Download Algebra 2 Normal Distribution Notes File

12.5 Notes—The Normal Distribution
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A normal distribution curve is a symmetrical, bell-shaped curve defined by the mean and
the standard deviation of a data set. The mean is located on the line of symmetry of the
curve.
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Areas under the curve represent probabilities associated with continuous distributions.

The normal curve is a probability distribution and the total area under the curve is 1.

For a normal distribution, approximately 68 percent of the data fall within one standard
deviation of the mean, approximately 95 percent of the data fall within two standard
deviations of the mean, and approximately 99.7 percent of the data fall within three
standard deviations of the mean.

The mean of the data in a standard normal distribution is 0 and the standard deviation is
1.

The standard normal curve allows for the comparison of data from different normal
distributions.

A z-score is a measure of position derived from the mean and standard deviation of data.

A z-score expresses, in standard deviation units, how far an element falls from the mean
of the data set.

A z-score is a derived score from a given normal distribution.

A standard normal distribution is the set of all z-scores.
Characteristics of the Normal Distribution
*The maximum occurs at the mean. The mean, median, and mode are equal.
*The distribution can extend from negative to positive infinity, but never touches the x-axis.
The Empirical Rule
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
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About 68% of the values are within 1 standard deviation of the mean.
About 95% of the values are within 2 standard deviations of the mean.
About 99.7% of the values are within 3 standard deviations of the mean.
Example 1: A normal distribution of the data has a mean of 34 and standard deviation of 5. Find the
probability that a random x value is greater than 24, that is, P(x>24).
Example 2: The useful life of a certain car battery is normally distributed with a mean of 100,000 miles
and a standard deviation of 10,000 miles. The company makes 20,000 batteries a month.
a. About how man batteries will last between 90,000 and 119,000 miles?
b. About how many batteries will last more than 120,000 miles?
c. About how many batteries will last less than 90,000 miles?
d. What is the probability that if you buy a car battery at random, it will last between 80,000 and
110,000 miles?