Chapter 5
Normal
Normal
Curve
Curve
Normal Curve
Normal Curve
Areas Under
the Normal
Curve
Areas Under
the Normal
Curve
Examples
Standard
Units
Standard
Units
Part II
Descriptive Statistics
Examples
Normal Approximation
for Data
Percentiles
and
Examples
Standard
Units
Standard
Units
Examples
Chapter 5
The Normal Approximation for Data
Normal Approximation
for Data
Percentile
Ranks
Percentiles
Percentiles
Percentile
Rank
Percentile
Rank
Quartiles
Quartiles
Change of
Change of
scale
scale
Data That
Data That
Don't Follow
Don't Follow
the Normal
the Normal
Curve
Curve
Curve
Curve
Areas Under
the Normal
Curve
Examples
Chapter 5
Normal curve
Normal
Normal
Normal Curve
2
2
and
Ranks
Normal Curve
The famous normal curve can often be used as an 'ideal'
histogram, to which histograms for data can be compared. Its
equation is
1 −x
y = √
e
× 100%
2π
but we will work with it through diagrams and tables, without
ever using the equation.
Percentiles
Percentile
Chapter 5
Normal curve
Chapter 5
It was discovered in 1720 by Abraham de Moivre, but is also
called Gaussian curve or bell curve.
Normal Curve
Areas Under
the Normal
Curve
Examples
Standard
Standard
Units
Units
Standard
Units
Standard
Units
Examples
Examples
Normal Approximation
for Data
Normal Approximation
for Data
Percentiles
Percentiles
and
and
Percentile
Percentile
Ranks
Ranks
Percentiles
Percentiles
Percentile
Rank
Percentile
Rank
Quartiles
Quartiles
Change of
Change of
scale
scale
Data That
Data That
Don't Follow
Don't Follow
the Normal
the Normal
Curve
Curve
The graph is symmetric about 0, and the total area equals
100%.
The curve is always above the horizontal axis.
The area under the normal curve between -1 and 1 is about
68%.
Chapter 5
Normal curve
Normal
Normal
Curve
Normal Curve
Areas Under
the Normal
Curve
Examples
Standard
The graph is symmetric about 0, and the total area equals
100%.
The curve is always above the horizontal axis.
Units
Curve
Normal Curve
Areas Under
the Normal
Curve
Examples
Standard
Standard
Units
Examples
Examples
Normal Approximation
for Data
Normal Approximation
for Data
Percentiles
Percentiles
and
and
Percentile
Percentile
Ranks
Ranks
Percentiles
Percentiles
Percentile
Rank
Percentile
Rank
Quartiles
Quartiles
Change of
Change of
scale
scale
Don't Follow
the Normal
Curve
Chapter 5
Normal
Curve
The graph is symmetric about 0, and the total area equals
100%.
The curve is always above the horizontal axis.
Units
Standard
Units
Data That
Normal curve
Chapter 5
The area under the normal curve between -2 and 2 is about
95%.
Areas under the normal curve
Other areas can be found using tables (page A104 in textbook).
Data That
Don't Follow
the Normal
Curve
The area under the normal curve between -3 and 3 is about
99.7%.
Examples
Chapter 5
Normal
Curve
Normal Curve
Normal Curve
Areas Under
the Normal
Curve
Areas Under
the Normal
Curve
Examples
Examples
Standard
Standard
Units
Units
Standard
Units
Standard
Units
Examples
Examples
Normal Approximation
for Data
Normal Approximation
for Data
Percentiles
Percentiles
and
and
Percentile
Percentile
Ranks
Ranks
Percentiles
Percentiles
Percentile
Rank
Percentile
Rank
Quartiles
Quartiles
Change of
Change of
scale
scale
Data That
Data That
Don't Follow
Don't Follow
the Normal
the Normal
Curve
Curve
1
2
3
Find the area between 0 and 1 under the normal curve.
Find the area to the right of 1 under the normal curve.
Find the area to the right of 0.45 under the normal curve.
Standard units
Chapter 5
Normal
Normal
Curve
Curve
Normal Curve
Normal Curve
Areas Under
the Normal
Curve
Areas Under
the Normal
Curve
Examples
Standard
Units
Standard
Units
Examples
Normal Approximation
for Data
Many histograms are similar in shape to the normal curve,
provided they are drawn to the same scale.
Making the horizontal scales match up between histogram and
normal curve involves standard units.
Examples
Standard
Units
Standard
Units
Examples
Normal Approximation
for Data
Percentiles
Percentiles
and
and
Percentile
Ranks
Percentiles
Percentile
Rank
Quartiles
A value is converted to standard units by checking how many
SDs it is above or below the average. Values above the average
get a plus sign, values below the average get a minus sign.
Percentile
Ranks
Percentile
Rank
Quartiles
Change of
scale
scale
Data That
Data That
Don't Follow
Don't Follow
the Normal
the Normal
Curve
Curve
Example 1
Chapter 5
Normal
Normal
Curve
Curve
Normal Curve
Normal Curve
Areas Under
the Normal
Curve
Areas Under
the Normal
Curve
Examples
Standard
Units
Standard
Units
Examples
Normal Approximation
for Data
Women age 18-74 in the HANES study
Average height is 63.5 inches
1
and
Percentile
Ranks
Percentiles
Percentile
Rank
Quartiles
Change of
Units
Standard
Units
Examples
68.5 inches = 63
inches} + 5| inches
| .5 {z
{z }
average
Percentiles
2
Examples
Standard
SD is 2.5 inches
2SD
In standard units, this is +2.
61 inches = 63
inches} − 2| .5 inches
| .5 {z
{z }
average
In standard units, this is -1.
1SD
• HANES: Health And Nutrition Examination Survey
(1976-1980)
• A representative cross-section of 20,322 Americans age 1
to 74 was examined.
• Data was obtained on
•
•
Normal Approximation
for Data
Percentiles
and
Percentile
Ranks
Percentiles
Percentile
Rank
Quartiles
Change of
scale
scale
Data That
Data That
Don't Follow
Don't Follow
the Normal
the Normal
Curve
Curve
demographic variables: age, education, income
physiological variables: height, weight, blood pressure,
cholesterol levels
Percentiles
Change of
Chapter 5
The HANES study
Chapter 5
•
•
•
dietary habits
levels of lead and pesticides in the blood
prevalence of diseases
68% - 95% rules
Chapter 5
Normal
Normal
Curve
Curve
Normal Curve
Normal Curve
Areas Under
the Normal
Curve
Areas Under
the Normal
Curve
Examples
Standard
Units
Standard
Units
Examples
Normal Approximation
for Data
Percentiles
and
Percentile
Ranks
Percentiles
Percentile
Rank
Quartiles
Change of
68% rule
Examples
For many lists 68% of the entries are between (average - SD)
and (average + SD).
Where does this rule come from?
• convert the interval to standard units: -1 to 1
• the area under the normal curve between -1 and 1 is 68%
• if the histogram follows the normal curve, the area under
the histogram is about 68%
Standard
Units
Standard
Units
Examples
Normal Approximation
for Data
Percentiles
and
Percentile
Ranks
Percentiles
Percentile
Rank
Quartiles
Change of
scale
scale
Data That
Data That
Don't Follow
Don't Follow
the Normal
the Normal
Curve
Curve
Normal approximation for data
Chapter 5
Chapter 5
Normal
Normal
Curve
Curve
Normal Curve
Normal Curve
Areas Under
the Normal
Curve
Areas Under
the Normal
Curve
Examples
Standard
Units
Standard
Units
Examples
Normal Approximation
for Data
Percentiles
and
Percentile
Ranks
Percentiles
Percentile
Rank
Quartiles
Change of
We have seen that many histograms are very similar to the
normal curve if we draw them in standard units.
Then the normal curve can be used to estimate the percentage
of entries in an interval as follows:
1 convert the interval to standard units
2 nd the corresponding area under the normal curve
Denition
This procedure is called the normal approximation.
68% - 95% rules
Chapter 5
95% rule
For many lists 95% of the entries are between
(average - 2 SDs) and (average + 2 SDs).
Where does this rule come from?
• convert the interval to standard units: -2 to 2
• the area under the normal curve between -2 and 2 is 95%
• if the histogram follows the normal curve, the area under
the histogram is about 95%
Example 1, continued
Examples
Standard
Units
Standard
Units
Examples
Normal Approximation
for Data
Percentiles
and
Percentile
Ranks
Percentiles
Percentile
Rank
Quartiles
Change of
scale
scale
Data That
Data That
Don't Follow
Don't Follow
the Normal
the Normal
Curve
Curve
Use the normal curve to
estimate the percentage
of women age 18-74 in
the HANES study sample
who's height is between
61 and 68.5 inches.
Example 2 (p. 88, Ex. Set C #2)
Chapter 5
Normal
Normal
In a law school class, the entering students averaged about
160 on the LSAT; the SD was
about 8. The histogram of
LSAT scores followed the normal curve reasonably well.
• About what percentage of
the class scored below
166?
• One student was 0.5 SDs
above average on the
LSAT. About what
percentage of the students
had lower scores?
Curve
Normal Curve
Areas Under
the Normal
Curve
Examples
Standard
Units
Standard
Units
Examples
Normal Approximation
for Data
Percentiles
and
Percentile
Ranks
Percentiles
Percentile
Rank
Quartiles
Change of
scale
Data That
Don't Follow
the Normal
Curve
Curve
Normal Curve
Areas Under
the Normal
Curve
Examples
Standard
Units
Standard
Units
Examples
Normal Approximation
for Data
Percentiles
and
Percentile
Ranks
Percentiles
Percentile
Rank
Quartiles
Change of
scale
Data That
Don't Follow
the Normal
Curve
Percentiles
Chapter 5
Normal
Curve
Curve
Normal Curve
Normal Curve
Areas Under
the Normal
Curve
Areas Under
the Normal
Curve
Examples
Units
Standard
Units
Examples
Normal Approximation
for Data
Percentiles
and
Percentile
Ranks
Percentiles
Percentile
Rank
Quartiles
Change of
We can also summarize data by looking at percentiles.
Denition
The k-th percentile is a number such that k% of the entries in a
list are smaller than the number, and (100-k)% are larger.
Examples:
• 1st percentile = number such that 1% of the entries are
smaller than the number, and 99% are larger
• 25th percentile = number such that 25% of the entries are
smaller than the number, and 75% are larger
Percentile rank
Chapter 5
Normal
Standard
Areas under the normal curve
Chapter 5
Examples
Standard
Units
Standard
Units
Examples
Normal Approximation
for Data
Percentiles
and
Percentile
Ranks
Percentiles
Percentile
Rank
Quartiles
Change of
scale
scale
Data That
Data That
Don't Follow
Don't Follow
the Normal
the Normal
Curve
Curve
Denition
The percentile rank of a value is the percentage of entries
smaller than that value.
Examples:
• the percentile rank of the highest homework score is 100%
• the percentile rank of the median homework score is 50%
Chapter 5
Normal
Curve
Example 3: percentile vs. percentile
rank
Chapter 5
Normal
Curve
Normal Curve
Normal Curve
Areas Under
the Normal
Curve
Areas Under
the Normal
Curve
Examples
Percentile and percentile rank are each other's opposites!
Standard
Standard
Units
Standard
Units
Examples
Normal Approximation
for Data
Percentiles
and
Percentile
Ranks
Percentiles
Percentile
Rank
Quartiles
Change of
scale
Data That
Don't Follow
the Normal
Units
Table: Selected percentiles for family income in the U.S. in 2004
1
10
25
50
75
90
99
$0
$15,000
$29,000
$54,000
$90,000
$135,000
$430,000
(see page 89 in textbook)
Curve
Chapter 5
Examples
Standard
Units
Examples
Normal Approximation
for Data
Percentiles
and
Percentile
Ranks
Percentiles
Percentile
Rank
Quartiles
scale
Data That
Don't Follow
the Normal
Chapter 5
Normal
Normal
Curve
Curve
Normal Curve
Normal Curve
Areas Under
the Normal
Curve
Areas Under
the Normal
Curve
Examples
Standard
Units
Standard
Units
Examples
Normal Approximation
for Data
Percentiles
and
Percentile
Ranks
Percentiles
Percentile
Rank
If a histogram follows the normal curve, then the normal curve
can be used to estimate percentiles
Method:
1 sketch a normal curve; nd the right value of z , using the
normal table
2 z is given in standard units; convert it back to the units in
the problem
Income data (table from previous slide):
• The 10th percentile is $15, 000
It is the number such that 10% of the entries are smaller
than that number
• The percentile rank of $15, 000 is 10%
It is the percentage of entries smaller than $15, 000
• Another way of saying this: an annual income of $15, 000
puts you at the 10th percentile of the income distribution
Change of
Curve
Percentiles and the normal curve
Example 3: percentile vs. percentile
rank
A percentile is a number
A percentile rank is a percentage
Example 4 (p.92, Ex Set E #2)
Examples
Standard
Units
Standard
Units
Examples
Normal Approximation
for Data
Percentiles
and
Percentile
Ranks
Percentiles
Percentile
Rank
Quartiles
Quartiles
Change of
Change of
scale
scale
Data That
Data That
Don't Follow
Don't Follow
the Normal
the Normal
Curve
Curve
Among all applicants to a
university, the Math SAT
scores averaged 535, the
SD was 100, and the
scores followed the normal curve. Estimate the
80th percentile.
Areas under the normal curve
Chapter 5
Normal
Normal
Curve
Curve
Normal Curve
Normal Curve
Areas Under
the Normal
Curve
Areas Under
the Normal
Curve
Examples
Examples
Standard
Standard
Units
Units
Standard
Units
Standard
Units
Examples
Examples
Normal Approximation
for Data
Normal Approximation
for Data
Percentiles
Percentiles
and
and
Percentile
Percentile
Ranks
Ranks
Percentiles
Percentiles
Percentile
Rank
Percentile
Rank
Quartiles
Quartiles
Change of
Change of
scale
scale
Data That
Data That
Don't Follow
Don't Follow
the Normal
the Normal
Curve
Curve
Quartiles
Chapter 5
Normal
Curve
Normal Curve
Areas Under
the Normal
Curve
Examples
Standard
Units
Standard
Units
Examples
Normal Approximation
for Data
Percentiles
and
Percentile
Ranks
Percentiles
Percentile
Rank
Quartiles
Change of
scale
Data That
Don't Follow
the Normal
Curve
Example 5 (p. 92, Ex. Set E #3)
Chapter 5
For a Berkely freshmen,
the average GPA is
around 3.0; the SD
is about 0.5.
The
histogram follows the
normal curve. Estimate
the 30th percentile of
the GPA distribution.
Quartiles
Chapter 5
Normal
Denition
• 1st quartile
= nr. such that 1/4 of the data are smaller and 3/4 are
larger
= 25th percentile
• 2nd quartile
= nr. such that 2/4 of the data are smaller and 2/4 are
larger
= 50th percentile
= median
• 3rd quartile
= nr. such that 3/4 of the data are smaller and 1/4 are
larger
= 75th percentile
Curve
Normal Curve
Areas Under
the Normal
Curve
Examples
Standard
Units
Standard
Units
Examples
Normal Approximation
for Data
Denition
Interquartile Range (IQR) is another measure of the spread of
the data. It is given by
Percentiles
and
Percentile
Ranks
Percentiles
Percentile
Rank
Quartiles
Change of
scale
Data That
Don't Follow
the Normal
Curve
Interquartile Range = 3rd quartile - 1st quartile
Chapter 5
Motivation
Normal
Normal
Curve
Curve
Normal Curve
Normal Curve
Areas Under
the Normal
Curve
Areas Under
the Normal
Curve
Examples
Standard
Units
Standard
Units
Examples
Normal Approximation
for Data
Percentiles
and
Percentile
Ranks
Percentiles
Percentile
Rank
Quartiles
Change of
scale
Changes of scale often occur:
• length: cm - inches - feet
• temperature: Fahrenheit - Celsius
Such changes of scale consist of:
• multiplying all entries by a constant
• adding a constant to all entries
• both of the above
How does this inuence the average, the SD and the standard
units?
Examples
Standard
Units
Standard
Units
Examples
Normal Approximation
for Data
Percentiles
and
Percentile
Ranks
Percentiles
Percentile
Rank
Quartiles
Change of
scale
Data That
Data That
Don't Follow
Don't Follow
the Normal
the Normal
Curve
Curve
Chapter 5
Example 6 (p. 93, Ex Set F #10)
Normal
Curve
Curve
Normal Curve
Normal Curve
Areas Under
the Normal
Curve
Areas Under
the Normal
Curve
Examples
Units
Standard
Units
Examples
Normal Approximation
for Data
Percentiles
and
Percentile
Ranks
Percentiles
Percentile
Rank
Quartiles
A group of people have average
temperature of 98.6 F, with SD
of 0.3 F.
1 Translate this to C.
2 How much is 1.5 SDs on
Fahrenheit in Celcius?
5
C =
F − 32
9
If we add a constant to all entries in a list, then
• the average increases by this constant
• the SD does not change
• the standard units do not change
If we multiply all entries in a list by a positive number, then
• the average is multiplied by this number
• the SD is multiplied by this number
• the standard units do not change
How to summarize data that don't
follow the normal curve?
Chapter 5
Normal
Standard
Eects of change of scale
Chapter 5
Examples
Standard
Units
Standard
Units
Examples
Normal Approximation
for Data
Percentiles
and
Percentile
Ranks
Percentiles
Percentile
Rank
Don't use the normal curve!
Good ways to summarize such data are:
• the histogram
• a table with percentiles (like for the income data)
• the 1st quartile, median and 3rd quartile
• box-plot:
•
50
Quartiles
Change of
Change of
scale
scale
Data That
Data That
Don't Follow
Don't Follow
the Normal
the Normal
Curve
Curve
box given by 1st and 3rd quartile: contains the middle
•
•
%
of the data
median is given as a line in the box
it also gives some information on entries that fall outside
the box
Chapter 5
*(
Chapter 5
Normal
Curve
Normal
Curve
Normal Curve
Areas Under
the Normal
Curve
Areas Under
the Normal
Curve
*'
Normal Curve
Examples
Examples
Standard
Units
Units
)(
Standard
Standard
Units
Standard
Units
Examples
Normal Approximation
for Data
Normal Approximation
for Data
)'
Examples
Percentiles
and
and
Percentile
Percentile
(
Percentiles
Ranks
Ranks
Percentiles
Percentiles
Percentile
Rank
Percentile
Rank
Quartiles
'
Quartiles
Change of
!
"
#
$
%
&
Data That
Don't Follow
Don't Follow
the Normal
the Normal
Curve
Curve
Long-left tail distribution?
Chapter 5
Chapter 5
Normal
Curve
Curve
0.05
Normal
Normal Curve
Areas Under
the Normal
Curve
Normal Curve
Areas Under
the Normal
Curve
Examples
0.04
Examples
Standard
Standard
0.03
Units
Standard
Units
Examples
Normal Approximation
for Data
0.02
Density
Units
Normal Approximation
for Data
Percentiles
and
and
Ranks
0.01
Percentiles
Percentile
Percentile
Ranks
Percentiles
Percentile
Rank
Percentile
Rank
Change of
scale
0.00
Percentiles
Quartiles
• Heights of STAT 220 students?
• Daily Rainfall?
• Family Income?
• Length of time spent on STAT 220 Quiz 1?
The Normal Approximation is discussed again in Chapter 18.
scale
Data That
Examples
• Long-left tail distribution?
Change of
scale
Standard
Units
Is the normal approximation
reasonable?
Quartiles
0
20
40
60
Points
80
100
Change of
scale
Data That
Data That
Don't Follow
Don't Follow
the Normal
the Normal
Curve
Curve
Heights of STAT 220 students?
Daily rainfall?
Chapter 5
Normal
Normal
Curve
Curve
Normal Curve
Normal Curve
Areas Under
the Normal
Curve
Areas Under
the Normal
Curve
...THE SEATTLE-TACOMA AIRPORT CLIMATE SUMMARY FOR THE MONTH OF
SEPTEMBER 2008...
Examples
Examples
Standard
Standard
Units
WEATHER
OBSERVED
VALUE
Standard
Units
NORMAL
DEPART
LAST YEAR`S
VALUE
FROM
VALUE
DATE(S)
NORMAL
Examples
PRECIPITATION
Normal Approximation
for Data
(INCHES)
5.95
1978
Normal Approximation
for Data
T
1991
MINIMUM
and
TOTALS
0.78
1.63
-0.85
3.16
and
Percentile
DAILY AVG.
0.03
0.05
-0.02
0.11
Percentile
Ranks
DAYS >= .01
5
MM
MM
10
DAYS >= .10
2
MM
MM
6
Percentile
Rank
DAYS >= .50
1
MM
MM
2
DAYS >= 1.00
0
MM
MM
1
Quartiles
GREATEST
Change of
Percentiles
Ranks
Percentiles
Percentile
Rank
Quartiles
24 HR. TOTAL
0.56
09/20 TO 09/21
1.20
scale
Data That
scale
(From http://www.weather.gov/climate/index.php?wfo=sew)
Data That
Don't Follow
the Normal
the Normal
Curve
Curve
Normal
Curve
Length of time spent on STAT 220
Quiz 1?
$0
$15,000
$29,000
$54,000
$90,000
$135,000
$430,000
Change of
Don't Follow
Chapter 5
1
10
25
50
75
90
99
Standard
Units
Examples
RECORD
MAXIMUM
Table: Selected percentiles for family income in the U.S. in 2004
Units
Percentiles
Percentiles
Family income?
Chapter 5
Chapter 5
Normal
Curve
Normal Curve
Normal Curve
Areas Under
the Normal
Curve
Areas Under
the Normal
Curve
Examples
Examples
Standard
Standard
Units
Units
(see page 89 in textbook)
Is the normal approximation
reasonable?
Standard
Units
Standard
Units
Examples
Examples
• Long-left tail distribution? No!
Normal Approximation
for Data
Normal Approximation
for Data
• Heights of STAT 220 students? Not Bad.
Percentiles
Percentiles
and
and
Percentile
Percentile
Ranks
Ranks
Percentiles
Percentiles
Percentile
Rank
Percentile
Rank
Quartiles
Quartiles
Change of
Change of
scale
scale
Data That
Data That
Don't Follow
Don't Follow
the Normal
the Normal
Curve
Curve
• Daily rainfall? No!
• Family income? No!
• Length of time spent on STAT 220 Quiz 1? No!
Chapter 5
Summary
Normal
Curve
Normal Curve
Areas Under
the Normal
Curve
Examples
Standard
Units
Standard
Units
Examples
Normal Approximation
for Data
Percentiles
and
Percentile
Ranks
Percentiles
Percentile
Rank
Quartiles
Change of
scale
Data That
Don't Follow
the Normal
Curve
• If a histogram has the shape of a normal curve, we can use
the normal approximation to obtain information about the
data behind the histogram based our knowledge of the
normal curve. For this, the data must be converted to
standard units.
• Percentiles, percentile ranks, and quartiles are numerical
summary information about our data that can be useful.
• Sometimes, we need to change the scale of the data. This
might inuence the average and the SD but not the
standard units.
• Normal approximation should only be used when the
histogram follows the normal curve. The summary
information can always be used.