Stat 101: Lecture 8
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
• Exam 2
– Mean:
– Mean:
y = 59.5
– Standard
Deviation:
s = 8.61
y = 50.1
– Standard
Deviation:
s = 11.86
3
Stat 101: Lecture 8
Standardizing
Look at the number of
standard deviations the
score is from the mean.
y− y
z=
s
4
Standardized Exam Scores
• Exam 1
– Score: 67
67 − 59.5
8.61
z = 0.87
z=
• Exam 2
– Score: 64
64 − 50.1
11.86
z = 1.17
z=
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
Stat 101: Lecture 8
Standardizing
• Shifts the distribution by
subtracting off the mean.
• Rescales the distribution by
dividing by the standard deviation.
7
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.
8
Normal Models
• Our conceptualization of what the
distribution of an entire population
of values would look like.
• Characterized by population
parameters: μ and σ.
9
Stat 101: Lecture 8
Sample Data
0.3
Density
0.2
0.1
0.0
85
90
95
100
Octane Rating
10
Normal Model
0.3
Density
0.2
σ
0.1
0.0
85
90
μ
95
100
Octane Rating
11
Normal Model
• Octane Rating
• Center:
– Mean, μ = 91
• Spread:
– Standard deviation, σ=1.5
12
Stat 101: Lecture 8
68-95-99.7 Rule
• 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.
13
Octane Rating
• 68% of the values fall between
89.5 and 92.5.
• 95% of the values fall between
88.0 and 94.0.
• 99.7% of the values fall between
86.5 and 95.5.
14
Normal Percentiles
• What percentage of octane ratings
fall below 92?
• Draw a picture.
• How far away from the mean is 92
in terms of number of standard
deviations?
15
Stat 101: Lecture 8
0.3
Density
0.2
Shaded
Area
0.1
0.0
90
85
92
95
100
Octane Rating
16
Standard Normal Model
z=
92 − 91
= 0.67
1.5
• Table Z: in your text.
z
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.7486
17
Standard Normal Model
z=
92 − 91
= 0.67
1.5
• http://davidmlane.com/hyperstat/z_tab
le.html
• 75% of octane values fall below 92.
18