Transcript Chapter 2: The Normal Distribution

A density curve is similar to a histogram, but there
are several important distinctions.
1. Obviously, a smooth curve is used to represent
data rather than bars. However, a density
curve describes the proportions of the
observations that fall in each range rather than
the actual number of observations.
2. The scale should be adjusted so that the total
area under the curve is exactly 1. This
represents the proportion 1 (or 100%).
3. While a histogram represents actual data
(i.e., a sample set), a density curve
represents an idealized sample or
population distribution. (describes the
proportion of the observations)
4. Always on or above the horizontal axis
5. We will still utilize mu m for mean and sigma s
for standard deviation.
Three points that have been previously made
are especially relevant to density curves.
1. The median is the "equal areas" point.
Likewise, the quartiles can be found by
dividing the area under the curve into 4
equal parts.
2. The mean of the data is the "balancing"
point.
3. The mean and median are the same for a
symmetric density curve.

We have mostly discussed right skewed,
left skewed, and roughly symmetric
distributions that look like this:
We could have a bi-modal distribution. For
instance, think of counting the number of
tires owned by a two-person family. Most
two-person families probably have 1 or 2
vehicles, and therefore own 4 or 8 tires.
Some, however, have a motorcycle, or
maybe more than 2 cars. Yet, the
distribution will most likely have a “hump”
at 4 and at 8, making it “bi-modal.”
We could have a uniform distribution.
Consider the number of cans in all six
packs. Each pack uniformly has 6 cans.
Or, think of repeatedly drawing a card
from a complete deck. One-fourth of the
cards should be hearts, one-fourth of the
cards should be diamonds, etc.
Many other distributions exist, and some do
not clearly fall under a certain label.
Frequently these are the most interesting,
and we will discuss many of them.
#1 RULE – ALWAYS MAKE A PICTURE
It is the only way to see what is really going on!
Curves that are symmetric, singlepeaked, and bell-shaped are often
called normal curves and describe
normal distributions.
 All normal distributions have the same
overall shape. They may be "taller" or
more spread out, but the idea is the
same.

The "control factors" are the mean μ and
the standard deviation σ.
 Changing only μ will move the curve
along the horizontal axis.
 The standard deviation σ controls the
spread of the distribution. Remember
that a large σ implies that the data is
spread out.

You can locate the mean μ by finding
the middle of the distribution. Because it
is symmetric, the mean is at the peak.
 The standard deviation σ can be found
by locating the points where the graph
changes curvature (inflection points).
These points are located a distance σ
from the mean.

In a NORMAL DISTRIBUTIONS with mean μ
and standard deviation σ:
 68% of the observations are within σ of
the mean μ.
 95% of the observations are within 2 σ of
the mean μ.
 99.7% of the observations are within 3 σ
of the mean μ.
1. They occur frequently in large data sets
(all SAT scores), repeated measurements
of the same quantity, and in biological
populations (lengths of roaches).
2. They are often good approximations to
chance outcomes (like coin flipping).
3. We can apply things we learn in studying
normal distributions to other distributions.

The distribution of heights of young
women aged 18 to 24 is approximately
normally distributed with mean m = 64.5
inches and standard deviation s = 2.5
inches.
Where do the middle 95% of heights fall?
 What percent of the heights are above
69.5 inches?
 A height of 62 inches is what percentile?
 What percent of the heights are
between 62 and 67 inches?
 What percent of heights are less than 57
in.?


Suppose, on average, it takes you 20
minutes to drive to school, with a
standard deviation of 2 minutes.
Suppose a normal model is appropriate
for the distribution of drivers times.
› How often will you arrive at school in less
than 20 minutes?
› How often will it take you more than 24
minutes?