Download Normal Distribution

Normal Distribution
• Many common statistics (such as human
height, weight, or blood pressure) gathered
from samples in the natural world tend to
have a normal distribution about their mean.
– A normal distribution has data that vary randomly
from the mean.
Characteristics of a Normal
Distribution
• The graph of the curve is continuous, bellshaped, and symmetric with respect to the
mean.
• The mean, median, and mode are equal and
located at the center.
• The curve approaches, but never touches, the
𝑥-axis.
• The total area under the curve is equal to 1 or
100%.
In a normal distribution,
• Approximately 68% of the population is within
1 standard deviation of the mean.
• Approximately 95% of the population is within
2 standard deviations of the mean.
• Approximately 99.7% of the population is
within 3 standard deviations of the mean.
• This is the Empirical or 68-95-99.7 Rule.
Empirical Rule
Example 1 A normal distribution has a mean of
21 and a standard deviation of 4.
a. Find the range of values that represent the
middle 68% of the distribution.
b. What percent of the data will be greater than
29?
Example 2 The heights of 1800 adults are
normally distributed with a mean of 70 inches
and a standard deviation of 2 inches.
a. How many adults are between 66 and 74
inches?
b. What is the probability that a random adult is
more than 72 inches tall?
Example 3 IQ scores are known to be normally
distributed with a mean of 100 and a standard
deviation of 15.
What percent of the population has a score
above 130?
What score cuts off the lowest 16% of the
population?
• The Empirical Rule is only useful for evaluating
specific values, such as 𝜇 + 𝜎 .
• Therefore, we standardize the data set by
converting the data to 𝑧-values.
z
X 

Standard Normal Distribution
• Has the following properties:
1. Its graph is bell-shaped and the total area under
the curve is 1 or 100%.
2. Almost all the area is between 𝑧 = −3 and 𝑧 =
3.
3. Its mean is equal to 0 (𝜇 = 0).
4. Its standard deviation is equal to 1 (𝜎 = 1)
Example 4 Find 𝑧 if 𝑋 = 18, 𝜇 = 22, and 𝜎 =
3.1. Indicate the position of 𝑋 in the
distribution.
X
Example 5 The U.S. Air Force requires that that
pilots have heights between 64 in. and 77 in.
Given that men have normally distributed
heights with a mean of 69.5 in. and a standard
deviation of 2.4 in., find the percentage of men
who satisfy that height requirement.
Use the z-score table!