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Z-Scores
Introduction to Statistics
Chapter 5
Feb 4-9, 2010
Classes #6-7
Z-Scores

Specifies the precise location of
each x score within a normal
distribution
Normal Distribution


A Normal Distribution is a distribution that
has a symmetric, unimodal and bell-shaped
density curve
The mean and standard deviation completely
specify the curve
Examples with approximate
Normal distributions






Height
Weight
IQ scores
Standardized test scores
Body temperature
Repeated measurement of same
quantity
Z-Scores

Main purposes



Make raw scores more meaningful
Tells us exactly where original scores are
located
Allows us to standardize an entire
distribution and thus allow for better
comparisons
The Standard Deviation and the
Normal Distribution


There are known percent ages of scores above or
below any given point on a normal curve
 34% of scores between the mean and 1 SD
above or below the mean
 An additional 14% of scores between 1
and 2 SDs above or below the mean
 Thus, about 96% of all scores are within 2
SDs of the mean (34% + 34% + 14% +
14% = 96%)
Note: 34% and 14% figures can be useful to
remember
The Normal Curve
Mean = 65
S=4
0.70
99.72% of cases
Relative Frequency
0.60
95.44% of cases
0.50
68.26% of cases
0.40
0.30
0.20
0.10
0.00
2%
14%
34%
34%
14%
2%
51 53 55 57 59 61 63 65 67 69 71 73 75 77 79
-3S
-2S
-1 S
0
+1 S
+2S
+3S
The 68-95-99.7 Rule



“A standard score
expresses a score’s
position in relation to the
mean of the distribution,
using the standard
deviation as the unit of
measurement”
“A z-score states the
number of standard
deviations by which the
original score lines above
or below the mean”
Any score can be
converted to a z-score
as follows
RelativeFrequency
The Standard Normal Curve
XX
z
S
z-score formula
For population:
z 
x

For a sample:
X M
z
s
z-score




One of the primary purposes of z-score is to
describe the exact location of a score within a
distribution
Pay particular attention to whether its +/This tells us whether the score is located
above (+) or below (-) the mean
The number tells us the distance between the
score and the mean in terms of standard
deviations
Finding Area When the Score is
Known

To find the proportion of the curve that lies
above or below a particular score

Convert raw score to z score
Finding Scores When the Area is
Known





Draw normal curve, shading
approximate area for the
percentage desired
Make a rough estimate of the
Z score where the shaded
area starts
Find the exact Z score using
normal curve table
Check to verify that it’s close
to your estimate
Convert Z score to raw score,
if desired
!!! Remember !!!

We can use the standard normal distribution
table (Table B.1 in Appendix B, pp. 584-587)
ONLY when our distribution of scores is normal.
 Using the standard normal table is not
appropriate if our distribution differs markedly
from normality
 e.g.,
 rectangular
 skewed
 bimodal
Comparing Scores from
Different Distributions


Again: The standard normal distribution
has a mean of 0 and standard deviation
of 1
Consider two sections of statistics


Gurnsey’s class has a mean of 80 and S of 5
Marcantoni’s class has a mean of 70 and S
of 5
Student 1 gets 80 in Gurnsey’s class
 Student 2 gets 75 in Marcantoni’s class


In relation to the rest of the class, which
student did better?
Percentile Ranks and the
Normal Distribution


When we ask what proportion of a
distribution lies below a particular z
score, we are actually asking what is
the percentile rank of the score
e.g., in a distribution with a mean of
100 and standard deviation of 15, 84%
of the distribution falls below a score of
115 [z = (115-100)/15 = 1].

Therefore, the percentile rank of 115 is 84%
Question


Example: Dave gets a 50 on his
Statistics midterm and a 50 on his
Calculus midterm. Did he do
equally well on these two exams?
Big question: How can we compare
a person’s score on different
variables?
Calculus
10
Statistics
•In one case, Dave’s exam
score is 10 points above
the mean
5
15
Example 1
•In the other case, Dave’s
exam score is 10 points
below the mean
0
•Standard deviation is 10.
•In an important sense, we
must interpret Dave’s grade
relative to the average
0
2 04 06 08 01 0 0
performance of the class
Mean Calculus
Mean Statistics
GRA DE
= 40
= 60
15
Example 1
Statistics
Calculus
Dave in Statistics:
10
(one SD above the
mean)
5
(50 - 40)/10 = 1
Dave in Calculus
(50 - 60)/10 = -1
0
(one SD below the
mean)
0
2 04 06 08 01 0 0
Mean Statistics = 40
GRA DE
Mean Calculus = 60
0 5 10 15 20 25 30
Example 2
Statistics
•The following week
Dave gets the same
grades (50 in each
class)
Calculus
0
•Both distributions have
the same mean (40),
but different standard
deviations (5 vs. 20)
2 04 06 08 0
100
GRA DE
0 5 10 15 20 25 30
Example 2
An example where the
means are identical, but
the two sets of scores
have different spreads
Statistics
Dave’s Stats Z-score
(50-40)/5 = 2.00
Calculus
Dave’s Calc Z-score
(50-40)/20 = 0.50
0
2 04 06 08 0
100
GRA DE
Example 2


In one case, Dave is performing better
than almost 95% of the class. In the
other, he is performing better than
approximately 68% of the class.
Thus, how we evaluate Dave’s
performance depends on how much
variability there is in the exam scores
Standard (Z) Scores

In short, we would like to be able to
express a person’s score with respect
to both (a) the mean of the group and
(b) the variability of the scores

how far a person is from the mean = X M
Standard score or
Z 
(X  M )
s
Example 3: Young Women’s Height

The heights of young women are
approximately normal with mean = 64.5
inches and std.dev. = 2.5 inches.
Example: Young Women’s Height

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% of young women between 62 and 67?
% of young women lower than 62 or taller than
67?
% between 59.5 and 62?
% taller than 68.25?
Example: Young Women’s Height

How about our class?
Three Properties of Standard
Scores

1. The mean of a set of z-scores is
always zero
Three Properties of Standard Scores


Why?
The mean has been subtracted from
each score. Therefore, following the
definition of the mean as a balancing
point, the sum (and, accordingly, the
average) of all the deviation scores
must be zero.
Three Properties of Standard Scores

2. The SD of a set of standardized
scores is always 1
The distribution of z-scores is always
equal to a SD of 1.0
M = 50
if x = 60,
SD = 10
60  50 10

1
10
10
x
20
30
40
50
60
70
80
z
-3
-2
-1
0
1
2
3
Three Properties of Standard
Scores

3. The distribution of a set of
standardized scores has the same
shape as the unstandardized scores

Z-score distribution is the same as raw
score distribution
The shape is the same
(but the scaling or metric is different)
STANDARDIZED
0
0.0
0.1
2
0.2
0.3
4
0.4
6
0.5
UNSTANDARDIZED
0.4
0.6
0.8
1.0
-6
-4
-2
0
2
Two Disadvantages of Standard
Scores
1.
A person’s score is expressed
relative to the group (X - M)
Example:
If Dave had taken his Calculus exam in a
class in which everyone knew math well
his z-score would be well below the
mean. If the class didn’t know math
very well, however, Dave would be
above the mean. Dave’s score depends
on everyone else’s scores.
Two Disadvantages of Standard
Scores
2. If the absolute score is meaningful
or of psychological interest, it will
be obscured by transforming it to a
relative metric
Credits
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http://wwwpsychology.concordia.ca/fac/gurnsey/PSYC315/M.Chapter6.ppt#2
95,17,Comparing Scores from Different Distributions
http://www.uic.edu/classes/psych/psych343f/lecture06.ppt#268,
19,Two Disadvantages of Standard Scores
http://www.unc.edu/~zhuz/teaching/Stat31/Notes/Lec05bb.ppt#
352,26,Homework 2.2
http://publish.uwo.ca/~pakvis/The%20Normal%20Curve.ppt#28
9,30,Probability: