COURSE: JUST 3900
INTRODUCTORY STATISTICS
FOR CRIMINAL JUSTICE
Chapter 5: Z-Scores
Location of Scores & Standardized Distributions
Instructor:
Dr. John J. Kerbs, Associate Professor
Joint Ph.D. in Social Work and Sociology
© 2013 - - DO NOT CITE, QUOTE, REPRODUCE, OR DISSEMINATE WITHOUT
WRITTEN PERMISSION FROM THE AUTHOR:
Dr. John J. Kerbs can be emailed for permission at kerbsj@ecu.edu
z-Scores and Location

By itself, a raw score or X value provides
very little information about how that
particular score compares with other
values in the distribution.

A score of X = 53, for example, may be a relatively
low score, or an average score, or an extremely high
score depending on the mean and standard deviation
for the distribution from which the score was obtained.

If the raw score is transformed into a z-score,
however, the value of the z-score tells exactly where
the score is located relative to all the other scores in
the distribution.
Distribution Examples:
Same μ and Different σ
If you received a
76 on an exam, in
which class
would you prefer
to have this
score?
z-Scores and Location (continued)

The process of changing an X value into a zscore involves creating a signed number,
called a z-score
 The sign of the z-score (+ or –) identifies whether the X


value is located above the mean (positive) or below the
mean (negative).
The numerical value of the z-score corresponds to the
number of standard deviations between X and the mean
of the distribution.
Thus, a score that is located two standard deviations
above the mean will have a z-score of +2.00.
And, a
z-score of +2.00 always indicates a location above the
mean by two standard deviations.
Relationship between z-score Values &
Locations in Population Distributions
Transforming
populations of
scores into zscores: Note that
distribution
shape does not
change below.
Note that
mean is
transformed
into a value
of 0 and the
standard
deviation is
transformed
into a value
of 1.
Practice Interpreting z-Scores

For the following z-scores, please describe the score’s
location in each distribution.


z = 1.75
z = - 0.50
z = 0.75
z = - 1.25
Identify the z-score value for the following locations in a
distribution.



Below the mean by 3 standard deviations
Above the mean by ¼ of a standard deviation
Below the mean by 1 standard deviations
Transforming Back and Forth
Between X and z

The basic z-score definition is usually
sufficient to complete most z-score
transformations. However, the definition can
be written in mathematical notation to create a
formula for computing the z-score for any
value of X.
X– μ
z = ────
σ
Practice z-Score Calculations

•
X– μ
z = ────
σ
With the formula above, please calculate:
• z for a distribution with μ = 20, σ = 6, X = 18
• z for a distribution with μ = 20, σ = 6, X = 22
• z for a distribution with μ = 20, σ = 6, X = 26
• z for a distribution with μ = 20, σ = 6, X = 32
Transforming Back and Forth
Between X and z (continued)

Also, the terms in the formula can be
regrouped to create an equation for
computing the value of X corresponding to
any specific z-score.
X = μ + zσ
Practice z-Score Calculations

•
X = μ + zσ
With the formula above, please calculate:
•
•
•
•
•
X for a distribution with μ = 20, σ = 6, z = 1.5
X for a distribution with μ = 20, σ = 6, z = -1.25
X for a distribution with μ = 20, σ = 6, z = 1/3
X for a distribution with μ = 20, σ = 6, z = -0.5
If μ = 50, X = 42 and z = - 2.00, what is the
standard deviation (σ) for the distribution?
Relationship between z-score Values &
Locations in Population Distributions
The distance that
is equal to 1
standard
deviation on the
x-axis (σ =10)
corresponds to 1
point on the zscore scale.
The Three Properties of z-Scores

1. Shape

The distribution of z-scores will have the exact
same shape as the original distribution

If the original distribution is negatively skewed, then the
z-scores distribution will be negatively skewed

If the original distribution is positively skewed, then the
z-scores distribution will be positively skewed

If the original distribution is normally distributed
(symetrical), then the z-scores distribution will be
normally distributed
The Three Properties of z-Scores
(Continued)

2. The Mean

The z-score distribution will always have a mean
of 0 (i.e., μ = 0).
 By definition, this is why all positive z-scores are
above the mean
 By definition, this is why all negative z-scores
are below the mean
The Three Properties of z-Scores
(Continued)

3. The Standard Deviation (σ)


The z-score distribution will always have a
standard deviation of 1 (i.e., σ = 1).
Because all z-score distributions have the same
mean and the same standard deviation, the zscore distribution is called a standardized
distribution.
 Standardized distributions are used to make
dissimilar distributions comparable.
z-scores and Locations



In addition to knowing the basic definition of a zscore and the formula for a z-score, it is useful to be
able to visualize z-scores as locations in a
distribution.
Remember, z = 0 is in the center (at the mean), and
the extreme tails correspond to z-scores of
approximately –2.00 on the left and +2.00 on the
right.
Although more extreme z-score values are possible,
most of the distribution is contained between z = –
2.00 and z = +2.00.

Remember: about 95% of all scores fall within + or – 2
standard deviations from the mean
z-scores and Locations
(Continued)

The fact that z-scores identify exact locations
within a distribution means that z-scores can
be used as descriptive statistics and as
inferential statistics.
As descriptive statistics, z-scores describe
exactly where each individual is located.
 As inferential statistics, z-scores determine
whether a specific sample is representative of its
population, or is extreme and unrepresentative.

z-Scores as a Standardized
Distribution
When an entire distribution of X values is
transformed into z-scores, the resulting
distribution of z-scores will always have a
mean of zero and a standard deviation of one.
 The transformation does not change the
shape of the original distribution and it does
not change the location of any individual
score relative to others in the distribution.

Transforming Raw Scores to z-scores:
No Change in Distribution Shape
z
z-Scores as a Standardized
Distribution (Continued)

The advantage of standardizing
distributions is that two (or more) different
distributions can be made the same.

For example, one distribution has μ = 100 and
σ = 10, and another distribution has μ = 40
and σ = 6.
 When
these distribution are transformed to zscores, both will have μ = 0 and σ = 1.
z-Scores as a Standardized
Distribution (Continued)
Please convert the following population of
N=6 scores (0, 12, 10, 4, 6, 4) into a
standardized distribution
 Step 1: Calculate the mean
 μ = ΣX/N
 Step 2: Calculate the standard deviation (σ)
 Step 3: Calculate z-score for each value of X

z-Scores as a Standardized
Distribution (Continued)
Please convert the following population of
N=6 scores (0, 12, 10, 4, 6, 4) into a
standardized distribution
 Step 1: Calculate the mean
 μ = ΣX/N = 36/6 = 6
 Because the mean is even, you can use
the definitional formula of the SS in Step 2

z-Scores as a Standardized
Distribution (Continued)
Please convert the following population of
N=6 scores (0, 12, 10, 4, 6, 4) into a
standardized distribution
 Step 2: Calculate the standard deviation (σ)






σ=
SS = Σ(X - μ)2 = (0-6)2 + (12-6)2 + (10-6)2 + (4-6)2 +
(6-6)2 + (4-6)2
= 36 + 36 + 16 + 4 + 0 + 4
= 96
σ=
z-Scores as a Standardized
Distribution (Continued)
Please convert the following population of
N=6 scores (0, 12, 10, 4, 6, 4) into a
standardized distribution
 Step 3: Calculate z-score for each value of X

X-score
(x-μ) where μ = 6
σ
z-score = (x-μ) /σ
0
-6
4
z = - 1.50
12
6
4
z = 1.50
10
4
4
z = 1.00
4
-2
4
z = - 0.50
6
0
4
z = 0.00
4
-2
4
z = - 0.50
z-Scores as a Standardized
Distribution (Continued)
Because z-score distributions all have the
same mean and standard deviation,
individual scores from different distributions
can be directly compared.
 A z-score of +1.00 specifies the same
location in all z-score distributions.

z-Scores and Samples
It is also possible to calculate z-scores for
samples.
 The definition of a z-score is the same for
either a sample or a population, and the
formulas are also the same except that the
sample mean and standard deviation are used
in place of the population mean and standard
deviation.

z-Scores and Samples
Example




Thus, for a score X from a sample, you can calculate the zscore as follows:
X–M
z = ─────
s
Using z-scores to standardize a sample also has the same
effect as standardizing a population.
Specifically, the mean of the z-scores will be zero (M z = 0)
and the standard deviation of the z-scores will be equal to
1.00 (s z = 1) provided the standard deviation is computed
using the sample formula (dividing n – 1 instead of n).
Each z-score can be transformed into an X value as
follows: X = M + z s
z-Scores and Samples
Example

Please use the formula below to calculate the following:
X–M
z = ─────
s
•
z for a distribution with M = 40, s = 12, X = 43
z for a distribution with M = 40, s = 12, X = 34
z for a distribution with M = 40, s = 12, X = 58
z for a distribution with M = 40, s = 12, X = 28
•
Answers: z = 0.25, - 0.50, 1.50, - 1.00
•
•
•
z-Scores and Samples
Example

Please use the formula below to calculate the following:
X = M+zs
•
•
•
•
•
X for a distribution with M = 80, s = 20, z = - 1.00
X for a distribution with M = 80, s = 20, z = 1.50
X for a distribution with M = 80, s = 20, z = - 0.50
X for a distribution with M = 80, s = 20, z = 0.80
Answers: X = 60, 110, 70, 96
Other Standardized Distributions
Based on z-Scores
Although transforming X values into zscores creates a standardized distribution,
many people find z-scores burdensome
because they consist of many decimal
values and negative numbers.
 Therefore, it is often more convenient to
standardize a distribution into numerical
values that are simpler than z-scores.

Other Standardized Distributions
Based on z-Scores (Continued)

To create a simpler standardized
distribution, you first select the mean and
standard deviation that you would like for
the new distribution. This is your choice:


e.g., μ = 50 and σ = 10 or μ = 100 and σ = 10
Then, z-scores are used to identify each
individual's position in the original
distribution and to compute the individual's
position in the new distribution.
Other Standardized Distributions
Based on z-Scores (Continued)



Suppose, for example, that you want to
standardize a distribution so that the new mean is
μ = 50 and the new standard deviation is σ = 10.
An individual with z = –1.00 in the original
distribution would be assigned a score of X = 40
(below μ by one standard deviation) in the
standardized distribution.
Repeating this process for each individual score
allows you to transform an entire distribution into a
new, standardized distribution.
Other Standardized Distributions
Based on z-Scores (Continued)


Suppose, for example, that you want to
standardize a distribution so that the new mean is
μ = 50 and the new standard deviation is σ = 10.
In the original distribution, μ = 68 and σ = 15

What is the z-score for an x value of 83 in the original
distribution: Z = (83-68)/15 = 15/15 = + 1.00

An individual with an 83 in the original distribution would
be given a z-score of + 1.00. In the new distribution
(μ = 50, σ = 10), the original score of 83 would be
assigned a value of 60 in the new standardized
distribution (μ+1 σ = 50 +10 = 60), which is one
standard deviation above the mean.