Where is center
of population???
• Median?
Mean and Median Applet
Toy Example 3: Bimodal distribution
• Two lumps of data
• Where is center
of population???
• Median?
(not compelling)
Mean and Median Applet
Toy Example 3: Bimodal distribution
• Two lumps of data
• Where is center
of population???
• Median?
(not compelling)
• Mean?
Mean and Median Applet
Toy Example 3: Bimodal distribution
• Two lumps of data
• Where is center
of population???
• Median?
(not compelling)
• Mean?
(more sensible)
Mean and Median Applet
Toy Example 3: Bimodal distribution
• Now add one
more data point
Mean and Median Applet
Toy Example 3: Bimodal distribution
• Now add one
more data point
• Small change
in mean
Mean and Median Applet
Toy Example 3: Bimodal distribution
• Now add one
more data point
• Small change
in mean
• Big change
in median
Mean and Median Applet
Toy Example 3: Bimodal distribution
• Now add one
more data point
• Small change
in mean
• Big change
in median
• Note tie color:
yellow = red + green
Mean and Median Applet
Toy Example 3: Bimodal distribution
• Now add one
more data point
Mean and Median Applet
Toy Example 3: Bimodal distribution
• Now add one
more data point
• Small change
in mean
• Big change
in median
Mean and Median Applet
Mean vs. Median:
• Which is “better”?
Mean and Median Applet
Mean vs. Median:
• Which is “better”?
• Not comparable
• Mean better sometimes
• Median better other times
Mean and Median Applet
Mean vs. Median:
• Which is “better”?
• Not comparable
• Mean better sometimes
• Median better other times
• Depends on Context (specific case)
Mean and Median Applet
Mean vs. Median:
• Which is “better”?
• Not comparable
• Mean better sometimes
• Median better other times
• Depends on Context (specific case)
• Best you can do:
Understand issues, and make choice
Mean and Median Applet
HW:
1.65
1.67
1.69
And now for something
completely different
More Lateral Thinking Puzzles:
And now for something
completely different
More Lateral Thinking Puzzles:
cycle
cycle
cycle
And now for something
completely different
More Lateral Thinking Puzzles:
cycle
cycle
cycle
tricycle
And now for something
completely different
More Lateral Thinking Puzzles:
0
________
M. D.
Ph. D.
And now for something
completely different
More Lateral Thinking Puzzles:
0
________
M. D.
Ph. D.
Two Degrees Below Zero
And now for something
completely different
More Lateral Thinking Puzzles:
knee
______________
light
And now for something
completely different
More Lateral Thinking Puzzles:
knee
______________
light
neon light
And now for something
completely different
More Lateral Thinking Puzzles:
ground
_______________________
feet feet feet feet feet feet
And now for something
completely different
More Lateral Thinking Puzzles:
ground
_______________________
feet feet feet feet feet feet
six feet underground
Big Picture
•
Margin of Error
•
Choose Sample Size
Need better prob tools
Start with visualizing probability distributions
Big Picture
Start with visualizing probability distributions,
Next exploit constant shape property of Binom’l
Centerpoint feels p
Spread feels n
Notions of Spread
Textbook: Sections 4.4 and 1.2
Recall parallel development:
(a)
Probability Distributions
(b)
Lists of Numbers
Study both together
Notions of Spread
Toy Example:
List 1
=
-1, 0, 1
List 2
=
-10, 0, 10
Notions of Spread
Toy Example:
•
List 1
=
-1, 0, 1
List 2
=
-10, 0, 10
Both have mean 0
Notions of Spread
Toy Example:
List 1
=
-1, 0, 1
List 2
=
-10, 0, 10
•
Both have mean 0
•
But List 2 is far more spread
Notions of Spread
Toy Example:
List 1
=
-1, 0, 1
List 2
=
-10, 0, 10
•
Both have mean 0
•
But List 2 is far more spread
•
How to measure this?
Notions of Spread
Approach:
Study deviations
Notions of Spread
Approach:
Study deviations = distances to center
Notions of Spread
Approach:
Study deviations = distances to center
(a) Probability Distributions
Based on Random Variable X
Notions of Spread
Approach:
Study deviations = distances to center
(a) Probability Distributions
Based on Random Variable X
Deviation = X – EX
Notions of Spread
Approach:
Study deviations = distances to center
(a) Probability Distributions
Based on Random Variable X
Deviation = X – EX
Recall Center of X Distribution
Notions of Spread
Approach:
Study deviations = distances to center
(a) Probability Distributions
Deviation = X – EX
Notions of Spread
Approach:
Study deviations = distances to center
(a) Probability Distributions
Deviation = X – EX
(b) Lists of Numbers
Notions of Spread
Approach:
Study deviations = distances to center
(a) Probability Distributions
Deviation = X – EX
(b) Lists of Numbers
Based on x1 , x2 ,, xn .
Notions of Spread
Approach:
Study deviations = distances to center
(a) Probability Distributions
Deviation = X – EX
(b) Lists of Numbers
Based on x1 , x2 ,, xn .
Deviations are: x1  x , x2  x ,, x  xn .
Notions of Spread
Approach:
Study deviations = distances to center
(a) Probability Distributions
Deviation = X – EX
(b) Lists of Numbers
Recall: Average,
i.e. center of list
Based on x1 , x2 ,, xn .
Deviations are: x1  x , x2  x ,, x  xn .
Notions of Spread
How to summarize deviations:
Mean Absolute Deviation
Notions of Spread
How to summarize deviations:
Mean Absolute Deviation
(a) Probability Distributions
E |X – EX|
Notions of Spread
How to summarize deviations:
Mean Absolute Deviation
(a) Probability Distributions
E |X – EX|
(# spaces to center)
Notions of Spread
How to summarize deviations:
Mean Absolute Deviation
(a) Probability Distributions
E |X – EX|
(make positive, thus distance to center)
Notions of Spread
How to summarize deviations:
Mean Absolute Deviation
(a) Probability Distributions
E |X – EX|
(Summarize as average distance)
Notions of Spread
How to summarize deviations:
Mean Absolute Deviation
(a) Probability Distributions
E |X – EX|
(b) Lists of Numbers n
1
n
 x x
i 1
i
.
Notions of Spread
How to summarize deviations:
Mean Absolute Deviation
(a) Probability Distributions
E |X – EX|
(b) Lists of Numbers n
1
n
 x x
i 1
i
(# spaces to center)
Notions of Spread
How to summarize deviations:
Mean Absolute Deviation
(a) Probability Distributions
E |X – EX|
(b) Lists of Numbers n
1
n
 x x
i 1
i
(make positive, thus distance to center)
Notions of Spread
How to summarize deviations:
Mean Absolute Deviation
(a) Probability Distributions
E |X – EX|
(b) Lists of Numbers n
1
n
 x x
i 1
i
(Summarize as average distance)
Notions of Spread
How to summarize deviations:
Mean Absolute Deviation
Problems:
Notions of Spread
How to summarize deviations:
Mean Absolute Deviation
Problems:
•
Hard to find distribution (i.e. measure error)
Notions of Spread
How to summarize deviations:
Mean Absolute Deviation
Problems:
•
Hard to find distribution (i.e. measure error)
•
No calculus for minimizing (later in course)
Notions of Spread
How to summarize deviations:
Mean Absolute Deviation
Problems:
•
Hard to find distribution (i.e. measure error)
•
No calculus for minimizing (later in course)
•
No “shortcut formula” (later)
Notions of Spread
How to summarize deviations:
Mean Absolute Deviation
Problems:
•
Hard to find distribution (i.e. measure error)
•
No calculus for minimizing (later in course)
•
No “shortcut formula” (later)
Hence will not study further here
Notions of Spread
How to summarize deviations:
Mean Absolute Deviation
Problems:
•
Hard to find distribution (i.e. measure error)
•
No calculus for minimizing (later in course)
•
No “shortcut formula” (later)
Hence will not study further here
(but is important in more advanced courses)
Notions of Spread
More common summary of deviations:
Standard Deviation
Notions of Spread
More common summary of deviations:
Standard Deviation
(a) Probability Distributions
  E X  EX 
2
Notions of Spread
More common summary of deviations:
Standard Deviation
(a) Probability Distributions
  E X  EX 
2
Greek “sigma” (lower case)
Notions of Spread
More common summary of deviations:
Standard Deviation
(a) Probability Distributions
  E X  EX 
2
Essentially root of average square deviations
Notions of Spread
More common summary of deviations:
Standard Deviation
(a) Probability Distributions
  E X  EX 
2
Square root makes units same as X
(e.g. ft, not ft2 = sq. ft)
Notions of Spread
More common summary of deviations:
Standard Deviation
(a) Probability Distributions
  E X  EX 
2
(b) Lists of Numbers
s
n
1
n 1
 x  x 
i 1
2
i
Notions of Spread
More common summary of deviations:
Standard Deviation
(a) Probability Distributions
  E X  EX 
2
(b) Lists of Numbers
s
n
1
n 1
 x  x 
i 1
2
i
Again root of average square deviations
Notions of Spread
More common summary of deviations:
Standard Deviation
(a) Probability Distributions
  E X  EX 
2
(b) Lists of Numbers
s
n
1
n 1
 x  x 
i 1
2
i
Again square root makes units same as X
Notions of Spread
More common summary of deviations:
Standard Deviation
(a) Probability Distributions
  E X  EX 
2
(b) Lists of Numbers
n
s
Reason for
1
n 1
instead of
1
n 1
1
n
 x  x 
i 1
2
i
discussed later
Notions of Spread
E.g. 1:
for list 10, 11, 12
Notions of Spread
E.g. 1:
for list 10, 11, 12,
x  11
Notions of Spread
E.g. 1:
for list 10, 11, 12,
deviations are: -1, 0, 1
x  11
Notions of Spread
E.g. 1:
x  11
for list 10, 11, 12,
deviations are: -1, 0, 1
s
1
2
1
2

 0 1  1
2
2
Notions of Spread
E.g. 1:
deviations are: -1, 0, 1
•
x  11
for list 10, 11, 12,
Same as for list -1, 0, 1
s
1
2
1
2

 0 1  1
2
2
Notions of Spread
E.g. 1:
x  11
for list 10, 11, 12,
deviations are: -1, 0, 1
s
•
Same as for list -1, 0, 1
•
s does not feel location
1
2
1
2

 0 1  1
2
2
Notions of Spread
E.g. 1:
x  11
for list 10, 11, 12,
deviations are: -1, 0, 1
s
•
Same as for list -1, 0, 1
•
s does not feel location
•
Can shift all numbers in list
1
2
1
2

 0 1  1
2
2
Notions of Spread
E.g. 1:
x  11
for list 10, 11, 12,
deviations are: -1, 0, 1
s
1
2
1
2
•
Same as for list -1, 0, 1
•
s does not feel location
•
Can shift all numbers in list, and

 0 1  1
not change spread
2
2
Notions of Spread
E.g. 1:
for list 10, 11, 12,
E.g. 2:
for list
0, 11, 22
Notions of Spread
E.g. 1:
for list 10, 11, 12,
E.g. 2:
for list
0, 11, 22
x  11
Notions of Spread
E.g. 1:
for list 10, 11, 12,
E.g. 2:
for list
0, 11, 22
Dev’ns are: -11, 0, 11
x  11
Notions of Spread
E.g. 1:
for list 10, 11, 12,
E.g. 2:
for list
x  11
0, 11, 22
Dev’ns are: -11, 0, 11
s
1
2
11
2

 0  11  11
2
2
Notions of Spread
E.g. 1:
for list 10, 11, 12,
E.g. 2:
for list
Dev’ns are: -11, 0, 11
•
x  11
0, 11, 22
s
1
2
11
2
s for E.g. 2 is 11 times larger

 0  11  11
2
2
Notions of Spread
E.g. 1:
for list 10, 11, 12,
E.g. 2:
for list
x  11
0, 11, 22
Dev’ns are: -11, 0, 11
s
1
2
11
2
•
s for E.g. 2 is 11 times larger
•
Since #s are 11 times more spread
(on number line)

 0  11  11
2
2
Notions of Spread
E.g. 3:
x
f(x)
0
0.1
1
0.2
4
0.7
Notions of Spread
E.g. 3:
x
f(x)
0
0.1
1
0.2
Expected Value (center)
4
0.7
Notions of Spread
E.g. 3:
x
f(x)
0
0.1
1
0.2
4
0.7
Expected Value (center)
EX = (0.1)0 + (0.2)1 + (0.7)4 = 3
Notions of Spread
E.g. 3:
x
f(x)
x-EX
0
0.1
-3
1
0.2
-2
4
0.7
1
EX = (0.1)0 + (0.2)1 + (0.7)4 = 3
Notions of Spread
E.g. 3:
x
f(x)
x-EX
(x-EX)2
0
0.1
-3
9
1
0.2
-2
4
4
0.7
1
1
Notions of Spread
E.g. 3:
x
f(x)
x-EX
(x-EX)2
0
0.1
-3
9
1
0.2
-2
4
4
0.7
1
1
So E(X – EX)2 = (0.1)9 + (0.2)4 + (0.7)1
Notions of Spread
E.g. 3:
x
f(x)
x-EX
(x-EX)2
0
0.1
-3
9
1
0.2
-2
4
4
0.7
1
1
So E(X – EX)2 = (0.1)9 + (0.2)4 + (0.7)1 = 2.4
Notions of Spread
E.g. 3:
x
f(x)
x-EX
(x-EX)2
0
0.1
-3
9
1
0.2
-2
4
4
0.7
1
1
So E(X – EX)2 = (0.1)9 + (0.2)4 + (0.7)1 = 2.4
So σ(X) =
2.4 = sqrt(2.4)
Notions of Spread
Quantity closely related to standard deviation
Notions of Spread
Quantity closely related to standard deviation
is its square
Notions of Spread
Quantity closely related to standard deviation
is its square, called the variance
(a) Probs: var(X)
Notions of Spread
Quantity closely related to standard deviation
is its square, called the variance
(a) Probs: var(X) = [σ(X)]2
Notions of Spread
Quantity closely related to standard deviation
is its square, called the variance
(a) Probs: var(X) = [σ(X)]2 = σ2
Notions of Spread
Quantity closely related to standard deviation
is its square, called the variance
(a) Probs: var(X) = [σ(X)]2 = σ2 = E(X - EX)2
Notions of Spread
Quantity closely related to standard deviation
is its square, called the variance
(a) Probs: var(X) = [σ(X)]2 = σ2 = E(X - EX)2
(b) Lists:
s 
2
n
1
n 1
 x  x 
i 1
2
i
Notions of Spread
Excel Computation for Lists:
Notions of Spread
Excel Computation for Lists:
•
Can do manually (using SUM & SQRT)
Notions of Spread
Excel Computation for Lists:
•
Can do manually (using SUM & SQRT)
•
But faster to use:
–
STDEV
–
VAR
Notions of Spread
Excel Computation for Lists:
•
Can do manually (using SUM & SQRT)
•
But faster to use:
•
–
STDEV
–
VAR
Application same as for other functions
Notions of Spread
HW:
C19:
Calculate the standard
deviation for the following lists, and
compare qualitatively in terms of spread:
(a) 1, 3, 3, 1
(1.15)
(b) -6, -4, -4, -6
(1.15)
(c) 1, 5, 5, 1
(2.31)
(d) 1, 1, 1, 1
(0)
Notions of Spread
HW:
1.79a
4.77
(2.689, 1.710)