5.1 Intro to Normal
Distributions
Statistics
Mrs. Spitz
Fall 2008
Objectives/Assignment
• How to interpret graphs of normal
probability distributions
• How to find areas under a normal curve,
and use them to find probabilities for
random variables with normal distributions.
• Assn: pp. 198-201 #1-26 all
Properties of a Normal
Distribution
• In 4.1, we learned that a continuous
random variable has an infinite number of
values that can be represented by an
interval on the number line. It’s probability
distribution is called a continuous
probability distribution. In this chapter, we
will be studying the most important
continuous probability distribution in
statistics—the normal distribution.
Guidelines: Properties of a Normal
Distribution
•
A normal distribution is a continuous
probability distribution for a random
variable, x. The graph of a normal
distribution is called the normal curve. A
normal distribution has the following
properties.
1. The mean, median and mode are equal.
2. The normal curve is bell-shaped and is
symmetric about the mean.
Guidelines: Properties of a Normal
Distribution
3. The total area under the normal curve is equal to 1.
4. The normal curve approaches, but never touches the xaxis as it extends farther and farther away from the
mean.
5. Between  -  and  +  (in the center of the curve) the
graph curves downward. The graph curves upward to
the left of  -  and to the right of  + . The points at
which the curve changes from curving upward to
curving downward are called inflection points.
Guidelines: Properties of a Normal
Distribution
Inflection points
Guidelines: Properties of a Normal
Distribution
• A normal distribution can have any mean
and any positive standard deviation.
These two parameters,  and  completely
determine the shape of a normal curve.
The mean gives the location of the line of
symmetry and the standard deviation
describes how much the data are spread
out.
Look at the three examples below:
See the line of symmetry for each? That’s the mean. However, if it is
fatter, then the standard deviation is greater. That’s the difference.
Ex. 1: Understanding Mean &
Standard Deviation
1.
2.
Which normal curve has a greater mean?
Which normal curve has a greater standard deviation?
SOLUTION:
1. The line of symmetry of curve A occurs at x = 15. The line of symmetry of
curve B occurs at x = 12. So, curve A has a greater mean.
Ex. 1: Understanding Mean &
Standard Deviation
1.
2.
Which normal curve has a greater mean?
Which normal curve has a greater standard deviation?
SOLUTION:
2. Curve B is more spread out than curve A, so curve B has a greater standard
deviation.
Try it yourself:
•
•
•
Consider the normal curves
shown below. Which normal
curve has the greatest mean?
Which normal curve has the
greatest standard deviation?
Justify your answers?
Find the location of the line of
symmetry for each curve. Make a
conclusion about which mean is
greatest.
Determine which normal curve is
more spread out. Make a
conclusion about which standard
deviation is the greatest.
Try it yourself:
1a. A: 45, B: 60, C: 45
1b. C has the most
spread, so the
greatest standard
deviation
Slide 11B
Ex 2: Interpreting graphs of Normal Distributions
• The heights (in feet) of fully grown white oak trees are
normally distributed. The normal curve shown below
represents this distribution. What is the mean height of
a fully grown white oak tree? Estimate the standard
deviation of this normal distribution.
Mean of
about 90 feet
with standard
deviation of
about 3.5
feet.
Normal Curves and Probability
• The total area under a probability curve is
equal to 1. The area of a region under a
probability curve gives the probability that
the random variable will have a value in
the corresponding interval. In this chapter,
you will learn several ways to find areas,
under normal curves. You have already
studied one of these ways in section 2.4—
The Empirical Rule.
Ex. 3: Estimating a Probability for a Normal Curve
• Adult IQ scores
are normally
distributed with 
= 100 and  =
15. Estimate the
probability that a
randomly
chosen adult
has an IQ
between 70 and
115.
Using the Empirical Rule, the area under the
normal curve between these two values is:
Solution: Draw the
curve.
Area = .135 + .34 + .34 = .815. So the
probability the adult has an IQ between 70 and
115 is about .815.
70
85
100
115
130