Prelude to Chapters 6 & 7: ZScores
Suppose you scored 24 (out of 30) on a test
How well did you do?
W/out knowing the average score & the spread of the scores, it is hard to
determine
Zscores, or standardized scores, can specifically describe the relative
standing of every score in a distribution.

serves as a reference point:
Are you above or below average?
How much are you above or below the average?
What uses do Zscores serve?
Tell exact location of a score in a distribution Johnny is 10 yrs old and weighs 45 lbs How does his wt. compare to other 10 yr old boys?

Compare scores across different distributions
Jill scored 63 on her chemistry test & 47 on her biology test
On which test did she perform better?
How are Zscores calculated?
If you know & of a distribution, you can calculate a Z
score for any value in that distribution
z =
Relative status, location, of a raw score (X)
Sign tells you if score is above (+) or below ()
2. Value tells you the magnitude of distance in SD units
Converting a “raw” score into a Zscore: Example
The average pregnancy lasts 266 days, w/ a standard deviation
of 16 days
Laura gave birth after 273 days
Let’s convert this to a Zscore:
z =
X = 273 = 266 = 16
z = = +0.4375
Converting a Zscore to a “raw” score: Example
The length of Ellen’s pregnancy results in a Zscore of –1.25
How many days was she pregnant?
X =
Z = 1.25 = 266 = 16
X = 266 + (1.25)(16)
X = 246 days
If ALL raw scores in a distribution are converted to Z’s, you have a Zdistribution
Important Features of a Zdistribution

Mean of distribution is 0

SD of distribution is 1

Shape of distribution is the SAME as the shape of the original
Z Distribution Example
X

X 

/

Z

26

26 – 19 = 7

7 / 5

+1.4

18

18  19 = 1

1 / 5

0.2

20

20 – 19 = 1

1 / 5

+0.2

12

12 – 19 = 7

7 / 5

1.4

= 19 = 0
= 5 = 1
= (1.4 + 0.2 + 0.2 + 1.4) / 4 = 0
Comparing Values from Different Distributions
George scored 64 on his Botany test Carl scored 52 on his Calculus test
Who did better?
Difficult to compare “raw” scores
Can convert both scores to Z’s to put them on equivalent scales
Express each score relative to its OWN &
Zscores are directly comparable—in the same “metric”
Botany test (George): = 60 Calculus test (Carl): = 45
= 4.5 = 5
Z = (64 – 60) / 4.5 = +0.89 Z = (52 – 45) / 5 = +1.4
Carl Did Better!
Other Types of Standard Scores
“Transformed standard scores”
Further transformation of a zscore
Done for convenience
Often used in psychological/achievement testing
Some common transformed standard scores:
IQ scores: = 100 = 15
SAT sores: = 500 = 100
You decide what and you want
Does NOT change shape of the distribution!
Steps to follow:
(1) Transform raw score to zscore
(2) Choose new (a convenient #)
(3) Choose new (a convenient #)

Compute transformed standard score (TSS)
TSS = _{new} + z _{new}
Example:
IQ Scores = 100 + (z) 15
z = 1.0 IQ = 100 + (1) 15 = 85
z = 2.0 IQ = 100 + (2) 15 = 130
Let’s choose: _{new} = 50
_{new} = 10

Student  X  Z  Standard Score (TSS)  Garth 
6

(6 – 8) / 2 = 1

50 + (1)(10) = 40

Peggy

11

(11 – 8) / 2 = +1.5

50 + (1.5)(10) = 65

Andy

8

(8 – 8) / 2 = 0

50 + (0)(10) = 50

Helen

9

(9 – 8) / 2 = 0.5

50 + (0.5)(10) = 55

Humphrey

5

(5 – 8) / 2 = 1.5

50 + (1.5)(10) = 35

Vivian

9

(9 – 8) / 2 = +0.5

50 + (0.5)(10) = 55

N = 6 _{new} = 50
= 8 _{new} = 10
= 2
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