Z Score,T Score, Percentile Rank and Box Plot Graph

1. Z-Score
T-Score
Percentile Rank
&
Box-Plot Graph
K.THIYAGU,
Assistant Professor,
Department of Education,
Central University of Kerala, Kasaragod
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2. Z-Score
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3. Z-Score
Z-Score (Z-value, Standard Score, and Normal Score)
Z-score is the
number of standard deviations
from the mean, a data point is
It’s a measure of how many
standard deviations
below or above the population mean
The basic Z score formula for a sample is:
Z =
(𝒙#𝝁)
𝝈 Deviation
Std
core
DeviationS
score
Z
.
=
-
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5. Z-score (Standard Deviations) p-value (Probability) Confidence level
< -1.65 or > +1.65 < 0.10 90%
< -1.96 or > +1.96 < 0.05 95%
< -2.58 or > +2.58 < 0.01 99%
https://pro.arcgis.com/en/pro-app/latest/tool-reference/spatial-statistics/what-is-a-z-score-what-is-a-p-value.htm
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6. Example # 1
Let’s say you have a test score of 190.
The test has a mean (µ) of 150 and a
standard deviation (σ) of 25.
Assuming a normal distribution,
Your Z-score would be:
Z = (x – µ) / σ = 190 – 150 / 25 = 1.6.
The Z score tells you how many standard deviations
from the mean your score is.
In this example,
Your score is 1.6 Standard Deviations above the Mean.
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7. Ahalya has secured 50 marks in English; the
class mean is 60, and the standard deviation is
10. What is the Z-score of Ahalya?
D
S
M
X
Z
.
-
= s
1
10
10
10
60
50
-
=
-
=
-
=
Z
Example # 2
Answer:
The raw score of 50 may be converted to Z-score by
applying the formula.
So, Ahalya’s Z-score is
Ahalya’s score is 1-Sigma distance below the Mean.
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8. T-Score
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9. T-Score
T-score is a type of standard score computed
by multiplying a Z-score by 10 and adding 50.
T-scores tell individuals
How far is their score from the mean?
T-scores have a mean of 50 and
a standard deviation of 10.
T-score was 70 it would, in turn, mean that their
score was 20 points above the mean.
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10. ThiyaguSuriya 10

11. Z-Score
vs
T-Score
https://www.educba.com/z-score-vs-t-score/
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12. https://www.educba.com/z-score-vs-t-score/
Z-Score
vs
T-Score
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13. T = (Z x 10) + 50.
T = 10 Z + 50.
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14. Example
Amartya secured 60 marks in English,
which has a mean of 40, and an S.D. of 8.
She secured 50 marks in the mother tongue
test, having a mean of 50 and an S.D. of 6.
Convert all raw scores into T-scores. In
which subject has Amartya done better?
T = 10 Z + 50
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15. Answer:
Amartya secures 60 in English.
Where Mean = 40 and S.D = 8
Answer:
Amartya secures 50 in Mother Tongue
Where Mean = 50 and S.D = 6
𝑇 = 10 [
𝑥 − 𝑀
𝑆𝐷
] + 50
T = 10 [
!"#!"
$
] + 50
T = 10 [
"
$
] + 50
T = 0 + 50
T = 50
𝑇 = 10 [
𝑥 − 𝑀
𝑆𝐷
] + 50 T = 10 [
$"#%"
&
] + 50
T = 10 [
/0
1
] + 50
T = 10 [
2
/
] + 50
T = 25 + 50 = 75
From the above two T-Scores, Amartya Secures better in English subject than Mother Tongue.
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16. T-Score vs. Z-Score: When to Use Each
https://www.statology.org/t-score-vs-z-score/
Do you know the population SD?
Use a t-score
Is the sample size (n) greater than 30?
N
o
Y
e
s
Use a t-score Use a z-score
N
o
Yes
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17. Percentile
Rank
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18. Percentile Rank
The nth percentile is that scale
value or score point below which
‘n’ percent of the cases in the
distribution fall. The scale value is
the percentile, while its
corresponding percentage value is
its percentile rank.
Percentile
Rank
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19. Percentile Rank
Calculation of Percentile Rank
from Ungrouped Data
ú
û
ù
ê
ë
é -
-
=
N
R
PR
50
100
100
Where,
PR = Percentile Rank
R = Rank position of the score of an individual
whose percentile rank is to be determined.
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20. Example
Find out the percentile rank of score ‘25’
from the following scores.
65, 46, 58, 32, 25, 14, 15, 10, 9, 7, 5, 3
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22. Group Data
ú
û
ù
ê
ë
é
÷
ø
ö
ç
è
æ -
+
= w
b f
i
L
K
cf
N
PR
100
• Cfb=cumulative frequency below the class interval which k
lies / containing k
• K = Score for which we want to find percentile rank
• L = Actual exact lower limit of the class interval containing
K
• Fw= frequency within the class interval containing k
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23. CI ECI /CCI f cf
91 – 100 90.5 – 100.5 3 30
81 – 90 80.5 – 90.5 7 27 K
71 – 80 70.5 – 80.5 8 20
61 – 50 60.5 – 50.5 5 12
51 – 60 50.5 – 60.5 4 7
41 – 50 40.5 – 50.5 3 3
Find out the percentile rank of 82 scores of the following frequency distribution.
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24. ThiyaguSuriya 24

25. Box Plot
Graph
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26. Box Plot
The box plot (box and whisker diagram) is a
standardized way of displaying the distribution of data
based on the five number summaries:
Minimum,
First Quartile,
Median,
Third Quartile,
and Maximum.
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27. Box Plot
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29. ThiyaguSuriya 29

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31. Boxplots allow us to evaluate whether a data set is symmetrical, right-skewed, or left-skewed.
https://www.labxchange.org/library/items/lb:LabXchange:d8863c77:html:1
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32. Histograms and boxplots of symmetric, right-skewed, and left-skewed unimodal data sets
https://www.labxchange.org/library/items/lb:LabXchange:d8863c77:html:1
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33. Thank You
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