Download 3.3 Density Curves and Normal Distributions

3.3 Density Curves and
Normal Distributions
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Density Curves
Measuring Center and Spread for Density Curves
Normal Distributions
The 68-95-99.7 Rule
Standardizing Observations
Using the Standard Normal Table
Inverse Normal Calculations
Normal Quantile Plots
Exploring Quantitative Data
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We now have a kit of graphical and numerical tools for describing
distributions. We also have a strategy for exploring data on a single
quantitative variable. Now, we’ll add one more step to the strategy.
Exploring Quantitative Data
1. Always plot your data: Make a graph.
2. Look for the overall pattern (shape, center, and spread) and
for striking departures such as outliers.
3. Calculate a numerical summary to briefly describe center
and spread.
4. Sometimes, the overall pattern of a large number of
observations is so regular that we can describe it by a
smooth curve.
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Density curves
A density curve is a mathematical model of a distribution…
The total area under the curve, by definition, is equal to 1, or 100%.
The area under the curve for a range of values is the proportion of
all observations for that range.
Area under Density Curve ~ Relative Frequency of Histogram
Histogram of a sample with the
smoothed, density curve
describing theoretically the
population.
rel. freq of left
histogram=287/947=.303
area = .293 under rt.
curve
Density Curves
A density curve is a curve that:
 Is always on or above the horizontal axis
 Has an area of exactly 1 underneath it
A density curve describes the overall pattern of a
distribution. The area under the curve and above any
range of values on the horizontal axis is the
proportion of all observations that fall in that range.
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Density curves come in many
shapes. Some are well known
mathematically and others aren’t
– but they all lie above the
horizontal axis and have total
area = 1.
Density Curves
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Our measures of center and spread apply to density
curves as well as to actual sets of observations.
Distinguishing the Median and Mean of a Density Curve
 The median of a density curve is the equal-areas point―the point that
divides the area under the curve in half.
 The mean of a density curve is the balance point, at which the curve
would balance if made of solid material.
 The median and the mean are the same for a symmetric density curve.
They both lie at the center of the curve. The mean of a skewed curve is
pulled away from the median in the direction of the long tail.
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Density Curves
 The mean and standard deviation computed from
actual observations (data) are denoted by x and s,
respectively, and are called the sample mean and
sample standard deviation.
 The mean and standard deviation of the idealized
distribution represented by the density curve are
denoted by µ (“mu”) and  (“sigma”), respectively,
and are sometimes called the population mean and
population standard deviation.
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Normal Distributions
One particularly important class of density curves are the Normal curves,
which describe Normal distributions.
 All Normal curves are symmetric, single-peaked, and bell-shaped.
 A specific Normal curve is described by giving its mean µ and
standard deviation σ.
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Normal Distributions
A Normal distribution is described by a Normal density curve.
Any particular Normal distribution is completely specified by two
numbers: its mean µ and standard deviation σ.
 The mean of a Normal distribution is the center of the
symmetric Normal curve.
 The standard deviation is the distance from the center to the
change-of-curvature points on either side, the points of
inflection of the density.
 We abbreviate the Normal distribution with mean µ and
standard deviation σ as N(µ,σ).
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Normal distributions
Normal – or Gaussian – distributions are a family of symmetrical, bellshaped density curves defined by a mean m (mu) and a standard
deviation  (sigma) : N(m,).
1
f (x) =
e
s 2p
2
1æ x -m ö
- ç
÷
2è s ø
x
e = 2.71828… The base of the natural logarithm
π = pi = 3.14159…
x
A family of density curves
Here, means are the same (m = 15)
while standard deviations are
different ( = 2, 4, and 6).
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
Here, means are different
(m = 10, 15, and 20) while
standard deviations are the
same ( = 3).
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
The 68-95-99.7 Rule
The 68-95-99.7 Rule
In the Normal distribution with mean µ and standard deviation σ:
 Approximately 68% of the observations fall within σ of µ.
 Approximately 95% of the observations fall within 2σ of µ.
 Approximately 99.7% of the observations fall within 3σ of µ.
Here’s a N(64.5”, 2.5”)
distribution of heights of
college-aged females.
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The 68-95-99.7 Rule
The distribution of Iowa Test of Basic Skills (ITBS) vocabulary scores for
7th-grade students in Gary, Indiana, is close to Normal. Suppose the
distribution is N(6.84, 1.55).

Sketch the Normal density curve for this distribution.

What percent of ITBS vocabulary scores are less than 3.74?

What percent of the scores are between 5.29 and 9.94?
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Standardizing Observations
If a variable x has a distribution with mean µ and standard
deviation σ, then the standardized value of x, or its z-score, is
z
x-μ

Note z= the number of s.d.’s away from mu that x is…(sorry about
the grammer!)
All Normal distributions are the same if we measure in units of size σ
from the mean µ as center.
The standard Normal distribution is the
Normal distribution with mean 0 and
standard deviation 1. That is, the standard
Normal distribution is N(0,1) – it is
represented by Z and we write Z ~ N(0,1)
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The Standard Normal Table
Because all Normal distributions are the same when we standardize, we
can find areas under any Normal curve from a single table.
The Standard Normal Table
Table A is a table of areas under the standard Normal curve. The
table entry for each value z is the area under the curve to the left
of z.
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The Standard Normal Table
Suppose we want to find the proportion of observations from the
standard Normal distribution that are less than 0.81.
We can use Table A:
P(z < 0.81) = 0.7910
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Z
0.00
0.01
0.02
0.7
0.7580
0.7611
0.7642
0.8
0.7881
0.7910
0.7939
0.9
0.8159
0.8186
0.8212
Tips on using Table A
To calculate the area between 2 z- values, first get the area under N(0,1) to the
left for each z-value from Table A.
Then subtract the smaller
area from the larger area.
A common mistake made by
students is to subtract both z
values - it is the areas that are
subtracted, not the z-scores!
area between z1 and z2
=
area left of z1 – area
left of z2
area right of z =
1
-
area left of z
Normal Calculations
How to Solve Problems Involving Normal Distributions
Express the problem in terms of the observed variable X.
Draw a picture of the distribution of X and shade the area of
interest under the curve.
Perform calculations.
 Standardize X to restate the problem in terms of a standard
Normal variable Z.
 Use Table A and the fact that the total area under the curve is
1 to find the required area under the standard Normal curve.
Write your conclusion in the context of the problem.
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Inverse Normal Calculations
According to the Health and Nutrition Examination
Study of 1976–1980, the heights X (in inches) of
adult men aged 18–24 are N(70, 2.8).
How tall must a man be (? below) to be in the lower 10% for men
aged 18–24?
N(70, 2.8)
0.10
?
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70
Inverse Normal Calculations
N(70, 2.8)
How tall must a man be in
the lower 10% for men aged
18–24?
0.10
? 70
Look up the closest probability
(closest to 0.10) in the table.
Find the corresponding z the
standardized score.
The value you seek is that many
standard deviations from the
mean.
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z
0.07
0.08
0.09
–1.3
0.0853
0.0838
0.0823
–1.2
.1020
0.1003
0.0985
–1.1
0.1210
0.1190
0.1170
Z = –1.28
Normal Calculations
How tall must a man be in the lower
10% for men aged 18–24?
N(70, 2.8)
0.10
Z = –1.28
? 70
We need to “unstandardize” the z-score to find the observed value (x):
z
xm

x  m  z
x = 70 + z(2.8)
= 70 + [(1.28 )  (2.8)]
= 70 + (–3.58) = 66.42
A man would have to be approximately 66.42 inches tall or less to
place
in the lower 10% of all men in the population.
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Normal Quantile Plots
One way to assess if a distribution is indeed approximately Normal is to
plot the data on a Normal quantile plot.
The data points are ordered from smallest to largest and their
percentile ranks are converted to z-scores with Table A. These z-scores
are then plotted against the data to create a Normal quantile plot.
 If the distribution is indeed Normal, the plot will show a straight
line, indicating a good match between the data and a Normal
distribution – in JMP the points fall within the dotted lines.
 Systematic deviations from a straight line indicate a non-Normal
distribution. Outliers appear as points that are far away from the
overall pattern of the plot – some points fall outside the dotted
lines in JMP.
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Normal Quantile Plots
Good fit to a straight line: the distribution
of rainwater pH values is close to
normal. The intercept of the line ~ mean
of the data and the slope of the line ~
s.d. of the data
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Curved pattern: The data are not
Normally distributed. Instead, it shows
a right skew: A few individuals have
particularly long survival times.
Normal quantile plots are complex to do by hand, but they are easy to
do in JMP – under the red triangle, choose Normal Quantile Plot – but
notice the difference when compared to the above plots…
HW:
•Finish reading section 3.3
•Work over all the examples!
•I’ve put up some videos on computing Normal
Probabilities as an online assignment… due
9/19 & 9/22 at 9:00am
•Work on #3.82-3.93, 3.95, 3.100-3.102,
3.104, 3.106-3.109, 3.110-3.128. Do as many
of these problems in order to really
understand what’s going on here!
•Quiz in class on Monday 9/22
•Test #1 on October 1, covering Chapts. 1-4