Download Chapter 13 - Department of Statistics

Reminders
• Homework due tomorrow
• Quiz tomorrow
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Warm Up - ACT Math Scores
Density
Distribution of ACT Math Scores
0
5
10
15
20
25
30
35
scores
What percent of scores are between 12 and 24?
Options: 38%, 50%, 53%, 68%, 77%, 86%, 90% , 95%, 99%
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Warm Up - ACT Math Scores
Density
Distribution of ACT Math Scores
0
5
10
15
20
25
30
35
scores
What percent of scores are between 6 and 30?
Options: 38%, 50%, 53%, 68%, 77%, 86%, 90% , 95%, 99%
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Warm Up - ACT Math Scores
Density
Distribution of ACT Math Scores
0
5
10
15
20
25
30
35
scores
What can we say about the relationship between the mean and
the median?
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Chapter 13: Normal Distributions
Aaron Zimmerman
STAT 220 - Summer 2014
Department of Statistics
University of Washington - Seattle
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Density Curves
• I’ve drawn a smooth
Distribution of ACT Math Scores
Density
density curve over the
histogram
• Curves show proportions
of observations in any
region by areas under the
curve
• The entire area under the
curve must be 1 and the
curve must be
non-negative.
0
5
10
15
20
25
30
35
scores
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Strategies for exploring data
• We have already discussed three “first steps” when you
get new data on a single quantitative variable
(1) Make a histogram
(2) Look for the overall pattern (shape, center, spread)
and for striking deviations such as outliers
(3) Choose either the five-number summary or the mean
and standard deviation to briefly describe the center and
spread in numbers
• Today, we are going to add one more strategy
(4) Sometimes the overall pattern of a large number of
observations is so regular that we can describe it by a
smooth curve
The smooth curve we will discuss is a Normal curve
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Motivating example
Distribution of ACT Math Scores
ACT Math scores is
symmetric with no
clear outliers
Density
• The distribution of
• This is the type of
distribution that is
well-described by a
Normal curve
0
5
10
15
20
25
30
35
scores
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Properties of Normal curves
• You can get new Normal
curves by sliding and
stretching other Normal
curves
• A specific Normal curve is
completely described by
giving its mean and
standard deviation
• The mean determines the
center of the distribution.
It is located at the center
of symmetry of the curve
?The mean ACT Math
score is 18
Distribution of ACT Scores
0
5
10
15
20
25
30
35
x
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Properties of Normal curves
• The mean determines the
center of the distribution.
It is located at the center
of symmetry of the curve
?The mean ACT Math
score is 18
• The standard deviation
determines the shape of
the curve. It is the
distance from the mean to
the change-of-curvature
point on either side
?The standard deviation
of ACT Math scores is 6
Distribution of ACT Scores
0
5
10
15
20
25
30
35
x
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Some cautions before we continue!
• There are many types of data that are Normally
distributed
?Human birth weight
?Heights of trees (within species)
?Sample means and proportions calculated from repeated
random samples from the same population
• However, there are many, many types of data that are
not even close to Normal (and we have seen many of
these already)
?Always remember to check a histogram of your
data before performing any Normal calculations!
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The 68-95-99.7 Rule
In any Normal distribution
• 68% of the observations
fall within one SD of the
mean
• 95% of the observations
fall within two SDs of the
mean
• 99.7% of the observations
fall within three SDs of
the mean
Mean = 0, SD=1
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The 68-95-99.7 Rule
Distribution of ACT Scores
ACT math scores are normally
distributed with a mean of 18
and a standard deviation of 6
• So, 68% of ACT math
scores are between 12
(one SD below the mean)
and 24 (one SD above the
mean)
0
5
10
15
20
25
30
35
x
Mean = 18, SD=6
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The 68-95-99.7 Rule
Distribution of ACT Scores
ACT math scores are normally
distributed with a mean of 18
and a standard deviation of 6
• 95% of ACT math scores
are between 6 (two SDs
below the mean) and 30
(two SDs above the mean)
0
5
10
15
20
25
30
35
x
Mean = 18, SD=6
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The 68-95-99.7 Rule
Distribution of ACT Scores
ACT math scores are normally
distributed with a mean of 18
and a standard deviation of 6
• Note: we already
calculated these first two
approximations using the
histogram on the warm
up!
0
5
10
15
20
25
30
35
x
Mean = 18, SD=6
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The 68-95-99.7 Rule
Distribution of ACT Scores
ACT math scores are normally
distributed with a mean of 18
and a standard deviation of 6
• 99.7% of ACT math
scores are between 0
(three SDs below the
mean) and 36 (three SDs
above the mean)
0
5
10
15
20
25
30
35
x
Mean = 18, SD=6
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Birth Weight
Distribution of Birth Weights
Human birth weight is
Normally distributed
with a mean of 3300
grams and a standard
deviation of 300 grams.
Approximately what
percent of babies have a
birth weight between
3000 grams and 3600
grams?
2500
3000
3500
4000
4500
x
Mean = 3300, SD=300
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Cherry Tree Heights
Distribution of Cherry Tree Heights
The heights of cherry
trees are Normally
distributed with a mean
of 10 feet and a
standard deviation of 2
feet. The heights of
approximately 95% of all
cherry trees lie between
what two values?
5
10
15
x
Mean = 10, SD=2
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Cherry Tree Heights
Distribution of Cherry Tree Heights
Approximately what
percentage of cherry
trees have heights
between 6 and 12 feet?
5
10
15
x
Mean = 10, SD=2
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Standard scores
• The 68-95-99.7 rule is convenient for quick calculations,
but we can calculate more precise percentages using
standard scores
?To calculate the standard score for any observation in a
Normal distribution, subtract the mean and then
divide by the standard deviation
standard score =
observation − mean
standard deviation
Key idea #1: The standard score tells you how many
standard deviations above or below the mean the observation lies
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Standard scores
Suppose that Grainne scored a 27 on the ACT Math test,
while Karthik scored a 15. Remember, the mean is 18 and the
SD is 6.
• To calculate Grainne’s standard score:
standard score =
observation − mean
27 − 18
=
= 1.5
standard deviation
6
? Conclusion : Grainne scored 1.5 SDs above the mean
• To calculate Karthik’s standard score:
standard score =
observation − mean
15 − 18
=
= −0.5
standard deviation
6
? Conclusion : Karthik scored 0.5 SDs below the mean
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Why do standard scores work?
Distribution of ACT Scores
We start back with the
distribution of ACT
math scores
0
5
10
15
20
25
30
35
x
Mean = 18, SD=6
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Why do standard scores work?
When we subtract the mean from every observation, the mean
of the distribution becomes zero
Distribution of ACT Scores
Distribution of ACT Scores
0
5
10
15
20
25
x
Mean = 18, SD=6
30
35
−15
−10
−5
0
5
10
15
x
Mean = 0, SD=6
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Why do standard scores work?
When we divide every observation by the SD, the SD of the
distribution becomes one
Distribution of ACT Scores
Distribution of ACT Scores
−15
−10
−5
0
5
10
x
Mean = 0, SD=6
15
−3
−2
−1
0
1
2
3
x
Mean = 0, SD=1
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Why do standard scores work?
? When we subtract the mean from every observation, the
mean of the distribution becomes zero
? When we divide every observation by the SD, the SD of the
distribution becomes one
Distribution of ACT Scores
Distribution of ACT Scores
+/− 1 SD, Area=68%
−20
−10
0
10
x
20
Mean=18, SD=6
30
Distribution of ACT Scores
+/− 1 SD, Area=68%
−20
−10
0
10
20
x
Mean=0, SD=6
30
+/− 1 SD, Area=68%
−20
−10
0
10
20
30
x
Mean=0, SD=1
? All of the blue areas are the same size
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Why do standard scores work?
Grainne’s score is blue while Karthik’s is green
Distribution of ACT Scores
0
5
10
15
20
25
30
35
x
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Why do standard scores work?
Grainne’s score is blue while Karthik’s is green
?Subtract the mean
Distribution of ACT Scores
−15
−10
−5
0
5
10
15
x
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Why do standard scores work?
Grainne’s score is blue while Karthik’s is green
?Subtract the mean
?Divide by the standard deviation
Distribution of ACT Scores
−3
−2
−1
0
1
2
3
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Why do standard scores work?
• The idea here is that although we’ve stretched and
shifted the Normal curve, we’ve done it to all points in
the same way
• Specifically, we’ve done it so areas under the curve are
the same
• As a result, once we have standard scores, we can use a
standard normal distribution to learn something about the
original distribution
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Birth Weight
Distribution of Birth Weights
Human birth weight is
Normally distributed
with a mean of 3300
grams and a standard
deviation of 300 grams.
If Aaron was 3450 grams
when he was born, what
was his standard score?
2500
3000
3500
4000
4500
x
30
Cherry Tree Heights
Distribution of Cherry Tree Heights
The heights of cherry
trees are Normally
distributed with a mean
of 10 feet and a
standard deviation of 2
feet. What is the
standard score for a tree
that is 7.5 feet tall?
5
10
15
x
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Homework
• Read Chapter 13
• Do problems 13.1, 13.2, 13.7 (use information at bottom
of page 280), 13.10a, 13.13, 13.14 (pay close attention to
the mean and SD for each age group), 13.16b
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