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Module Assessment 4
Bachelor of secondary education (University of Eastern Philippines)
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4
Analysis and Interpretation
of Assessment Results
Overview
Statistics is an important tool in the analysis and interpretation of assessment results.
This is necessary in describing and interpreting the performance of the learners in the
assessment procedures. Teachers should have statistical knowledge in order to use the proper
methods to collect data, employ the correct analyses, and effectively present the results. This will
also guide them in making decisions that will benefit the learners.
In this module, some important tools in analyzing and interpreting assessment results are
discussed. Some of these tools are measure of central tendency, measures of variability and
measures of relative position. Different ways of presenting data are also included.
Learning Outcomes
After learning this module, you should be able to:


interpret assessment results accurately and utilize them to help learners improve their
performance and achievement; and
utilize assessment results to make informed-decisions to improve instruction.
Lesson 1. Presentation
The reader’s sustained interest should be the primary concern in presenting the gathered
data. The presentation may be done in different manners like textual presentation, graphical
presentation, and tabular presentation.
Textual Presentation


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
The data are presented in paragraph form.
This kind of representation is useful when we are looking to supplement
qualitative statements with some data.
For this purpose, the data should not be voluminously represented in tables or diagrams.
It just has to be a statement that serves as a fitting evidence to our qualitative evidence
and helps the reader to get an idea of the scale of a phenomenon.
If the data under consideration is large then the text matter increases substantially. As a
result, the reading process becomes more intensive, time-consuming and cumbersome.
Example:
There are about 540, 000 Filipinos who joined the ranks of job seekers. According to
government data, the number of jobless Filipinos last July reached 4.35 million, an increase
of more than half-a-million Filipinos from the same period last year. The unemployment rate
last July was 12.7% compared to 11.2% of July last year.
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Tabular Presentation

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
The data are presented in tables to show relation between the column and row quantities.
Tables are useful to highlight precise numerical values; proportions or trends are better
illustrated with charts or graphics.
Tables summarize large amounts of related data clearly and allow comparison to be
made among groups of variables.
Generally, well-constructed tables should be self-explanatory with four main parts: title,
columns, rows and footnotes.
A table facilitates representation of even large amounts of data in an attractive, easy to read
and organized manner. The data is organized in rows and columns.
This is one of the most widely used forms of presentation of data since data tables are easy
to construct and read
The advantages of tabular presentation includes: ease of representation; ease of
analysis; helps in comparison, and economical.
Frequency distribution is the most common tabular presentation.
Frequency Distribution
Frequency distribution is a tabular arrangement of data into appropriate categories
showing the number of observations in each category or group. Using frequency distribution
encompasses the size of the table and it makes the data more interpretive.
Steps in Constructing Frequency Table
1. Compute the value of the range (R). Range is the difference between the highest score and the
lowest score.
2. Determine the class size (c.i.). The class size is the width of each class interval. It is the
quotient when you divide the range by the desired number of classes or categories. If the
desired number of classes is not identified, find the value of k, where k = 1 + 3.3 log n.
3. Set up the class limits of each category. Class limit is the groupings or categories defined by
the lower and upper limits. Lower class limit represents the smallest number in each group
while upper class limit represents the highest number in each group. Use the lowest score as
the lower limit of the first class.
4. Set up the class boundaries if needed. This can be computed by getting the difference of the
lower limit of the second class and the upper limit of the first class divided by 2.
5. Tally the scores in the appropriate classes.
6. Find the other parts if necessary such as class marks, class boundaries among others. Class
marks are the midpoint of the lower and the upper class limits. Class boundaries are the
numbers used to separate each category in the frequency distribution but without gaps created
by the class limits. Add 0.5 to the upper limit to get the upper class boundary and subtract 0.5
to the lower limit to get the lower class boundary in each group or category.
Example: Raw scores of 40 students in a 50-item mathematics quiz is given, construct a
frequency distribution table following the steps given.
17
27
44
50
22
25
35
22
47
33
30
45
46
34
44
33
48
26
26
38
25
20
36
37
46
45
38
29
25
41
23
39
15
33
37
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18
21
49
32
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1. Find the range.
R = H.S. – L.S.
R = 50 – 15
R = 35
2. Solve the value of k.
k = 1 + 3.3 log n
k = 1 + 3.3 log 40
k = 1 + 3.3 (1.602)
k = 1 + 5.29
k = 6. 29 or 6
3. Find the class size
4. Construct the class limit starting with the lowest score as the lower limit of the first
category. The last category should contain the highest score in the distribution. Each category
should contain 6 as the size of the width (x). In some books, the last category contains the lowest
score. Count the number of scores that falls in each category (f). Find the class boundaries and
class marks of the given score distribution.
X
Tally
Frequency (f)
Class Boundaries
15 -20
21 - 26
27 – 32
33 – 38
39 – 44
45 - 50
////
/////-////
///
/////-/////
////
/////-/////
4
9
3
10
4
10
n = 40
14.5 - 20.5
20.5 – 26.5
26.5 – 32.5
32.5 – 38.5
38.5 – 44.5
44.5 – 50.5
Class Marks
(XM)
17.5
23.5
29.5
35.5
41.5
47.5
Graphical Presentation
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Graphics are particularly good for demonstrating a trend in the data that would not be
apparent in tables.
It provides visual emphasis and avoids lengthy text description.
The data are presented in visual form.
It is a picture that displays numerical information
However, presenting numerical data in the form of graphs will lose details of its precise
values which tables are able to provide.
The scores expressed in frequency distribution can be meaningful and easier to interpret
when they are graphed.
There are methods of graphing frequency distribution: bar graph or histogram and
frequency polygon and smooth curve.
A. Histogram
It consists of a set of rectangles having bases on the horizontal axis which centers at the
class marks. The base widths correspond to the class size and the height of the
rectangles corresponds to the class frequencies. Histogram is best used for graphical
representation of discrete data or non-continuous.
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B. Frequency Polygon
It is constructed by plotting the class marks against the class frequencies. The x-axis
corresponds to the class marks and the y-axis corresponds to the class frequencies.
Connect the points consecutively using a straight line. Frequency polygon is best used
in representing continuous data such as scores of students in a given test.
Example: Construct a histogram and a frequency polygon using the frequency distribution of 40
students in a 50-item mathematics quiz.
X
15 -20
21 - 26
27 – 32
33 – 38
39 – 44
45 - 50
Tally
////
/////-////
///
/////-/////
////
/////-/////
Frequency (f)
4
9
3
10
4
10
n = 40
Class Boundaries
14.5 - 20.5
20.5 – 26.5
26.5 – 32.5
32.5 – 38.5
38.5 – 44.5
44.5 – 50.5
Class Marks (XM)
17.5
23.5
29.5
35.5
41.5
47.5
Frequency
Polygon
Histogram
12
12
10
10
88
66
44
22
00
17.5
23.5
29.529.535.5
41.547.547.5
17.5 23.5
35.5 41.5
Assessment Task 4.1
1. Consider the given data below. Construct a frequency distribution then draw
frequency polygon and histogram using the frequency distribution.
37
16
24
24
33
47
39
28
18
34
35
22
19
43
15
23
26
33
35
29
46
34
38
24
20
18
33
34
40
39
21
15
16
32
35
20
25
the
25
30
25
Lesson 2. Quantitative Analysis and Interpretation
Levels of Measurement
The data can be classified into two types. These are the continuous and discontinuous or
discrete data. Continuous data are measures like feet, pounds, kilos, minutes, and meters.
These kind of data can be made into measurement of varying degree of precision, for example, 1
yard equals 3 feet, or 1 foot equals 12 inches. Discontinuous or discrete data are
measurement expressed in whole units. Counting of people, number of objects, number of
houses, number of students, and so on. According to Stevens, these are four levels of
measurement:
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
A. Nominal. In nominal measurement, the numerical values just “name” the attribute
uniquely. No ordering of the cases is implied. For example, jersey numbers in
basketball are measures at the nominal level. A player with number 30 is not more of
anything than a player with number 15, and is certainly not twice whatever
number 15 is.
B. Ordinal. In ordinal measurement, the attributes can be rank-ordered. Here, distances
between attributes do not have any meaning. For example, on a survey you might
code Educational Attainment as 0 = less than high school; 1 = some high school; 2 =
high school degree; 3 = some college; 4 = college degree; 5 = post college. In this
measure, higher numbers mean more education. The interval between values is not
interpretable in an ordinal measure.
C. Interval. In interval measurement, the distance between attributes does have
meaning. For example, when we measure temperature, the distance from 30-40 is
same as distance from 70-80. The interval between values is interpretable. Because
of this, it makes sense to compute an average of an interval variable, where it doesn’t
make sense to do so for ordinal scales. But note that in interval measurement ratios
don’t make any sense - 80 degrees is not twice as hot as 40 degrees.
D. Ratio. In ratio measurement there is always an absolute zero that is meaningful. This
means that you can construct a meaningful fraction (or ratio) with a ratio variable.
Weight is a ratio variable. In applied social research most “count” variables are ratio,
for example, the number of clients in past six months.
It’s important to recognize that there is a hierarchy
implied in the level of measurement idea. At lower
levels of measurement, assumptions tend to be
less restrictive and data analyses tend to be less
sensitive. At each level up the hierarchy, the
current level includes all of the qualities of the one
below it and adds something new. In general, it is
desirable to have a higher level of measurement
rather than a lower one.
Measures of Central Tendency
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Measure of central tendency provides a very convenient way of describing a set of scores
with a single number that describes the performance of the group.
It is also defined as a singles value that is used to describe the “center” of the data.
It is thought of as a typical value in a given distribution.
Mean, median, and mode are the three measures of central tendency.
A. Mean
Mean is the most commonly used and the most important measure of the center data and
it is also referred as the arithmetic average. Mean is used if sampling stability is desired and if
other measures are to be computed such as standard deviation, coefficient of variation and
skewness. The properties of the mean includes:
a. It measures stability. Mean is the most stable among other measures of central
tendency because every score contributes to the value of the mean.
b. The sum of each score’s distance from the mean is zero.
c. It is easily affected by the extreme scores.
d. It may not be an actual score in the distribution.
e. it can be applied to interval level of measurement.
f. It is very easy to compute.
The mean of an ungrouped set of data is equal to the sum of the quantities, divided by
the number of quantities under consideration.
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Example: Compute the mean of the following scores of 15 students in math quiz consisting of 25
items: 25, 20, 18, 18, 17, 15, 15, 15, 14, 14, 13, 12, 12, 10, and 10.
Solution:
Analysis: The average performance of 15 students who participated in a math quiz
consisting of 25 items is 15.2. The implication of this is that student who got scores
below 15.2 did not perform well in the said exam. Students who got scores higher
than 15.2 performed well compared to the performance of the whole class.
The mean for grouped data can be computed in two ways: using short method and
midpoint method. In the example we will use the midpoint method. Grouped data are the data in a
frequency distribution. The midpoint formula for solving the mean is:
, where = mean value
F = frequency in each class or category
Xm = midpoint of each class or category
= summation of the product of fXm
Steps in Solving Mean for Grouped Data
1. Find the midpoint or class mark (Xm) of each class or category using the formula .
2. Multiply the frequency and the corresponding class mark fX m.
3. Find the sum of the results in step 2.
4. Solve the mean using the formula .
Example: Scores of 40 students in a science class consist of 60 items and they are tabulated
below.
X
f
Xm
fXm
10 - 14
5
12
60
15 – 19
2
17
34
20 – 24
3
22
66
25 – 29
5
27
135
30 – 34
2
32
64
35 – 39
9
37
333
40 – 44
6
42
252
45 – 49
3
47
141
50 - 54
5
52
260
n = 40
= 1 345
Solution:
Analysis: The mean performance of 40 students in science quiz is 33.63. Those students
who got scores below 33.63 did not perform well in the said examination while those
students who got scores above 33.63 performed well.
B. Median
Median is what divides the scores in the distribution into two equal parts. Fifty percent lies
below the median and 50% lies above the median value. It is also known as the middle score or
the 50th percentile. Median is used when the exact midpoint of the score distribution is desired
and if there are extreme scores in the distribution. The properties of the median includes:
a. It may not be an actual observation in the data set.
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b. It can be applied in ordinal level.
c. It is not affected by extreme values because median is a positional measure.
To find the median of ungrouped data, arrange the scores (from lowest to highest or
highest to lowest) and determine the middle most score in a distribution if n is an odd number or
get the average of the two middle most scores if n is an even number.
Example: a. Find the median score of 7 students in PE1 class with the following scores: 2, 10, 5,
19, 16, 17, 15.
Solution:
Arrange the scores from highest to lowest 19, 17, 16, 15, 10, 5, 2.
Analysis: The median score is 15. Fifty percent or three of the scores are above 15 (19,
17, 16) and 50% of the scores are below 15 (10, 5, 2).
b. Find the median score of 8 students in an English class with the following scores: 30,
2, 19, 5, 10, 17, 15, and 16.
Solution:
Arrange the scores from highest to lowest 30, 19, 17, 16, 15, 10, 5, 2. Get the
mean of 17 and 16 by adding the two numbers and dividing the sum by 2. So, .
Analysis: The median score is 15.5 which means that 50% of the scores in the
distribution are lower than 15.5, those are 15, 10, 5, 2; and 50% are greater than
15.5, those are 30, 19, 17, 16 which means that 4 scores are below 15.5 and 4
scores are above 15.5.
To find the median for grouped data, the formula that can be used is:
, where:
= median value
MC = median class is a category containing the
LB = lower boundary of the median class (MC)
cfp = cumulative frequency before the median class if the
scores are arranged form lowest to highest value
fm = frequency of the median class
c.i. = size of the class interval
Steps in Solving Median for Grouped Data
1. Complete the table for cf˂.
2. Get of the scores in the distribution so that you can identify MC.
3. Determine LB, cfp, fm, and c.i.
4. Solve the median using the formula.
Example: Scores of 40 students in a science class consist of 60 items and they are tabulated
below. The highest score is 54 and the lowest score is 10. Solve for the median of the
given scores in the frequency distribution table.
X
10 - 14
f
5
cf˂
5
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15 – 19
20 – 24
25 – 29
30 – 34
35 – 39
40 – 44
45 – 49
50 - 54
2
3
5
2
9 (fm)
6
3
5
n = 40
7
10
15
17 (cfp)
26
32
35
40
Solution:
The category containing is 35 -39.
MC = 35-39
LL of the MC = 35
LB = 34.5
cfp = 17
fm = 9
c.i. = 5
Analysis: The median value is 36.17, which means that 50% or 20 scores are less than
36.17.
C. Mode
The mode or the modal score is a score or scores that occurred most in the distribution. It
is classified as unimodal, bimodal, trimodal, and multimodal. Unimodal distribution if it contains
only one mode. Bimodal distribution if it contains two modes. Trimodal distribution if it contains
three modes or Multimodal distribution if it contains more than two modes. Mode can be used
when the typical value is desired and when the data set is measured on a nominal scale. The
properties of the mode are:
a. It can be used when the data are qualitative as well as quantitative.
b. It may not be unique.
c. It is not affected by extreme values.
d. It may not exist.
Example: a. Find the mode of the following ungrouped data; 10, 9, 9, 7, 6, 5, 3, 2, 1.
Solution: The mode of the data is 9 since 9 occurs twice. The distribution has only 1
mode so it is a unimodal distribution.
b. Find the mode of the following ungrouped data: 18, 18, 17, 17, 17, 16, 16, 16, 15, 15.
Solution: The mode of the data is 17 and 16 since 17 and 16 occurs thrice. The
distribution is bimodal since it has two modes.
In solving the mode for grouped data, use the formula:
,
where: = mode value
= lower boundary of the modal class
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d 1 = difference between the frequency of the modal class and the
frequency above it, when the scores are arranged from lowest to
highest
d 2 = difference between the frequency of the modal class and the
frequency below it, when the scores are arranged from lowest to
highest
c.i. = size of the class interval
Example: Scores of 40 students in a science class consists of 60 items and they are tabulated
below.
X
f
10 - 14
5
15 – 19
2
20 – 24
3
25 – 29
5
30 – 34
2
35 – 39
9
40 – 44
6
45 – 49
3
50 - 54
5
n = 40
Solution:
Modal class = 35-39
LL of MC = 35
LB = 34.5
d1 = 9 – 2 = 7
d2 = 9 – 6 = 3
c.i. = 5
Analysis: The mode of the score distribution that consists of 40 students is 38, because
38 occurred several times.
Measures of Variability
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Measure of variability is a single value that is used to describe the spread of the scores in
a distribution or the spread of values about the mean.
The term variation is also known as variability, dispersion, or spread.
Intuitively, a smaller dispersion of scores arising from the comparison often indicates
more consistency and more reliability.
Range, mean deviation, and standard deviation are the most common measures of
variability.
A. Range
Range is simply the difference of the highest (H) and the lowest (L) scores in a set of data
under consideration. It is the simplest and the crudest measure of variation. The properties of
range includes:
a. It is quick and easy to understand.
b. It is a rough estimation of variations.
c. It is easily affected by the extreme scores.
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To solve for the range of ungrouped data, simply subtract the highest score by the lowest
score. The formula is: R = H.S. – L.S.
Example: Find the range of the two groups of score distribution.
Group A
10
12
15
17
25
26
28
30
35
Group B
15
16
16
17
17
23
25
26
30
Solution:
Range of Group A
R = H.S. – L.S.
R = 35 – 10
R = 25
Range of Group B
R = H.S. – L.S.
R = 30 – 15
R = 15
Analysis: The range of Group A = 25 which is greater than the range of Group B = 15.
The implication of this is that the scores in group A are more spread out than the
scores in group B or the scores in group B are less scattered than the scores in
group A.
To solve for the range for grouped data, the formula is R = HSUB – LSLB, where R = range
value, HSUB = upper boundary of the highest score, and LS LB = lower boundary of the lowest
score.
Example: Find the value of the range of the scores of 50 students in Mathematics achievement
test.
X
25 – 32
33 – 40
41 – 48
49 – 56
57 – 64
65 – 72
73 – 80
81 – 88
89 - 97
f
3
7
5
4
12
6
8
3
2
n = 50
Solution:
LL of the LS = 25
LSLB = 25 - .05 = 24.5
UL of the HS = 97
HSUB = 97 + .05 = 97.5
R = HSUB – LSLB
R = 97.5 – 24.5
R = 73
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B. Mean Deviation
Mean deviation measures the average deviation of the values from the arithmetic mean.
It gives equal weight to the deviation of every score in the distribution. The formula for solving the
mean deviation for ungrouped data is
where, MD = mean deviation value
x = individual score
= sample mean
n = number of cases
Steps in Solving Mean Deviation for Ungrouped Data
1. Solve the mean value.
2. Subtract the mean value from each score.
3. Take the absolute value of the difference in step 2.
4. Solve the mean deviation using the formula.
Example: Find the mean deviation of the scores of 10 students in a Mathematics test. Given the
scores: 35, 30, 26, 24, 20, 18, 18, 16, 15, and 10.
Solution:
x
35
13.8
13.8
30
8.8
8.8
26
4.8
4.8
24
2.8
2.8
20
-1.2
1.2
18
-3.2
3.2
18
-3.2
3.2
16
-5.2
5.2
15
-6.2
6.2
10
-11.2
11.2
= 60.4
Mean:
Mean Deviation:
Analysis: The mean deviation of the 10 scores of students is 6.04. This means that on
the average, the value deviated from the mean of 21.2 is 6.04.
To solve for the mean deviation for grouped data, use the formula: , where, MD = mean
deviation value
f = class frequency
Xm = class mark or midpoint of each category
= mean value
n = number of cases
Steps in Solving Mean Deviation for Grouped Data
1. Solve for the value of the mean.
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2. Subtract the mean value from each midpoint or class mark.
3. Take the absolute value of each difference.
4. Multiply the absolute value and the corresponding class frequency.
5. Find the sum of the results in step 4.
6. Solve for the mean deviation using the formula for grouped data.
Example: Find the mean deviation of the given scores below.
X
f
Xm
fXm
Xm / Xm - /
10-14
5
12
60
-21.63
21.63
15-19
2
17
34
-16.63
16.63
20-24
3
22
66
-11.63
11.63
25-29
5
27
135
-6.63
6.63
30-34
2
32
64
-1.63
1.63
35-39
9
37
333
3.37
3.37
40-44
6
42
252
8.37
8.37
45-49
3
47
141
13.37
13.37
50-54
5
52
260
18.37
18.37
n = 40
1 345
Solution:
Mean:
f/ Xm - /
108.15
33.26
34.89
33.15
3.26
30.33
50.22
40.11
91.85
425.22
Mean Deviation:
Analysis: The mean deviation of the 40 scores of students is 10.63. This means that on
the average, the value deviated from the mean of 33.63 is 10.63.
C. Variance and Standard Deviation
 Variance is one of the most important measures of variation. It shows variation about the
mean.
 Standard Deviation is the square root of the variance. It is the average distance of all
the scores that deviates from the mean value.
 The formula for the population and sample variance and the population and sample
standard deviation for ungrouped data are given below:
Population Variance (Ungrouped Data)
Sample Variance (Ungrouped Data)
Population Standard Deviation
Sample Standard Deviation
Steps in Solving Variance and Standard Deviation of Ungrouped Data
1. Solve for the mean value.
2. Subtract the mean value from each score.
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3. Square the difference between the mean and each score.
4. Find the sum of the results in step 3.
5. Solve for the population variance or sample variance using the formula of ungrouped data.
6. NOTE: If the variance is already solved, take the square root of the variance to get the value of
the standard deviation.
Example: Using the data below, find the variance and standard deviation of the scores of 10
students in a science quiz.
x
19
17
16
16
15
14
14
13
12
10
Ʃx = 146
(
19.36
5.76
1.96
1.96
0.16
0.36
0.36
2.56
6.76
21.16
Ʃ(= 60.40
4.4
2.4
1.4
1.4
0.4
-0.6
-0.6
-1.6
-2.6
-4.6
Solution:
a. Population Variance
=
b. Population Standard Deviation
=
c. Sample Variance
=
d. Sample Standard Deviation
=
Steps in Solving the Variance and Standard Deviation of Grouped Data
1. Solve for the mean value.
2. Subtract the mean value from each midpoint or class mark.
3. Square the difference between the mean value and midpoint or class mark.
4. Multiply the squared difference and the corresponding class frequency.
5. Find the sum of step 4.
6. Solve the population variance or sample variance using the formula of grouped data.
7. NOTE: If the variance is already solved, take the square root of the variance to get the value of
the standard deviation.
Population Variance (Grouped Data)
Sample Variance (Grouped Data)
Population Standard Deviation
Sample Standard Deviation
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Example: Score distribution of the test results of 40 students in a Physical Education class
consisting of 50 items. Solve the variance and standard deviation for grouped data.
X
15-20
21-26
27-32
33-38
39-44
45-50
f
3
6
5
15
8
3
n=40
Xm
17.5
23.5
29.5
35.5
41.5
47.5
fXm
52.5
141
147.5
532.5
332
142.5
Ʃ fXm = 1 348
Xm -16.2
-10.2
-4.2
1.8
7.8
13.8
33.7
33.7
33.7
33.7
33.7
33.7
(Xm - 2
262.44
104.04
17.64
3.24
60.84
190.44
f(Xm - 2
787.32
624.24
88.2
48.6
486.72
571.32
Ʃf(Xm - 2=2 606.4
Solution:
a. Population Variance
=
b. Population Standard Deviation
=
c. Sample Variance
=
d. Sample Standard Deviation
=
Interpretation of Standard Deviation
1. If the value of standard deviation is large, on the average, the scores in the distribution will be
far from the mean. Therefore, the scores are spread out around the mean value. The
distribution is also known as heterogeneous.
2. If the value of standard deviation is small, on the average, the scores in the distribution will be
close to the mean. Hence, the scores are less dispersed or the scores in the distribution are
homogeneous.
Measures of Relative Position
The measures of position are used to locate the relative position of a specific data value
in relation to the rest of the data. The most popular measures of position are standard scores or
z-scores, percentiles, deciles, and quartiles,
A. Standard Score or z-scores




A z-score (also called a standard score) gives an idea of how far from the mean a data
point is.
Technically it’s a measure of how many standard deviations below or above
the population mean a raw score is.
It can be placed on a normal distribution curve.
Z-scores range from -3 standard deviations (which would fall to the far left of the normal
distribution curve) up to +3 standard deviations (which would fall to the far right of the
normal distribution curve).
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

In order to use a z-score, you need to know the mean μ and also the population standard
deviation σ. Z-scores are a way to compare results to a “normal” population.
It is defined as .
For population:
___ ________
_
_
For sample:
___
_ _______ ___
_ ___ ____
_
___ __ _______ _
___
_____
Example: A student scored 65 in calculus test that had a mean of 50 and a standard deviation of
10. She scored 30 in a history test with a mean of 25 and a standard deviation of 5.
Compare her relative positions in the two tests.
Solution:
First find the z-scores:
For calculus
=
For history
=
Analysis: Since the z-scores for calculus is larger, her relative position in calculus is
higher than that of history.
B. Percentiles








Percentiles are the values of the variable that divide a set of observations into 100 equal
parts. Each set of observations has 99 percentiles and are denoted by P 1, P2,…, P99.
A percentile is a value in the data set.
A percentile rank of a given value is a percent that indicates the percentage of data is
smaller than the value.
The kth percentile, Pk is a value such that k% of the observations are smaller than or
equal to Pk and (100-k)% of the observations are larger than Pk. For example, if a value is
located at the 80th percentile, it means that 80% of the values that fall below the value
and 20% of the values fall above it.
C. Deciles
Deciles are the values of the variable that divide a set of observations into 10 equal parts.
Each set of observations has 9 deciles which are denoted by D 1, D2,…,D9.
The first decile D1 is a value in the data set that 10% of the values fall below D 1 and 90%
of the values fall above D1.
Similarly, the second decile D2 is a value in the data set that 20% of the values fall below
D2 and 80% of the values fall above D2 and so on.
Note: D1 = P10, D2 = P20, D3 = P30,… D9 = P90.
D. Quartiles


Quartiles are the values of the variable that divide a set of observations into 4 equal
parts. Each set of observations has 3 quartiles and they are denoted by Q 1, Q2, and Q3.
The first quartile Q1 is a value in the data set that 25% of the values fall below Q 1 and
75% of the values fall above Q1.
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



The second quartile Q2 is a value in the data set that 50% of the values fall below Q 2 and
50% of the values fall above Q2.
The third quartile Q3 is a value in the data set that 75% of the values fall below Q 3 and
25% of the values fall above Q3.
Note: Q1 = P25, Q2 = P50, Q3 = P75.
The 50th percentile, 5th decile and second quartile of a distribution are equal to the same
value and are referred to as the median. That is, median = Q2 = D5 = P50.
Calculations of the Percentiles, Deciles, and Quartiles
Ungrouped Data Formulas
Quartiles:
Deciles:
Percentiles:
where:
D = decile value
k = 1, 2, 3,…, 9
n = no. of observation
where:
P = percentile value
k = 1, 2, 3,…, 99
n = no. of observation
Example: The following are the test scores of 12 students in a statistic class: 70, 77, 65, 56, 99,
62, 79, 73, 85, 87, 92, and 82.
A. Find the value of 80th percentile. Give a brief interpretation of it.
B. Find Q1.
C. Find D5.
Solution:
A. First, arrange the scores in ascending order.
56, 62, 65, 70, 73, 77, 79, 82, 85, 87, 92, 99
The value of 9.6th term can be approximated by the average of 9 th and 10th
terms in the ranked/arranged data. Therefore,
Analysis: Thus approximately 80% of the scores are less than 86 and 20% are greater
than 86 in the given data.
B. . Since the 3rd term in the ranked data is 65, therefore 65.
C. . Since the 6th term in the ranked data is 77, So
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Grouped Data Formulas
Quartiles:
, where: LB = lower boundary of the median class (MC)
K = values form 1 to 3
n = number of observation
cfp = cumulative frequency before the median class if the
scores are arranged form lowest to highest value
fq = frequency of the quartile class
c.i. = size of the class interval
Deciles:
, where: LB = lower boundary of the median class (MC)
K = values form 1 to 9
n = number of observation
cfp = cumulative frequency before the median class if the
scores are arranged form lowest to highest value
fd = frequency of the decile class
c.i. = size of the class interval
Percentiles:
Example: The data for the scores of fifty students in Science class are given below. For the
following distribution find:
A. the first quartile
B. the 5th decile
C. 82nd percentile
X
25-32
33-40
41-48
49-56
57-64
65-72
73-80
81-88
89-96
f
3
7
5
4
12
6
8
3
2
n=50
X
f
Solution:
cf˂
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25-32
33-40
41-48
49-56
57-64
65-72
73-80
81-88
89-96
3
7
5
4
12
6
8
3
2
n=50
3
10
15
19
31
37
45
48
50
A.
The category containing is 41-48
LL = 41
LB = 40.5
cfp = 10
fq = 5
c.i. = 8
Analysis: Therefore, 25% of the scores of 50 students who participated in the test are
less than 44.50.
B.
The category containing
LL = 57
LB = 56.5
cfp = 19
fd = 12
c.i. = 8
is 57-64
Analysis: Therefore, 50% of the scores of 50 students are less than 60.50.
C.
The category containing is 73-80
LL = 73
LB = 72.5
cfp = 37
fp = 8
c.i. = 8
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Analysis: Therefore, 82% of the scores of 50 students are less than 76.5.
Assessment Task 4.2
1. The data below are the scores of 50 students in a 50-item test in Chemistry. Using the
data, compute the following and make an analysis.
a. Mean
f. Variance
b. Median
g. Standard Deviation
c. Mode
h. Q2
d. Range
i. D8
e. Mean Deviation
j. P75
X
15-19
20-24
25-29
30-34
35-39
40-44
45-49
f
1
6
11
10
12
5
5
n = 50
Feedback
How are you doing so far? Were you able to follow the different computations for the
quantitative analysis? These computations are necessary for data interpretation. This will guide
teachers in making decisions regarding the performance of their students. If you are having a
hard time on some lessons, you can always go back. Read the examples given and practice
solving.
Summary
To aid you in reviewing the concepts in this module, here are the highlights:
 The data presentation may be done in different manners
like textual presentation,
graphical presentation, and tabular presentation. In textual presentation, the data are
presented in paragraph form. For tabular presentation, the data are presented in tables
to show relation between the column and row quantities. While for graphical
presentation, the data are presented in visual form like histogram and frequency
polygon.
 Frequency distribution is a tabular arrangement of data into appropriate categories
showing the number of observations in each category or group.
 Nominal scales are used as measures of identity. Ordinal scale is used in
measurement like ranking of individuals or objects. Interval scales are numbers that
reflect differences among items. Ratio scale is the highest type of scale.
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 Measure of central tendency provides a very convenient way of describing a set of
scores with a single number that describes the performance of the group. The most
commonly used measures of central tendency are mean, median, and mode. Mean is
also referred as the arithmetic average, median is a score that divides the distribution
into two equal parts, while mode is the score that occurred most in the distribution.
 Measure of variability is a single value that is used to describe the spread of the scores
in a distribution or the spread of values about the mean. Range, mean deviation,
variance, and standard deviation are the most common measures of variability.
 The measures of position are used to locate the relative position of a specific data
value in relation to the rest of the data. The most popular measures of position are
standard scores or z-scores, percentiles, deciles, and quartiles.
Suggested Readings
If you want to learn more about the topics in this module, you may log on to the following
links:
https://conjointly.com/kb/levels-of-measurement/
https://www.toppr.com/guides/economics/presentation-of-data/textual-and-tabular-presentationof-data/
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4453119/
https://www.statisticshowto.com/probability-and-statistics/z-score/
http://www.tihe.org/courses/it133/IT%20133%20Lectures/IT133%20-%20Lecture%2004.pdf
References
Broto, A. S. (2006). Statistics Made Simple. 2nd ed. University of Eastern Philippines, Northern
Samar, Philippines.
Gabuyo, Y.A. (2012) Assessment of Learning I. Rex Book Store, Inc., Manila, Philippines.
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