value. If the score to be ”standardized” is larger then the mean, the
deviation will be a positive one and the value of the zscore will be
positive; the reverse is also true.
(2)
the magnitude of the
z
score communicates an observation’s relative
distance to the mean, as compared to other data values.
(3)
we can use
z
scores to standardize entire distributions. By converting
each score in a distribution to a
z
score we obtain a standardized
distribution.Such a standardized distribution, regardless of its original
values, will have a mean of 0.0 and a variance and standard deviation
of 1.0.
(4)
they can be mathematically related to probabilities. If the standardized
distribution is a normal distribution we can state the probability of
occurrence of an observation with a given
z
score.
Lecture Number 3
Data Description
October 19, 2016
74 / 99
Measures of Position  Standard Scores
Example 12:
A student scored 65 in a Biometry test that had a mean of
50 and a standard deviation of 10; she scored 30 in a Soil Science test
with a mean of 25 and a standard deviation of 5. Compare her relative
positions in the two tests.
Solution for Example 12:
First, find the
z
scores. For Biometry the
z
score is:
z
=
x

¯
x
s
=
65

50
10
= 1
.
5
For Soil Science, the
z
score is:
z
=
x

¯
x
s
=
30

25
5
= 1
.
0
Since the
z
score for Biometry is larger, her relative position in the class is
higher than in the Soil Science class.
Lecture Number 3
Data Description
October 19, 2016
75 / 99
Measures of Position  Percentiles
Percentiles divide a data set into 100 equal parts.
Definition of Percentile
The
p
th percentile of a set of
n
measurements arranged in order of
magnitude is that value that has at most
p
% of the measurements below
it and at most (100

p
)% above it.
Percentiles are frequently used to describe the results of achievement,
test scores and the ranking of a person in comparison to the rest of
the people taking an examination.
Note that percentiles are not the same as percentages! e.g.If a
student gets 73% out of 100% in a Biometry test, there is no
indication of his/her position with respect to the rest of the class.
The computation of percentiles is illustrated on the next slide.
Lecture Number 3
Data Description
October 19, 2016
76 / 99
Measures of Position  Percentiles
Each data value corresponds to a percentile for the percentage of the
data values that are less than or equal to it.
Let
x
(1)
,
x
(2)
,
x
(3)
, ...,
x
(
n
)
denote the ordered observations for a data
set; that is,
x
(1)
≤
x
(2)
≤
x
(3)
≤
...
≤
x
(
n
)
.
The
i
th ordered observation,
x
(
i
)
, corresponds to the 100(
i

0
.
5)
/
n
percentile.
Percentile
=
100(
i

0
.
5)
n
(11)
Alternative formula:
Percentile
=
(
No
.
of values below x
i
)+0
.
5
Total No
.
of values
*
100%
These formulas are used in place of assigning the percentile 100
i
/
n
so
that we avoid assigning the 100th percentile to
x
(
n
)
, which would
imply that the largest possible data value in the population was
observed in the data set, an unlikely happening.
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 F. TAILOKA
 Standard Deviation, Mean