1.3 Density Curves and
Normal Distributions
What is a density curve?
Idea of density
Density curve describes an overall pattern.
Area under the curve should give a
probability- hence, total area under a
density curve is always 1.
We would like to model real life data by a
density curve.
Population vs. Sample
Whenever we use a histogram to approximate a
density curve, we must be careful to distinguish
between the mean of the sample vs. the mean of
the population. Likewise for standard deviation.
We use x for the mean of the sample and the
Greek letter μ (mu) for the mean of the
population. Note that while we may never know
μ, it does exist. We use s for standard deviation
of sample and σ (sigma) for the population
standard deviation.
Normal Distributions
When a density distribution is symmetric,
we say this is a normal distribution.
Normal distributions can have different
shapes.
It turns out that a normal curve is
completely determined by the mean and
standard deviation.
Mean is at the maximum point.
1 Standard deviation is where the curve
changes “concavity.”
68%-95%-99.7% rule
The general formula
(which you will not
need to know or use
for this class) for a
normal distribution is
given by
Notice that the mean
and standard
deviation completely
characterize a normal
curve.
 ( x   ) /( 2  )
2
f ( x) 
e

2
2
Keeping our discussion of the normal curve in
mind, we consider a special normal curve (and
hence special continuous distribution) by letting
μ=0 and σ=1. This is the standard normal
variable, usually denoted z.
Table A in the back of the book gives “z-scores”
for a standard normal distribution.
We will see that any continuous distribution may
be “scaled” to standard by shifting the numbers
so that their mean is 0 and declaring that the
standard deviation constitutes a single unit
First an example.
1.
2.
3.
4.
For the standard normal random
variable, find
P(0<z<1.23)
P(-1.04<z)
P(-.65<z<1.82)
P(z>1.25 or z<-1.25)
We can also answer the following
question: P(z>?)=.22
Any normal distribution
can be “scaled” and
thought of as a standard
normal distribution. To
see how this is done,
consider a normal
variable x with mean μ
and standard deviation σ.
We define the standard
score or z-score to be
z
x

This does exactly what we want. For
example, when x=μ, we get a value of 0.
Let μ=50 and σ=5. If x=60, we can see
that x is 2 standard deviations above the
mean. This is also seen by plugging in the
values into the z-score formula. Hence,
the z-score also tells us how many
standard deviations away from the mean a
particular value is.
Examples
1.
2.
3.
4.
Consider a normal random variable with
mean 40 and standard deviation 15.
What percentage of values are
Larger than 60?
Less than 45?
Between 33 and 43?
Within 1.3 standard deviations of the
mean?
The lengths x of nails in a large shipment
received by a carpenter are approximately
normally distributed with mean 2 inches and
standard deviation .1 inch.
1. If a nail is randomly selected, find
P(1.8<x<2.07).
2. What proportion of nails have lengths that lie
within one standard deviation of the mean?
3. The carpenter can not use a nail shorter than
1.75 inches or longer than 2.25 inches. What
percentage of the shipment of nails will the
carpenter be able to use?
1.
2.
3.
Companies who designed furniture for elementary
school classrooms produce a variety of sizes for kids
of different ages. Suppose the heights of kindergarten
children can be described by a normal model with a
mean of 38.2 inches and standard deviation of 1.8
inches.
What fraction of kindergarten kids should the company
expect to be less than 3 feet tall?
In what height interval should the company expect to
find the middle 80% of kindergarteners?
At least how tall are the biggest 5% of kindergartners?
Normal?
How can we tell if data is approximately
normal?
Use a normal quantile plot.
Idea: Plot each point against its normal
score. The closer to a straight line, the
more likely the data is normal.
Best done on a computer.