Download Normal Distribution

Avon High School
Section:
12.3
ACE COLLEGE ALGEBRA II - NOTES
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Normal Distribution
Uses of Normal Distribution
Mr. Record: Room ALC-129
Semester 2 - Day 54
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. The graph of a normal distribution is a
normal curve.
Sometimes an extraordinary factor affects data that would otherwise be normally distributed. A coin, for
example, may be somehow weighted unevenly so that heads tends to come up more frequently than tails. In
such a case, the data set could have a distribution that is skewed  an asymmetric curve where one end
stretches out further than the other end.
Example 1
Analyzing Normally Distributed Data
The bar graph gives the weights of a population of female brown bears. The curve shows how the
weights are normally distributed about the mean, 115 kg.
a. Approximately what percent of female brown bears weight between 100 and 129 kg?
b. Approximately what percentage of female brown bears weight less than 120 kg?
c. The standard deviation in the weights of female brown bears is about 10kg. Approximately what
percent of female brown bears have weights that are within 1.5 standard deviations of the mean?
Sketching a Normal Curve
Example 2
For a population of male European eels, the mean body length and one positive and one negative standard
deviation is shown below. Sketch a normal curve showing the eel lengths at one, two, and three standard
deviations from the mean.
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Example 3
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Analyzing a Normal Curve
The heights of adult American males are approximately normally distributed with mean 69.5 in. and
standard deviation 2.5 in.
a. What percent of adult American males are between 67 in. and 74.5 in. tall?
b. In a group of 2000 adult American males, about how many would you expect to be taller than 6 ft?
Uses of Normal Distribution
Fitting data to a normal distribution curve is not appropriate for all data sets. When a data set is not
symmetrical about the mean, is skewed, or has gaps or other unusual features, a normal curve may not be a
good model for the data.
Example 4
Determining if a Data Set is Normally Distributed
The data below represents the ages at inauguration for 44 U.S. Presidents.
a. Use your calculator to make a histogram of the data. Is the shape of the graphed data uniform,
symmetric, or skewed? Are there any unusual features?
b. The mean of this data set is 54.7. Where is the mean located on the histogram?
c. Is a normal curve an approprite model for the data? Explain.
Example 5
Determining if a Data Set is Normally Distributed
The data below represents the times spent waiting in line (in minutes) to ride a popular roller coaster at an
amusement park.
a. Use your calculator to make a histogram of the data. Is the shpe of the graphed data uniform, symmetric,
or skewed? Are there any unusual features?
b. The mean of this data set is 10.5. Where is the mean located on the histogram?
c. Is a normal curve an approprite model for the data? Explain.