Transcript Z score - rci.rutgers.edu

Chapter 4
Translating to and from Z
scores, the standard error
of the mean and confidence
Welcome Back!
intervals
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Concepts behind Z scores
Z scores represent standard deviations above
and below 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, then you can look up in
the Z table the proportion of the population
between the mean and score.
Z scores continued
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
scores.
Definition
If we know mu and sigma, any score
can be translated into a Z score:
Z=
score - mean
standard deviation
=
X-

Definition
Conversely, as long as you know mu and
sigma, a Z score can be translated into
any other type of score:
Score =  + ( Z *  )
Raw scores to Z scores
Given mu = 100 and sigma = 4.00, convert the following
raw scores to Z scores.
X

(X-)

(X-)/ 
106
87
100 6.00 4.00
100 -13.00 4.00
1.50
-3.25
95
100 -5.00
-1.25
4.00
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
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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?
Verbal SAT Scores
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Standard
deviations
Proportion mu to Z for .30
= .1179
Z score = ( 470 - 500) / 100
= -30 / 100
= -0.30
Proportion 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?
Convert to Z scores to find the
proportion of scores between two
raw scores.
Given mu = 100 and sigma = 15, what proportion of
the population 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 population 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
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
Concepts behind Scale
Scores
Scale scores are raw scores expressed in a
standardized way.
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 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 use other means and standard deviations.
Examples:
IQ -
 =100;  = 15
SAT/GRE -
 =500;  = 100
Normal scores -  =50;  = 10
Convert Z scores to IQ scores
Z

(Z*)

+2.67 15 40.05 100
-.060
15 -9.00
100
IQ= + (Z * )
140
91
Scale score translation: first
always 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.
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
Reverse the order: %tile 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
SAT / GRE scores - Examples
How many people out of 400 can be expected to score
between 550 and 650 on the SAT?
SAT

550 500
650 500
(X-) 
50
150
100
100
(X-)/ 
0.50
1.50
Proportion mu to Z0.50 = .1915
Proportion mu to Z1.50 = .4332
Proportion difference = .4332 - .1915 = .2417
Expected frequency = .2417 * 400 = 96.68 people
Normal scores - examples
1. Convert a normal score of 37 to a Z score.
X

(X-)

37
50
-13.0
(X-)/ 
10
-1.30
EASY
2. Convert a Z score of 0.00 to a normal score.
Z

(Z*)

0.00
10
0
50
 + (Z * )
50
Midterm type problems:
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
Double translations again
On the GRE-Advanced Psychology exam, there are 225 questions.
The mean is 125.00 correct with a standard deviation of 12.00.
Joan gets 116. What is her GRE score on this test?
Raw

(X- )

Scale Scale Scale
score (raw) (raw) (raw) Z

 score
116 125.00 -9.00 12.00 -0.75 500 100 425