Prelude to Chapters 6 & 7: z-scores Suppose you scored 24 (out of 30) on a test




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Prelude to Chapters 6 & 7: Z-Scores
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


Z-scores, 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?


  •  serves as a yardstick:

How much are you above or below the average?
What uses do Z-scores serve?

  1. 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?





  1. 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 Z-scores 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)





  • z-score has 2 parts:


  1. 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 Z-score: 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 Z-score:


z =
X = 273  = 266  = 16




z = = +0.4375
Converting a Z-score to a “raw” score: Example

The length of Ellen’s pregnancy results in a Z-score of –1.25

How many days was she pregnant?


X =
Z = -1.25  = 266  = 16
X = 266 + (-1.25)(16)

X = 246 days

Z Distribution: A Standardized Distribution


If ALL raw scores in a distribution are converted to Z’s, you have a Z-distribution


Important Features of a Z-distribution


  1. Mean of distribution is 0

  2. SD of distribution is 1

  3. 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  & 


Z-scores 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 z-score
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 z-score
(2) Choose new (a convenient #)
(3) Choose new (a convenient #)


  1. 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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