Transcript I. What are z scores?

Outline
I. What are z-scores?
II.Locating scores in a distribution
A. Computing a z-score from a raw
score
B. Computing a raw score from a zscore
C. Using z-scores to standardize
distributions
III. Comparing scores from
different distributions
I. What are z scores?
You scored 76
How well did you perform?
 serves as reference point:
Are you above or below average?
 serves as yardstick:
How much are you above or below?
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Convert raw score to a z-score
z-score describes a score relative to  & 
Two useful purposes:
Tell exact location of score in a distribution
Compare scores across different
distributions
II. Locating Scores in a
distribution
z
X

Deviation from  in SD units
Relative status, location, of a raw
score (X)
z-score has 2 parts:
1. Sign tells you above (+) or
below (-) 
2. Value tells magnitude of
distance in SD units
A. Converting a raw score
(X) to a z-score:
X
z

Example:
Spelling bee:  = 8
Garth X=6
Peggy X=11


=2
z=
z=
Example
• Let’s say someone has an IQ of
145 and is 52 inches tall
– IQ in a population has a mean of
100 and a standard deviation of
15
– Height in a population has a mean
of 64” with a standard deviation of
4
• How many standard deviations
is this person away from the
average IQ?
• How many standard deviations
is this person away from the
average height?
B. Converting a z-score to
a raw score:
X    z
Example:
Spelling bee:  = 8
Hellen z = .5 
Andy z = 0 
=2
X=
X=
raw score = mean + deviation
C. Using z-scores to
Standardize a Distribution
Convert each raw score to a z-score
What is the shape of the new dist’n?
Same as it was before!
Does NOT alter shape of dist’n!
Re-labeling values, but order stays
the same!
What is the mean?
=0
Convenient reference point!
What is the standard deviation?
=1
z always tells you # of SD units from
!
An entire population of scores is transformed into z-scores. The
transformation does not change the shape of the population but the
mean is transformed into a value of 0 and the standard deviation is
transformed to a value of 1.
Example:
Student
X
Garth
6
Peggy
11
Andy
8
Hellen
9
Humphry
5
Vivian
9
X-
z
N=6
N=6
=8
= 0
=2
=1
So, a distribution of z-scores
always has:
=0
=1
A standardized distribution
helps us compare scores from
different distributions
III. Comparing Scores
From Different Dist’s
Example:
Jim in class A scored 18
Mary in class B scored 75
Who performed better?
Need a “common metric”
Express each score relative to it’s
own  & 
Transform raw scores to z-scores
Standardize the distributions
 they will now have same  & 
Example:
Class A: Jim scored 18
 = 10
=5
1810
z
1.6
5
Class B: Mary scored 75
 = 50
 = 25
7550
z
25
1
Who performed better? Jim!
Two z-scores can always be
compared
Homework
• Chapter 5
– 7, 8, 9, 10, 11