Assessment of Student Learning
Module
10
Module 10
MEASURES OF VARIABILITY
Objectives

Describe the measures of variability/dispersion

Compute the quartile, percentile and standard deviation of a set of scores
Introduction
When we summarize and describe a set of scores, or a frequency distribution, we
are also interested to report how variable the scores are, or how much they spread out
from high to low scores. For example, two groups of children, both with the median age
of 10 years would represent quite different educational situations if one had a spread ages
9 to 11 while the other ranged from 6 to 14. Another example would be two groups of
children have a mean score of 75 on an achievement test, but one group has a score
ranging from 50 to 93 while the other group has scores ranging from 60 to 82. A
measure of this spread or dispersion is an important statistic for describing a group
Range
A very simple measure of variability is to get the score difference between the
highest and the lowest score and this measure is called the range of the distribution. If in
Page 1 of 12
Assessment of Student Learning
Module
10
a reading test for example, the highest score is 90 and the lowest is 41, the range is 49.
However, the range depends only upon the 2 extreme scores in the total group. This
makes this measure a very unreliable because it can be changed a good bit by the
inclusion or omission of a single extreme case. The example below illustrates that the
range of a set of scores is affected by a single extreme score
Group 1
45
50
76
77
80
81
90
95
Group 2
25
50
76
77
80
81
90
95
Semi-Interquartile Range or Quartile
Another measure of variability is the range of scores that includes a specified part
of the total group – usually the middle fifty percent. The middle fifty percent of the
group are scores lying between the 25th and 75th percentiles. The 25th(Q1)and 75th (Q3)
percentiles are called quartiles since they cut off the bottom quarter and the top quarter of
the group respectively. The score distance between them is called the interquartile range
.The statistic that is often reported as a measure of variability is the semi-quartile range
(Q), which is half of the interquartile range.
So,
Q = Q3-Q1
2
Page 2 of 12
Assessment of Student Learning
Module
10
How to compute for Q 1and Q3
X
95-99
90-94
85-89
80-84
75-79
70-74
65-69
60-64
55-59
50-54
45-49
40-44
f
3
4
5
8
6
10
4
4
2
0
1
3
cf
50
47
43
38
30
24
14
10
6
4
4
3
Q3
Q1
Q1 = LQ1+ i [(N/4 – cf)/ f Q1 ]
Where : Q1 = the 25th percentile
LQ1 = lower limit of the median class
N = total number of frequencies in the distribution
cf = cumulative frequency of the median class
f Q1 = frequency of the median
i = size of the interval of the median class
So, Q1 = LQ1+ i (N/4 – cf )/ f Q1
= 64.5 + 5 (50/4 - 10)/4
= 64.5 + 5 (12.5-10)/4
= 64.5 + 5 (2.5/4)
= 64.5 + 5(.63)
= 64.5 + 3.15
= 67.65
Page 3 of 12
Assessment of Student Learning
Module
10
Q3 = LQ3+ i (3N/4 – cf)/ f Q3
Where : Q3 = the 75th percentile
LQ3 = lower limit of the median class
N = total number of frequencies in the distribution
cf = cumulative frequency of the median class
fQ3 = frequency of the median
i = size of the interval of the median class
So, Q3 = LQ3+ i (3N/4 – cf)/ f Q3
= 79.5 + 5 (37.5- 30)/8
= 79.5 + 5 (7.5/8)
= 79.5 + 5 (.94)
= 79.5 + 4.7
= 84.2
Therefore:
Semi-quartile range (Q) = 84.2 – 67.65
= 8.275
2
Percentiles
The same procedure may be used when we find the score below which any
percentage of the group falls. These values are called percentiles. The median is the 50th
percentile, i.e., the score below which 50 percent of individuals fall. If we want to find
Page 4 of 12
Assessment of Student Learning
Module
10
the 40th percentile, we must find the score below which 40 percent of the cases fall. Any
other percentiles can be found in the same way. Percentiles have many uses, especially
in connection with test norms and interpretation of scores
Below is an illustration how a particular percentile is computed.
X
75-77
72-74
69-71
66-68
63-65
60-62
57-59
54-56
51-53
48-50
45-47
42-44
39-41
36-38
33-35
f
3
5
2
4
3
0
2
4
6
3
4
6
5
1
2
cf
50
47
42
40
36
33
33
31
27
21
18
14
8
3
2
N=50
For example, we are looking for the 20th percentile or the score in which 20 percent of the
cases falls below it, then
20th percentile or P20 = LP20+ i (20%N – cf)/ f P20
= 41.5 + 3 [(.20x50)-8]/6
= 41.5 + 3 (10-8)/6
= 41.5 + 1.00
= 42.50
Page 5 of 12
Assessment of Student Learning
Module
10
Standard Deviation or SD
Standard Deviation or SD is a measure of dispersion among all scores in the distribution
rather than through extreme scores of or only a proportion of the scores. It is the square root of
the average of the squared deviation from the Mean.
Finding the Standard Deviation of Ungrouped Scores
Steps:
1. Find the Mean
2. Subtract the Mean from the scores
3. Square the deviation
4. Find the sum of the squared deviation (∑d2)
5. Divide the sum of the squared deviation by the number of cases
6. Find the square root of the answer in step 5
In symbol, SD =
Where:
d = deviation from the mean
d2 = squared deviation
∑d2 = sum of the squared deviation
Page 6 of 12
Assessment of Student Learning
Module
10
Given these scores, let us find the S.D.
Score
92
75
85
83
90
73
79
80
88
83
d
+9
-8
+2
0
+7
-10
-4
-3
+5
0
d2
81
64
4
0
49
100
16
9
25
0
∑d2= 348
N = 10
Mean = 83
SD =
SD =
SD =
SD = 5.59
Finding the Standard Deviation of Grouped Scores
For ungrouped data, we compute for the standard deviation by computing first for the
actual Mean of the set of scores. For grouped data, standard deviation in computed from an
“assumed mean”. To illustrate, study the example given.
Page 7 of 12
Assessment of Student Learning
Module
10
X
75-77
72-74
69-71
66-68
63-65
60-62
57-59
54-56
51-53
48-50
45-47
f
3
4
6
5
8
9
5
8
3
2
2
N=55
d
5
4
3
2
1
0
-1
-2
-3
-4
-5
fd
15
16
18
10
8
0
-5
-16
-9
-8
-10
+67
- 48
∑fd = +19
fd2
75
64
54
20
8
0
5
32
27
32
50
∑fd2 = 367
Steps:
1. Choose any step interval for the assumed mean as the arbitrary starting point or
“origin”. In the example given, the interval 60-62 has been chosen. Call this interval zero, and
the next higher interval +1, the lower interval -1, etc. These are shown in the column labeled d.
(Note: Any interval can be chosen, and the final result will be the same)
2. Multiply frequency (f) by the number of deviations (d) and the resulting product is
shown in column labeled fd. Get the sum of fd by taking into account the plus and minus signs.
3. To get fd2, multiply d by the fd. Then get the sum of fd2
Page 8 of 12
Assessment of Student Learning
Module
10
Substitute the corresponding values to the formula:
2
S.D. = i
Where,
i= interval
N = number of cases
∑fd = summation of frequency deviation
∑fd2= summation of frequency x squared deviation
S.D. = 3
=3
2
2
=3
=3
= 3 x 2.56
= 7.68
Page 9 of 12
Assessment of Student Learning
Module
10
Summary
To represent the spread of scores, the range is the quickest measure. However, the range
is an unreliable measure because it greatly depends on extreme scores. Statisticians have
developed the semi-interquartile range, half the distance between the 25th and 75th percentile as a
more reliable measure of dispersion.
We can also use percentiles to determine the score by
which certain percent of the cases fall below it. Finally, a measure of variability/dispersion that
involves all scores in the distribution is the standard deviation, a type of average of the
deviations of the scores away from the average (Mean). For ungrouped data, standard deviation
is computed from the actual mean while for the grouped data, it is computed from an assumed
mean of the distribution.
Page 10 of 12
Assessment of Student Learning
Module
10
Activity 1
1. Compute for the Semi-quartile range (Q) and the 60th percentile of the frequency
distribution
X
90-94
85-89
80-84
75-79
70-74
65-69
60-64
55-59
50-54
45-49
f
2
2
5
7
10
11
8
3
0
2
2. Using the set of scores to illustrate finding the standard deviation of grouped scores,
choose another step interval for the assumed mean and show that the standard deviation
of the distribution is also equal to 7.68.
3. Compute for the standard deviation of both Section A and Section B. Compare
results, discuss differences if any.
Page 11 of 12
Assessment of Student Learning
Module
10
Section A
X
90-94
85-89
80-84
75-79
70-74
65-69
60-64
55-59
50-54
45-49
Section B
X
90-94
85-89
80-84
75-79
70-74
65-69
60-64
55-59
50-54
45-49
40-44
35-39
30-34
f
2
2
5
7
10
11
8
3
0
2
f
2
2
5
7
10
11
5
3
0
2
1
1
1
Activity 2. Share significant insights gained from the module.
Page 12 of 12