Use of standardized scores or z scores

2. USE OF STANDARDIZED SCORES OR Z-SCORES
Z Scores allow you to compare values from different data
sets. A Z scores measures how many standard deviations an
observation is above or below the mean.
1. Z Scores is unitless.
2. A positive Z Scores measures the number of standard
deviations an observation is above the mean.
3. A negative Z Scores measures the number of standard
deviations an observation is below the mean.
NOTE:
Care needs to be taken to determine whether a higher or
lower z- score is preferable.
In some situation high z- score is better and
In some situation low z- score is better.

3. ∑ HOW TO FIND A Z SCORE
Example-1
Suppose Jawwad score 75 in a test. The test has an average
60 with a standard deviation of 10. Find Z score
Solution:
Z = = =1.5
∑ How TO FIND A RAW SCORE WHEN A Z SCORE IS GIVEN
Example-2
Suppose you had an average marks in maths 60 with a
standard deviation of 10.How much you score in maths if
your z score is 1.5.
Solution:
Z =
Z× = −
= + (Z × )
= 60 + (1.5 × 10)
= 75

4. ∑ TO COMPARE TWO SCORES THAT ARE FROM DIFFERENT SAMPLES
Example-3.
Jawwad and Humza’s Maths marks are shown in the table
below along with the mean and standard deviation for the
whole class.
Jawwad
Marks
Humza
Marks
Mean
Marks
Standard
Deviation
Maths 75 65 60 10
Who has the higher relative score
Solution:
Jawwad Z Score is = =1.5
Humza Z Score is = = 0.5
Comments:
Jawwad has the higher relative score.
(In this case far from the mean is better)

5. Example-4.
Jawwad and Humza’s Maths marks are shown in the table
below along with the mean and standard deviation for the
whole class.
Jawwad
Marks
Humza
Marks
Mean
Marks
Standard
Deviation
Maths 55 50 60 10
Who has the higher relative score
Solution:
Jawwad Z Score is = = −0.5
Humza Z Score is = = -1
Comments:
Jawwad has the higher relative score.
(In this case nearest from the mean is better)

6. Example-5.
Jawwad and Humza’s Maths and English marks are shown
in the table below along with the mean and standard
deviation for the whole class.
Jawwad
Marks
Humza
Marks
Mean
Marks
Standard
Deviation
Maths 75 50 60 10
English 65 40 50 05
Work out Jawwad and Humza’s standardized scores
and comment on their performance.
Solution:
Jawwad’s standardized score can be shown in a table
X Jawwad Z Score
Maths 75 Z= = =1.5
English 65 = = =3
Humza’s standardized score can be shown in a table
X Humza Z Score
Maths 50 = = = - 1
English 40 = = = - 2
Comments:
Jawwad did better than Humza overall. The best mark is
Jawwad’s English mark of 65 because it has the highest
standardized score, while worst mark is Humza’s English
mark of 40.

7. Example-6.
Humza compared his running times in the 100m and 200m
events. In 100m he ran in16 seconds, while the age average
was 20 seconds with a standard deviation of 4 seconds. In
200 he ran in 39 seconds, while the age average was 44
seconds with a standard deviation of 2.5 seconds. In which
event did he do better, relative to other runners his age?
Solution:
For 100m
x = 16 sec.
µ = 20 sec. and = 4 .
=
−
=
16 − 20
4
= −1
For 200m
x = 39 sec.
µ = 44 sec. and = 2.5 .
=
−
=
39 − 44
2.5
= −2
Comments:
In 200m he did better.
(In this case far from the mean is better)

8. Example-7.
Humza free style swiming record is 34 seconds with
mean 42 seconds and a standard deviation of 3.5 seconds
but in butterfly his record is 38 seconds with mean 45
seconds and a standard deviation of 2.5 seconds. Decide in
which style his performance is better ?
Solution:
Free style
x = 34 sec.
µ = 42 sec. and = 3.5 .
=
−
=
34 − 42
3.5
= −2.29
Butterfly
x = 38 sec.
µ = 45 sec. and = 2.5 .
=
−
=
38 − 45
2.5
= −2.80
Comments:
In butterfly he did better.
(In this case far from the mean is better)