Download Chapter 13: Normal Distributions

Chapter 13: Normal Distributions
Thought Question
Numerical data can distinguish different types of writing, and sometimes even individual authors. Here are data collected by students on the percentages of words of 1 to 15 letters used
in articles in Popular Science magazine:
Length:
Proportion:
1
.036
2
.148
3
.187
4
.160
5
.125
6
.082
7
.081
Make a histogram of this distribution. Then
• Draw a smooth curve around your histogram.
• Indicate the location of the median word length.
• Indicate the location of the mean word length.
1
8
.059
9
.044
10
.036
11
.021
12
.009
13
.006
14
.004
15
.002
Density Curves
How can we represent distributions by a smooth curve?
Terminology
• Density Curve: a curve where the total area underneath it is equal to 1.
• Median of a Density Curve: the point that divides the area of the curve in half.
• Mean of a Density Curve: the point at which the curve would balance.
Histogram vs. Density Curve
Histogram:
• Shows the counts in each class by the height/area of the bars.
• Is a plot of data from a sample.
Density Curve:
• Shows the proportion of observations in any region by areas under the curve.
• Reflects the idealized shape of the population distribution.
How are the mean and median of a symmetric density curve related?
They are equal.
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Characteristics of Normal Distributions
• Any value can occur.
• The fraction of values in the population which lie in an interval is given by the area under
the curve inside the interval.
• The total area under the curve is 1.
• The mean µ determines the center of the distribution. The curve is symmetric about its
mean.
• The standard deviation σ determines the spread of the distribution. It is the distance
from the mean to the change-of-curvature point on either side.
µ−σ
µ
µ+σ
How does increasing the mean affect the normal curve?
The center is shifted to the right.
How does increasing the standard deviation affect the normal curve?
The curve is more spread out.
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The 68-95-99.7 Rule
µ − 3σ
µ − 2σ
µ−σ
µ
µ+σ
µ + 2σ
µ + 3σ
In Any Normal Distribution
• 68.2% of the observations fall within σ of the mean µ.
• 95.4% of the observations fall within 2σ of the mean µ.
• 99.7% of the observations fall within 3σ of the mean µ.
Example: ACT Scores
The distribution of ACT scores are approximately normally distributed with mean µ = 18 and
standard deviation σ = 6. Between what values will
68% of the scores lie? 12 and 24
95% of the scores lie? 6 and 30
99.7% of the scores lie? 0 and 36
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Standard Scores
How can we compare values from different distributions?
Terminology
• Standard Scores: observations expressed in standard deviations above or below the
mean.
How to Calculate the Standard Score
standard score =
observation − mean
standard deviation
If the observation is x, then the standard score =
5
x−µ
.
σ
Example
Suppose X takes values from a normal distribution with mean µ = 5 and standard deviation
σ = 1.5. Calculate the corresponding standard scores Z:
(a) X > 6
Values of X
6
5
Z=
X−µ
σ
?
Values of Z
0
(b) 3.21 < X < 4.76
Values of X
5
?
Values of Z
0
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Example: SAT and ACT Scores
The distribution of SAT scores are approximately normally distributed with mean µ = 500 and
standard deviation σ = 100. Jennie scored a 600 on the SAT math portion. Gerald scored a 21
on the ACT math portion. Who did better?
To determine this, calculate Jennie and Gerald’s standard scores.
Jennie’s standard score is 600−500
=
100
So Jennie did better than Gerald.
100
100
= 1 and Gerald’s standard score is
How many standard deviations above the mean SAT score is Jennie’s score?
1
How many standard deviations above the mean ACT score is Gerald’s score?
0.5
7
21−18
6
=
3
6
= 0.5.
Percentiles
Terminology
• cth Percentile: value such that c percent of the observations lie below it and the rest
lie above it.
What percentile is the median? 50th percentile
What percentile is the first quartile? 25th percentile
What percentile is the third quartile? 75th percentile
Example: SAT and ACT Scores
What percentile is Jennie’s SAT score?
84.13 percentile.
What percent of people scored below a 600 on the SAT math portion?
84.13 %.
What percentile is Gerald’s ACT score?
69.15 percentile.
What percent of people score below a 21 on the ACT math portion?
69.15 %.
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Example
For a normal distribution with mean 0 and standard devation 1, find the number z such that
(a) 95% of the values lie between −z and z
0.95
−z
0
z
(b) 90% of the values lie between −z and z
0
(c) 1% of the values are greater than z
0
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Chapter 13 Exercises
1. A company that sells annuities must base the annual payout on the distribution of the
lifetimes of the participants in the plan. Suppose the lifetimes of participants are approximately
normally distributed with µ = 77 years and σ = 7.5 years. What proportion of the plan
participants would receive payments beyond age 70?
Distribution of Participants’ Lifetimes
77
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