Z Scores and Normal
Distribution (12-6)
Objective: Draw and label normal
distributions, compute z-scores,
and interpret and analyze data
using z-scores.
Distribution
• When there are a large number of values
in a data set, the frequency distribution
tends to cluster around the mean of the
set in a distribution (or shape) called a
normal distribution.
• The graph of a normal distribution is
called a normal curve.
• Since the shape of the graph resembles a
bell, the graph is also called a bell curve.
Distribution
Example 1
• The mean height of 15-year-old boys in the
city where Isaac lives is 67 inches, with a
standard deviation of 2.8 inches. Use
normal distribution to represent these data.
1. Use the mean value and the standard deviation
to find your minimum and maximum values.



Xmin = 67 – 3(2.8) = 58.6
Xmax = 67 + 3(2.8) = 75.4
Ymax = 1 ÷ (2 x 2.8) = 0.1786
Example 1
• The mean height of 15-year-old boys in the
city where Isaac lives is 67 inches, with a
standard deviation of 2.8 inches. Use
normal distribution to represent these data.
2. Go to WINDOW. Set the Xmin, Xmax, and Ymax
using the values from Step 1. The Xscl should
be the standard deviation. The Ymin should be
0. The Yscl should be 1.
Example 1
• The mean height of 15-year-old boys in the
city where Isaac lives is 67 inches, with a
standard deviation of 2.8 inches. Use
normal distribution to represent these data.
3. By entering the mean and standard deviation
into the calculator, we can graph the
corresponding normal curve. Enter the values
using the following keystrokes: Y= 2nd VARS
ENTER X,T,Θ,n , 67 , 2.8 ) GRAPH .
Example 1
• The mean height of 15-year-old boys in the
city where Isaac lives is 67 inches, with a
standard deviation of 2.8 inches. Use
normal distribution to represent these data.
58.6 in
61.4 64.2
μ
67
69.8 72.6 75.4 in
Z-Scores
• The distance between a data value and the
mean value is called the z-score.
• A z-score is the number of standard
deviations a data value is from the mean.
• The unit of measure for a z-score is a
standard deviation.
• The z-score for a data value x is given by
x 
z=
, where μ is the mean and σ is the

xx
standard deviation. For samples, use z 
S
Example 2
• Find the z-score for a height of 74 inches
using the following data set.
Heights: 70, 66, 72, 74, 70, 73, 68, 70, 67, 70,
71, 69, 68, 66, 69, 69, 64
• μ = 69.2
• σ = 2.5
x   74  69.2 4.8
 1.92

z

2.5

2.5
• The z-score for 74 inches is 1.92.
Example 3
• Find the z-score of someone who is 73
inches tall.
x   73  69.2 3.8
 1.52
z



2.5
2.5
Example 4
• Find the z-score of someone who is 66
inches tall.
x   66  69.2 3.2
 1.28

z

2.5

2.5
Z-Scores
 A negative z-score is the number of
standard deviations to the left or less
than the mean.
 A positive z-score is the number of
standard deviations to the right or more
than the mean.
 A z-score is always measured in standard
deviations.
Example 5
•
•
Z-scores can be used to compare scores from two different data sets.
In the year 2000, Barry Bonds hit 42 home runs. In the year 1950, Joe
Dimaggio hit 38 home runs. Which player was the better homerun hitter
for their corresponding years? In other words, which one broke farther
from the pack?
Year Mean Total Home Runs Standard Deviation
1950
27.6
5.2
2000
31.4
5.8
42  31.4
10.6

= 1.83
5.8
5.8
38  27.6
10.4

 Zdimaggio =
=2
5.2
5.2
 Zbonds =
•
Since Dimaggio has a higher z-score, he was the better homerun hitter for
his time.