This document discusses various types of scaling scores used in educational testing and assessment. It begins by defining raw scores and derived scores. It then discusses three main types of derived scores: Z-scores (standard scores), T-scores (transformed scores), and C-scores. For each score type, it provides the meaning, calculation method, and examples of calculating the scores from raw scores. It concludes by discussing the utilities of each score type and providing some example problems for practice.

1. SCALING:
Z- SCORES (Standard Scores)
T- SCORES (Transformed Scores)
C- Scores
SURBHI SHARMA
Department of Education
UNIVERSITY OF LUCKNOW
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• What are SCORES?
• What are RAW SCORES?
• Meaning of Derived/Converted/ Standard Scores.
1. Z-Scores
1.1 Meaning of Z- Scores
1.2 Calculation of Z- Scores
1.3 Utility of Z- Scores
2. T- Scores
2.1 Meaning of T- Scores
2.2 Calculation of T- Scores
2.3 Utility of T- Scores
3. C- Scores
3.1 Meaning of C- Scores
3.2 Calculation of C- Scores
3.3 Utility of C- Scores
• Problem Sums
• References

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What are Scores?
According to Ross –
“A score on any test is simply a numerical
description of an individual’s performance on
that task.”

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What are RAW Scores?
According to Smith –
“Raw Scores are direct counts of the number of items a specified
event has occurred, such as -
• the number of test items answered correctly,
• the number of books read,
• the number of errors made in solving a problem,
• the number of statements endorsed on a questionnaire, etc.”

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I scored 60 in English & 60 in
Mathematics. Are my scores in both
subjects significantly meaning the
same?

6. What should I
do now?
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Derived Scores / Converted Scores / Standard Scores
Meaning:
The scores which have a definite reference
point or norm, and this norm is the Mean (M)
of the scale. Besides, it has a definite standard
deviation ().
…contd.

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Derived Scores / Converted Scores / Standard Scores
Types of Derived Scores:
• Z- Scores
• T- Scores
• C-Scores
• Stanine Scores
• Hull Scores , etc.

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In midterm exam, Ram has obtained a score of 55 in
mathematics.
The Maximum
Marks can be scored
is 85
for the class
The Mean is 50 for
the class
SITUATION A SITUATION B

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1. Z-Scores (Standard Scores):
Meaning of Z- Scores:
It tells how many standard deviations
someone is above or below the mean.
Mean(M) = 0
Standard Deviation () = 1
Its value is generally from
-3 to +3
The unit of Z-Score is similar to the
standard deviation ()

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Calculation of Z- Scores/ Sigma-scores (Standard Scores)
Where,
Z= Z-Score or sigma-score
X= Raw Score
M= Mean
 = Standard Deviation

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Ques. 1.
The marks obtained by student A in the
Mathematics and Language tests of maximum
marks 100, each are 80 and 70 respectively.
If the mean (M) of mathematics scores of the
class students is 60 and standard deviation ()
8, and in the language test 40 and 10
respectively; then find out in which subject the
student A is more able as compared to other
students?

13. Contd….
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Analysis:
Because student A’s Z-score in language is more
than that of mathematics, so he is more able in
language in the class as compared to
mathematics.

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Utility of Z-Scores: In Education for meaningful analysis of scores of a students, etc.

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2. T-Scores (Transformed Scores):
Meaning of T- Scores:
T-scores are those converted scores of raw scores
of which-
Mean(M) = 50
Standard Deviation () = 10
Distribution is Normal.

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Calculation of T- Scores:
Where,
T= T-Score
X= Raw Score
M= Mean
 = Standard Deviation

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Calculation of T- Scores:
Where,
T= T-Score
Z= Z-Score

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Ques. 2.
In Ques. 1 The score in mathematics of
student A=80 and Z-Score= 2.5; and the score
in language= 70 and Z-Score= 3. Convert these
scores into T-Scores.

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Contd. …

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Row Scores Standard Scores
( Z-Scores)
T-Scores
80 2.5 75
70 3 80
Tabulation of Scores :

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Utility of T-Scores:
If the range of general distribution is taken to be 100, then range of T-Scale becomes from 0 to
100 as a result, therefore the measurement of scores can be done more accurately.

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Comparison of Z- Scores & T- Scores

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3. C-Scores :
Meaning of C- Scores:
C-scores are those converted scores of raw scores of which-
Mean(M) = 5
Standard Deviation () = 2
 Range of Raw Score is displayed on a 11- point scale from 0
to 10.

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Calculation of C- Scores:
Where,
C= C-Score
Z= Z-Score
Method 1: Using Z-Score

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Calculation of C- Scores:
Method 2: Using Percentile Rank,
that is, a raw score into C-Score by first changing raw scores into percentile rank and
then into C-Scores by using the following table:

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Utility of C-Scores:
In Education, the use of C-Scores is more useful than the use
of Z-Scores and T-Scores.
However,
• C-Score has demerit of being awarding 0- Score.
•Also, there comes much difference in % if C-Score goes
up/down by even 1- point.

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Sums For Practice:

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Ques. A.
In a testing survey of class X students with
N=500, their mean was observed as 80 and
standard deviation as 8. Assuming normality of
distribution of their scores, how many students
would be below the score of a student getting
88? (UGC-Net, 2019)
a) 171
b) 421
c) 84
d) 16

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Ques. B.
A researcher gives two tests-one in English language and another in
mathematics. He/She finds the mean & standard deviation of each of
the two groups n=100, as follows:
Assuming the curve for the two distributions overlapping, how many
students in English language are below the mean of those in mathematics?
a) 34
b) 84
c) 16
d) 32
English Mathematics
Mean(M) 50 60
Standard Deviation() 10 12

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Answers :
Ques. A 421
Ques. B 34

34. Z-TableofAreas
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REFERENCES:
1] Asthana,B.(2015) Measurement and evaluation in
psychology and evaluation,Agarwal Publication, Agra.
2] Garrett, H.E.(1952) Statistics in psychology and education,
Longmans, Green & Co., London.
3] Gupta, S.C. & Kapoor, V.K.(2011) Fundamentals of
Mathematics Statistics, Sultan Chand & Sons, New Delhi.

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