Download Describing Data - Mrs Michele Murphy Carlisle Math Students

The Normal Distribution
Section 11.10

Discrete Probability Distribution
- a finite number of possible events,
or values
Continuous Probability Distribution
- events for this can be any value in
an interval of real numbers
Normal Distribution

Has data that vary randomly from the
mean.

The graph of a normal distribution is a
normal curve
Normal Distribution
For data with a (symmetric) bell-shaped
distribution, the standard deviation has the
following characteristics:
• About 68% of the data lie within one standard
deviation of the mean.
• About 95% of the data lie within two standard
deviations of the mean.
• About 99.7% of the data lie within three standard
deviations of the mean.
Normal Distribution
3 standard deviations
2 standard deviations
1 standard
deviation
34%
2.35%
x  3s
34%
13.5%
x  2s
13.5%
x s
x
xs
2.35%
x  2s
x  3s
A normal distribution has a symmetric bell shape centered on the mean.
The normal distribution and standard
deviations
In a normal distribution:
Approximately 68% of scores will fall within one
standard deviation of the mean
The normal distribution and standard
deviations
In a normal distribution:
Approximately 95% of scores will fall within two
standard deviations of the mean
The normal distribution and standard
deviations
In a normal distribution:
Approximately 99.7% of scores will fall within three
standard deviations of the mean
The Shape of Distributions
Sometimes data are not normally distributed. They can be
Skewed, an asymmetric curve where one end stretches
out further than the other end.
Normal distributions (bell
shaped) are a family of
distributions that have the
same general shape. They
are symmetric (the left side is
an exact mirror of the right
side) with scores more
concentrated in the middle
than in the tails. Examples of
normal distributions are
shown to the right. Notice
that they differ in how spread
out they are. The area under
each curve is the same.
The normal distribution and standard
deviations
34%
2.35%
34%
13.5%
In a normal distribution:
The total area under the curve is 1.
13.5%
2.35%
Analyze Normally Distributed
Data
The bar graph gives the weights of a population of female brown
bears. The red curve shows how the weights are normally
distributed about the mean, 115kg. Approximately what percent of
female brown bears weigh between 100 and 129 kg?
Example

For a population of male European eels, the mean body
length is shown below. Sketch a normal curve showing the
eel lengths at one, two, and three standard deviations from
the mean.
For a population of male European eels, the mean body length is
shown below. Sketch a normal curve showing the eel lengths at
one, two, and three standard deviations from the mean.
Example

The height of adult American males are
approximately normally distributed with mean of
69.5 in and standard deviation 2.5 in. What percent
of adult males are between 67 in and 74.5 in tall?
Example

In a group of 2000 adults American males.
About how many would you expect to be
taller than 6 ft?
Exit Ticket
The heights of men in a survey are normally
distributed about their mean.
1. About what percent of men aged 25 to 34 are
69 – 71 inches tall?
2. About what percent of men aged 25 to 34 are
less than 70 in tall?
3. Suppose the survey included data on 100 men.
About how many would you expect to be 69 – 71 in tall?
4. The scores on the Algebra 2 final are approximately normally distributed with a
mean of 150 and a standard deviation of 15.
a. What percentage of students who took the test scored above 180?
b. If 250 students took the final exam, approximately how many scored above 135?
5. For a population of female European eels, the mean body length is 21.1 in. The
standard deviation is 4.7 in. Sketch a normal curve showing eel lengths at one, two,
and three standard deviations from the mean.