Unit 6 – Data Analysis and
Probability
6.2 – USE NORMAL
DISTRIBUTIONS
Georgia Performance Standards
 MM3D2a – Determine intervals about the mean that
include a given percent of data.
 MM3D2b – Determine the probability that a given
value falls within a specified interval
 MM3D2c – Estimate how many items in a
population fall within a specified interval.
Vocabulary
 A normal distribution is modeled by a bell-
shaped curve called a normal curve that is
symmetric about the mean. A normal distribution
with mean x and standard deviation 𝜎 has the
following properties. The total area under the
related normal curve is 1. the percent of the area
covered by each standard deviation is shown in the
graph.
Normal Distribution
What is Standard Deviation?
 The standard deviation is a statistic that tells you
how tightly all the various examples are clustered
around the mean in a set of data. When the examples
are pretty tightly bunched together and the bellshaped curve is steep, the standard deviation is
small. When the examples are spread apart and the
bell curve is relatively flat, that tells you you have a
relatively large standard deviation.
Vocabulary
 The standard normal distribution is the normal
distribution with mean 0 and standard deviation 1.
the formula below can be used to transform x-values
from a normal distribution with mean x and
standard deviation 𝜎 into z-values having a standard
normal distribution.
Z-scores
 The z-value for a particular x-value is called the z-
score for the x-value and is the number of standard
deviations the x-value lies above or below the mean
x. To find the probability that z is less than or equal
to some given value, use the standard normal table
(next slide)
Standard Normal Table
Example 1: Find a normal probability
 A normal distribution has mean x and standard
deviation 𝜎 . For a randomly selected x-value from
the distribution, find P(x ≤ x ≤ x + 3 𝜎 ).
Finding Normal Probabilities
 A normal distribution has mean x and standard
deviation . Find the indicated probability for a
randomly selected x-value from the distribution.
P(x ≤ x ≤ x - 2𝜎 ).
P(x ≥ x + 2𝜎 ).
P(x -2𝜎 ≤ x ≤ x+𝜎 ).
Example 2: Interpret normally distributed data
 The heights of 3000 women at a particular college
are normally distributed with a mean of 65 inches
and a standard deviation of 2.5 inches. About how
many of these women have heights between 62.5
inches and 67.5 inches?
Use a z-score and the standard normal table
 In Example 2 (women going to college), find the
probability that a randomly selected college woman
has a height of at most 68 inches.
Z
.0
.1
.2
-3
.0013
.0010
.0007
-2
.0228
.0179
.0139
-1
.1587
.1357
.1151
-0
.5000
.4602
.4207
0
.5000
.5398
.5793
1
.8413
.8643
.8849
Guided Practice for Examples 2 and 3
 Guided Practice (Page 221)
 4-6