The Standard Deviation
as a Ruler
 A student got a 67/75 on the first exam
and a 64/75 on the second exam. She
was disappointed that she did not score
as well on the second exam.
 To her surprise, the professor said she
actually did better on the second exam,
relative to the rest of the class.
1
The Standard Deviation
as a Ruler
 How can this be?
 Both exams exhibit variation in the
scores.
 However, that variation may be
different from one exam to the next.
 The standard deviation provides a
ruler for comparing the two exam
scores.
2
Summarizing Exam Scores
 Exam 1
– Score: 67
– Mean:
y  59.5
– Standard
Deviation:
s  8.61
 Exam 2
– Score: 64
– Mean:
y  50.1
– Standard
Deviation:
s  11.86
3
Standardizing
Look at the number of
standard deviations the
score is from the mean.
y y
z
s
4
Standardized Exam Scores
 Exam 1
 Exam 2
– Score: 67
– Score: 64
67  59.5
z
8.61
z  0.87
64  50.1
z
11.86
z  1.17
5
Standardized Exam Scores
On exam 1, the 67 was 0.87
standard deviations better than the
mean.
On exam 2, the 64 was 1.17
standard deviations better than the
mean.
6
Standardizing
Shifts the distribution by
subtracting off the mean.
Rescales the distribution by
dividing by the standard deviation.
7
Distribution of Low Temps
20
10
Count
15
5
-10
0
10
20
30
40
50
Low Temperature (o F)
8
Shifting the Distribution
20
10
Count
15
5
-40
-30
-20
-10
0
10
Low Temperature – 32 (o F)
20
9
Shifting
 Temperature (o F)  Temp – 32 (o F)
– Median: 24.0o F
– Median: –8o F
– Mean: 24.4o F
– Mean: –7.6o F
– IQR: 16.0o F
– IQR: 16.0o F
– Std Dev: 11.22o F
– Std Dev: 11.22o F
10
Shifting
When adding (or subtracting) a
constant:
– Measures of position and center
increase (or decrease) by that
constant.
– Measures of spread do not change.
11
Rescaling
10
Count
15
5
-20
-15
-10
-5
0
5
10
Low Temperature (o C)
12
Rescaling
 Temp – 32 (o F)
 Temperature (o C)
– Median: –8o F
– Median: –4.4o F
– Mean: –7.6o F
– Mean: –4.2o F
– IQR: 16.0o F
– IQR: 8.9o F
– Std Dev: 11.22o F
– Std Dev: 6.24o F
13
Rescaling
When multiplying (or dividing) by
a constant:
– All measures of position, center and
spread are multiplied (or divided)
by that constant.
14
Standardizing
 Standardizing does not change the
shape of the distribution.
 Standardizing changes the center by
making the mean 0.
 Standardizing changes the spread by
making the standard deviation 1.
15
Normal Models
Our conceptualization of what the
distribution of an entire population
of values would look like.
Characterized by population
parameters: μ and σ.
16
30
Percent
20
10
0
40
45
50
55
60
65
70
75
80
Height
17
Describe the sample
Shape is symmetric and mounded
in the middle.
Centered at 60 inches.
Spread between 45 and 75 inches.
30% of the sample is between 60
and 65 inches.
18
Normal Models
Our conceptualization of what the
distribution of an entire population
of values would look like.
Characterized by a bell shaped curve
with population parameters
– Population mean = μ
– Population standard deviation = σ.
19
Sample Data
0.08
0.07
Density
0.06
0.05
0.04
0.03
0.02
0.01
0.00
40
45
50
55
60
65
70
75
80
Height
20
Normal Model
0.08
0.07
Density
0.06
0.05

0.04
0.03
0.02
0.01
0.00
40
45
50
55
60
65
70
75
80
Height (inches)
21
Normal Model
0.08
0.07
0.05
0.04
0.03
0.02
0.01
0.00
40
45
50
55
60
65
70
75
80
Height (inches)
0.08
0.07
0.06
Density
Sample – a
few items from
the population.
Example: 550
children.
0.06
Density
Population – all items
of interest.
Example: All children
age 5 to 19.
Variable: Height
0.05
0.04
0.03
0.02
0.01
0.00
40
45
50
55
60
65
70
75
80
Height
22
Normal Model
Height
Center:
– Population mean, μ = 60 in.
Spread:
– Population standard deviation, σ = 6
in.
23
68-95-99.7 Rule
For Normal Models
– 68% of the values fall within 1
standard deviation of the mean.
– 95% of the values fall within 2
standard deviations of the mean.
– 99.7% of the values fall within 3
standard deviations of the mean.
24
Normal Model - Height
68% of the values fall between
60 – 6 = 54 and 60 + 6 = 66.
95% of the values fall between
60 – 12 = 48 and 60 + 12 = 72.
99.7% of the values fall between
60 – 18 = 42 and 60 + 18 = 78.
25
From Heights to Percentages
What percentage of heights fall
above 70 inches?
Draw a picture.
How far away from the mean is 70
in terms of number of standard
deviations?
26
Normal Model
0.08
0.07
Density
0.06
0.05
0.04
Shaded
area?
0.03
0.02
0.01
0.00
40
45
50
55
60
65
70
75
80
Height (inches)
27
Standardizing
z
y

70  60
z
 1.67
6
28
Standard Normal Model
 Table Z: Areas under the standard
Normal curve in the back of your
text.
 On line:
http://davidmlane.com/hyperstat/z_table.html
29
From Percentages to Heights
What height corresponds to the
75th percentile?
Draw a picture.
The 75th percentile is how many
standard deviations away from the
mean?
30
Normal Model
0.08
25%
0.07
Density
0.06
0.05
50%
0.04
0.03
25%
0.02
0.01
0.00
40
45
50
55
60
65
70
75
80
Height (inches)
31
Standard Normal Model
 Table Z: Areas under the standard
Normal curve in the back of your
text.
 On line:
http://davidmlane.com/hyperstat/z_table.html
32
Reverse Standardizing
z
y

y  60
0.67 
6
y  6 * 0.67   60  64.02
33
Do Data Come from a
Normal Model?
 The histogram should be mounded in
the middle and symmetric.
 The data plotted on a normal
probability (quantile) plot should
follow a diagonal line.
– The normal quantile plot is an option in
JMP: Analyze – Distribution.
34
Do Data Come from a
Normal Model?
 Octane ratings – 40 gallons of
gasoline taken from randomly selected
gas stations.
 Amplifier gain – the amount
(decibels) an amplifier increases the
signal.
 Height – 550 children age 5 to 19.
35
.99
2
.95
.90
.75
.50
1
0
.25
.10
.05
.01
Normal Quantile Plot
3
-1
-2
-3
6
4
Count
8
2
85
90
Octane Rating
95
36
.99
2
.95
.90
.75
.50
1
0
.25
.10
.05
.01
Normal Quantile Plot
3
-1
-2
-3
25
15
Count
20
10
5
7.5
8
8.5
9
9.5
10 10.5 11 11.5 12
Amplifier Gain (dB)
37
.99
2
.95
.90
1
.75
0
.50
.25
Normal Quantile Plot
3
-1
.10
.05
-2
.01
-3
100
Count
150
50
45
50
55
60
65
70
75
38
Nearly normal?
Is the histogram basically
symmetric and mounded in the
middle?
Do the points on the Normal
Quantile plot fall close to the red
diagonal (Normal model) line?
39