Download Normal Distributions and theempirical rule

Normal Distributions and
the Empirical Rule
Objective:
The students will apply the Empirical
Rule to normally distributed data to
solve problems
Properties of a Normal
Distribution

x
•The mean, median, and mode are
equal
•Bell shaped and is symmetric about
the mean
•The total area that lies under the curve is one or 100%
2
The 68-95-99.7 Rule (Also
called the Empirical Rule)
• In the normal distribution with
mean  and standard deviation
• 68% of the observations fall
within
of the mean 
• 95% of the observations fall
within 2 of the mean 
• 99.7% of the observations fall
within 3 of the mean 


Empirical Rule
Example
1
Use the Empirical Rule to Analyze Data
A. A normal distribution has a mean of 45.1
and a standard deviation of 9.6. Find the
values that represent the middle 99.7% of
the distribution.
μ = 45.1 and σ = 9.6
The middle 99.7% of data in a normal
distribution is the range from μ – 3σ to μ +
3σ.
45.1 – 3(9.6) = 16.3
45.1 + 3(9.6) = 73.9
Answer: Therefore, the range of values in
the middle 99.7% is 16.3 < X < 73.9.
Example 2
Use the Empirical Rule to Analyze Data
B. A normal distribution has a mean of 45.1
and a standard deviation of 9.6. What
percent of the data will be greater than
54.7?
The value 54.7 is 1σ more than μ.
Approximately 68% of the data fall between
μ – σ and μ + σ, so the remaining data values
represented by the two tails covers 32% of
the distribution. We are only concerned with
the upper tail, so 16% of the data will be
greater than 54.7.
Answer: 16%
Example 3
A normal distribution has a mean of 38.3 and
a standard deviation of 5.9. What percent of
the data will be less than 26.5?
A. 0.3%
B. 2.5%
C. 5%
D. 97.5%
Example 4
Use the Empirical Rule to
Analyze a Distribution
A. PACKAGING Students counted the
number of candies in 100 small packages.
They found that the number of candies per
package was normally distributed, with a
mean of 23 candies per package and a
standard deviation of 1 piece of candy.
About how many packages have between 22
and 24 candies?
22 and 24 are 1σ away from the mean.
Therefore, about 68% of the data are
between 22 and 24.
Since 100 × 68% = 68 we know that about 68
of the packages will contain 22 to 24 pieces.
Answer: about 68 packages
Example 5
Use the Empirical Rule to
Analyze a Distribution
B. PACKAGING Students counted the number
of candies in 100 small packages. They found
that the number of candies per package was
normally distributed, with a mean of 23 candies
per package and a standard deviation of 1 piece
of candy. What is the probability that a package
selected at random has more than 25 candies?
Values greater than 25 are more than 2σ from the
mean. The values that are more than 2σ from the mean
cover two tails and 5% of the distribution. We are only
concerned with the upper tail, so 2.5% of the data will
be greater than 25.
Answer: about 2.5%
Example 6
DRIVER’S ED The number of students per
driver’s education class is normally distributed,
with a mean of 26 students per class and a
standard deviation of 3 students. What is the
probability that a driver’s education class selected
at random will have between 23 and 32 students?
A. 17%
B. 34%
C. 68%
D. 81.5%