Transcript Chapter 4 Section 3

Chapter 2: The Normal
Distribution
Section 1: Density Curves and the
Normal Distribution
Density Curves
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%).
Density Curves
3. While a histogram represents actual data (i.e., a
sample set), a density curve represents an
idealized sample or population distribution.
Density Curves: Mean & Median
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.
Greek 101
• Since the density curve represents
"idealized" data, we use Greek letters: mu m
for mean and sigma s for standard
deviation.
Shapes of Density Curves
• We have mostly discussed right skewed, left
skewed, and roughly symmetric
distributions that look like this:
Bimodal Distributions
We could have a bi-modal distribution. For
instance, think of counting the number of tires
owned by a two-person family. Most twoperson 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.”
Uniform Distributions
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.
Other Distributions
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.
Normal Curves
• Curves that are symmetric, single-peaked,
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.
What does it look like?
Normal Curves: μ and σ
• 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.
Finding μ and σ
• 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.
The 68-95-99.7 Rule
In a normal distribution 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.
The 68-95-99.7 Rule
Why Use the Normal
Distribution???
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.
Heights of Young Women
• 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.
The 68-95-99.7 Rule
Use the previous chart...
• 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.?
But...
However, NOT ALL DATA are normal or
even close to normal. Salaries, for instance,
are generally right skewed. Nonnormal data
are common and often interesting.