Z-Scores are measurements of how far from the
center (mean) a data value falls.
Adult Male Heights
Ex: A man who stands
71.5 inches tall is 1
standard deviation ABOVE
the mean.(z-score = 1)
Ex: A man who stands 64
inches tall is 2 standard
deviations BELOW the
mean. (z-score = -2)
Based upon the Empirical Rule, we know the approximate
percentage of data that falls between certain standard deviations on
a normal distribution curve.
Example:
If a class of test scores has a
mean of 65 and standard
deviation of 9, then what
percent of the students
would have a grade below
56?
Standardized Scores (aka z-scores)
Z-score represents the exact number of
standard deviations a value, x, is from the mean.
xx
z
s
observation (value)
mean
standard deviation
Example: (test score problem)
What would be the z-score for a student that received a
70 on the test?
Example: The mean speed of vehicles along a particular section
of the highway is 67mph with a standard deviation of 4mph. What
is the z-score for a car that is traveling at 72 mph?
What is the z-score for a car that is traveling 60mph?
Mark the z-scores on the number line below
|----------|----------|----------|----------|----------|----------|
(z =) -3
-2
-1
0
1
2
3
55
59
63
67
71
75
79
Z-scores are also called standardized scores.
To calculate any z-score:
𝑧=
𝑥−𝑥
𝑠
American Adult Males: mean of 69 inches and standard
deviation of 2.5 inches.
1) What is the standardized score for a male with a height of
72 inches?
2) What is the standardized score for a male with a height of
62 inches?
To find the proportion or probability that a certain interval is
possible, we use the z-score table.
The z-score table
always tells the
proportion to the
LEFT of that z-score
value.
What percentage of the data will have a z-score of less than 1.15?
P(z < 1.15)
Other Examples:
1) Find P(z < 0.22)
2) Find P(z > 0.22)
3) Find P(0.22 < z < 1.15)