Transcript (Z* ) Score= +

Chapter 4 Part 1
Translating to and from Z
scores, the standard error
of the mean, and
confidence intervals around
muT
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Where we have been:
Z scores
If you know the proportion from the mean to the
score, then you can easily calculate:
The proportion above or below the score.
The percentile rank equivalent.
The proportion of scores between two Z scores.
The expected frequency of scores between two
Z scores
Concepts behind Z scores
 Z scores represent standard deviations above and below
the mean.
 Positive Z scores are scores higher than the mean.
Negative Z scores are scores lower than the mean.
 If you know the mean and standard deviation of a
population, then you can always convert a raw score to
a Z score.
 If you know a Z score, the Z table will show you the
proportion of the population between the mean and that
Z score.
Synonyms
Scores above the mean can be called
positive Z scores or high scores. The
terms are synonymous.
Similarly, scores below the mean can be
called negative Z scores or low scores.
The terms are synonymous.
Raw scores to Z scores
If we know the mean and standard
deviation of a population (mu and
sigma), any score in that population can
be translated into a Z score:
Z=
score - mean
standard deviation
=
X-

Z scores to other scores
Conversely, as long as you know the
mean and standard deviation of a
population, any Z score can be translated
into equivalent score in that population.
Score =  + ( Z *  )
Calculating z scores
Z=
score - mean
standard deviation
What is the Z score for someone 6’ tall, if the mean is
5’8” and the standard deviation is 3 inches?
6’ - 5’8”
Z=
3”
72 - 68
4
=
=
= 1.33
3
3
Production
F
r
e
q
u
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n
c
y
Standard
deviations
Z score = ( 2100 - 2180) / 50
= -80 / 50
= -1.60
units
2100
3
2
1
0
1
2
3
2030
2080
2130
2180
2230
2280
2330
What is the Z score for a daily production of 2100, given
a mean of 2180 units and a standard deviation of 50 units?
If you know a Z score, you
can determine theoretical
relative frequencies and
expected frequencies
using the Z table.
You often start with raw or scale scores
and have to convert them to Z scores.
Scale scores are public relations versions
of Z scores. They are Z scores with
different means and standard deviations.
Concepts behind Scale
Scores
Scale scores are Z scores expressed in a
different way.
In a sense, the most basic scale score is the Z
score itself, with mu = 0.00 and sigma = 1.00.
Raw scores can be converted to Z scores, which
in turn can be converted to other scale scores.
And other scale scores can be converted to Z
scores, that in turn can be converted to raw
scores.
You need to memorize
these scale scores
Z scores have been standardized so that they always have
a mean of 0.00 and a standard deviation of 1.00.
Other scales that are Z scores in disguise use other
means and standard deviations for historic or public
relations reasons. Three frequently used scales are:
IQ -
 =100;  = 15
SAT/GRE -
 =500;  = 100
Normal scores -  =50;  = 10
For example: To solve the problem below,
convert an SAT Score of 470 to a Z score,
then use the Z table as usual.
F
r
e
q
u
e
n
c
y
Standard
deviations
Proportion mu to Z for Z score
of -.30 = .1179
Z score = ( 470 - 500) / 100
= -30 / 100
= -0.30
Proportion at/below score
= .5000 - .1179
= . 3821 = 38.21%
score
3
2
1
200
300
400
470
0
1
2
3
500
600
700
800
What percentage of test takers obtain a verbal score of
470 or less, given a mean of 500 and a standard deviation of 100?
SAT to percentile – first
transform to a Z scores
If a person scores 592 on the SATs, what percentile is she at?
SAT

592 500
(X-)

(X-)/ 
92
100
0.92
Proportion mu to Z = .3212
Percentile = (.5000 + .3212) * 100 = 82.12 = 82nd
Convert to IQ scores to Z scores
to find the proportion of scores
between two IQ scores.
IQ scores have mu = 100 and sigma = 15. What
proportion of the scores falls between 85 and 115?
Z score = (85 - 100) / 15 = -15 / 15 = -1.00
Z score = (115 - 100) / 15 = 15 / 15 = 1.00
Proportion = .3413 + .3413 = .6826
What proportion of the scores falls between 95 and 110?
Z score = (95 - 100) / 15 = -5 / 15 = -0.33
Z score = (110 - 100) / 15 = 10 / 15 = 0.67
Proportion = .2486 + .1293 = .3779
NOTICE: Equal sized intervals, close to
and further from the mean: More
scores close to the mean!
Given mu = 100 and sigma = 15, what proportion of
the population falls between 95 and 105?
Z score = (95 - 100) / 15 = -5 / 15 = -.33
Z score = (115 - 100) / 15 = 5 / 15 = .33
Proportion = .1293 + .1293 = .2586
What proportion of the population falls between 105 and 115?
Z score = (105 - 100) / 15 = 5 / 15 = 0.33
Z score = (115 - 100) / 15 = 105/ 15 = 1.00
Proportion = ..3413 - .1293 = .2120
Percentile equivalents of IQ
scores: first translate to Z
scores
Convert IQ scores of 120 & 80 to percentiles.
X

(X-)

(X-)/ 
120 100 20.0 15 1.33
80 100 -20.0 15 -1.33
mu-Z = .4082, .5000 + .4082 = .9082 = 91st percentile,
Similarly 80 = .5000 - .4082 = 9th percentile
Convert an IQ score of 100 to a percentile.
An IQ of 100 is right at the mean and that’s the 50th percentile.
Going the other way –
Z scores to scale scores
Remember:
Score =  + ( Z *  )
Convert Z scores to IQ scores:
Individual scale scores get rounded
to nearest integer.
Z

(Z*)

+2.67 15 40.05 100
-.060
15 -9.00
100
IQ= + (Z * )
140
91
You try one. Let’s use
normal scores (mu=50,
sigma = 10).
Translate a Z score of –2.67 to a normal
score.
Convert Z scores to normal scores:
Individual scale scores get rounded
to nearest integer.
Z

(Z*)

-2.67 10 –26.7 50
Score= + (Z * )
23
Tougher problems – like
online quiz or midterm
If someone scores at the
58th percentile on the
verbal part of the SAT,
what is your best estimate
of her SAT score?
Percentile to scale score
If someone scores at the 58th percentile on the SAT-verbal,
what SAT-verbal score did he receive?
58th Percentile is above the mean. This will be a
positive Z score. The mean is the 50th percentile. So the
58th percentile is 8% or a proportion of .0800 above
mu. So we have to find the Z score that gives us a
proportion of .0800 of the scores between mu and Z.
Look at Column 2 of the Z table on page 54. Closest
Z score for area of .0800 is 0.20
Z

0.20 100
(Z*)
20

500
SAT= + (Z * )
520
Note that the only hard
part is finding the Z score.
First, ask how far and which direction you are
from mean of the scale score.
Second, use that percentage, expressed as a
proportion to enter the Z table, looking in
Column 2, 4, or 6 for a similar proportion.
Third, choose the Z score closest to that value
Finally, convert that to a scale score,
(remembering on which side of the mean you
are).
Slightly tougher –below the
mean
Percentile to scale score
If someone scores at the 38th percentile on the SAT-verbal,
what SAT-verbal score did he receive?
38th percentile is below the mean. This will be a
negative Z score. The mean is the 50th percentile. So
the 38th percentile is 12% or a proportion of .1200
below mu. So we have to find the Z score that gives us
a proportion of .1200 of the scores between mu and Z.
Look at Column 2 of the Z table on page 54. Closest
Z score for area of .1200 is 0.31. Z is negative
Z
-0.31

(Z*)
100
-31

500
SAT= + (Z * )
469
Double translations
On the verbal portion of the Wechsler IQ test, John scores
35 correct responses. The mean on this part of the IQ test is
25.00 and the standard deviation is 6.00. What is John’s
verbal IQ score?
Raw

(X- )
Scale Scale Scale
score (raw) (raw) 
Z


score
35
25.00 10.00 6.00 1.67 100
15 125
Z score = 10.00 / 6.00 = 1.67
IQ score = 100 + (1.67 * 15) = 125