Numerical Measures of Position
Standardization
With the x & s
a clear picture of the distribution of a set
of values is possible.
Often, an individual value is of interest rather
than the entire set of values.
2
Standardization
Using the x & s, a standard score can be
calculated.
Standardization allows for
• the comparison of individual scores,
• even those with different scales
Ex. The ability to compare the results of 3
different writing measures
Ex. Weights of an individual calculated in lbs
and kg
3
Standardization and z Score
Standardization is simply a transformation of a value to
a linear standard score
most frequently a z score.
A z score tells us how far a raw value is above or below
(±) the mean of a distribution (the group of values).
The distance is expressed using standard deviation
units.
Standard Score
z = (x - x ) / s
4
z Score
z = (x - x ) / s
Ex. Ashley has two test scores on listening
comprehension; Test A = 49 and Test B = 99.
Test A = 49 Test A has a M = 44 and a s = 4. What would
her standard score be for Test A?
zA = (49 - 44 ) / 4 = 5/4 = 1.25
Test B = 99 Test B has a M = 104 and a s = 3. What
would her standard score be for Test B?
zB = (99 - 104 ) / 3 = -5/3 = -1.67
5
z Scores and Normal Curve
z scores can be plotted on the Normal Curve.
There is a direct 1:1 ratio between standard deviation
and z scores.
± 1s = ± 1z
± 2s = ± 2z
± 3s = ± 3z
6
z Scores and Normal Curve
z scores can be
plotted on the normal
curve
S
-3
-2
-1
M
0
z
-3
-2
-1
0
Test A
= 1.25 z
Test B
= -1.67 z
+1
+2
+3
+1
+2
+3
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z Score
Regardless of the scale, z scores remain the same
Patients’ z scores remain the same even though the
measurement scale changes
Weight in kg
Student
x
z
A
48
B
Weight in lbs.
x
z
-1.52
105.2
-1.52
52
-0.17
114.4
-0.17
C
54
-0.51
118.8
-0.51
D
60
1.18
123.2
1.18
8
Percentile Rank
Percentile ranks (%ile) are based on the percentage
areas under the curve.
A percentile rank (PRx) describes the percentage of
scores in the distribution that fall above and below the
given standard score.
Ex.
A value at the 65th percentile falls at or above 65% of
the values in the distribution.
A values at the 43rd percentile falls at or above 43% of
the values in the distribution.
9
Percentile Rank
The formula for Percentile Rank is as follows:
PRx = B + 0.5 fx 100 =
n
where B = number of values below x
where fx = number of values equal to x
With this measure, rank order is a must!!
10
Percentile Rank
Shoe Sizes for a sample of 12 male students
13 11 10 13 11 10 8 12 9 9 8 9
What is the percentile rank for a shoe size of 12?
Rank Order
8 8 9 9 9 10 10 11 11 12 13 13
PR12 = 9 + 0.5(1)
12
X 100
PR12 = 79.17
A value of 12 corresponds approx. to the 79th percentile.
79% of the sampled male students have a shoe size of 12 or
smaller.
11
Distributions
Percent of values
under portions of
the normal curve
34.13%
34.13%
13.59%
13.59%
2.14%
2.14%
0.13%
S
0.13%
-3
-2
-1
0
1
2
3
M
12
Percentile Rank and Normal Curve
Percentile Rank can be
plotted on the normal
curve
Size 12
= 79th
Size 10
= 50th
Size 11
= 67th
S
-3
-2
-1
M
0
z
-3
-2
-1
0
+1
+2
+3
%ile
0.1
2
16
50
84
98
99.9
+1
+2
+3
13