Assessment in Learning 2

1. LESSON 8 : ANALYSIS,
INTERPRETATION, AND
USE OF TEST DATA
Prepared by: Group 8

2. DESIRED
SIGNIFICANT
LEARNING
OUTCOME
In this lesson, you are expected to:
Analyze, interpret, and use test data
applying
 a.) measures of central tendency,
 b.) measures of variability,
 c.) measures of position, and
 d.) measures of covariability.

3. SignificantCulminating
PerformanceTask and
Success Indicators
At the end of the
lesson, you are expected to
analyze test scores using
the measures of central
tendency, variability,
position and co-
variability. Understanding
of these measures is helpful
to know how to improve
teaching and learning. Your
success in this performance
will be determined if you
have done the following.
Tasks Success Indicators
Calculate the mean, median, and
make de of test scores from the
given set of test scores.
From own and actual set of test data,
compute at least two measures of
central tendency in two ways.
Write a general statement of
interpretation from computer mean,
median, and mode of test data, and
the relationship of the three
measures and shape the skewness
of the test distribution of scores.
Give at least three interpretation
statements of your own test work
output aligned with the assessment
objectives.
Identify the level of measurement for
a given variable.
Explain in a statement or two the
level of measurement that applies to
variable measured by your own test
data.
Explain the concept of variability on
test results and calculate the different
measures of variability.
From own test data sheet, compute
the standard deviation and give a
statement or two to describe the test
results on the basis of such
measures.

4. Significant
Culminating
Performance
Task and
Success
Indicators
Tasks Success Indicators
Explain how measures of variability
relate to skewness and kurtosis of
frequency polygon of test data.
Present empirical evidence of how
measures of variation affects the
shape of frequency polygon of test
scores.
Determine the different measures of
position.
Apply the different measures of
position in interpreting test scores in
a distribution.
Convert raw scores in standard
scores.
Use standard scores to compare two
scores in a distribution in at least two
practical and real teaching-learning
situations.
Determine the measures of co-
variability between two tests.
From a given test data, compute the
occurrence of co-variability of two
test scores.

5. Prerequisite
of this Lesson
The discussion in this lesson, will build upon the
concepts and examples presented in lesson 7, which
focused on the tabular and graphical presentation and
interpretation of test results. In this lesson, other ways
of summarizing test data using descriptive statistics,
which provides a more precise means of describing a
set of scores will be discussed. The word “measures”
is commonly associated with numerical and quantitative
data. Hence, the prerequisite to understanding the
concepts contained in this lesson is your basic
knowledge of mathematics, e.g., summation of values,
simple operations on integers, squaring and finding the
square roots, and etc.

6. What are
measures of
central
tendency?
 Means the central location or point of
convergence of a set of values.
 Test scores have a tendency to converge
at a central value.
 This value is the average of the set of
scores.
 Three commonly-used measures of
central tendency of measures of central
location are the MEAN, MEDIAN, and the
MODE.

7. What are
measures of
central
tendency?
Mean
• Most preferred measure of central tendency for
use with test scores.
• Also referred to as the arithmetic mean.
• The computation is very simple.
𝑋
That is, 𝑋 = ______ where 𝑋 = the mean,
N
𝑋 = the sum of all the scores, and N = the number
of scores in the set.

8. What are
measures of
central
tendency?
- Mean
Consider again the test scores of students given in Table 8.1, which is
the same set of test scores in the previous lesson.
Table 8.1 Scores of 100 college student in a Final Examination
The mean is the sum of all the scores from 53 down to the last score,
which is 35.
That is, 𝑋
𝑋 = ______ = 53+36+57+…..+60+49+35
N
53 30 21 42 33 41 42 45 32
36 51 42 49 64 46 57 35 45
57 38 49 54 61 36 53 48 52
41 58 42 43 49 51 42 50 62
33 43 37 57 35 33 50 42 62
75 66 78 52 58 45 53 40 60
46 45 79 33 46 43 47 37 33
37 36 36 46 41 43 42 47 56
50 53 49 39 52 52 50 37 53
34 43 43 57 48 43 42 42 65

9. What are
measures of
central
tendency?
- Mean
Table 8.2 Frequency Distribution of Grouped Test Scores
Class
Interval
Midpoint
(X)
f X¡ F Cumulative
Frequency
(cf)
Cumulative
Percentage
75-80 77 3 231 100 100
70-74 72 0 0 97 97
65-69 67 2 134 97 97
60-64 62 8 496 95 95
55-59 57 8 456 87 87
50-54 52 17 884 79 79
45-49 47 18 846 62 62
40-44 42 21 882 44 44
35-39 37 13 481 23 23
30-34 32 9 288 10 10
25-29 27 0 0 1 1
20-24 22 1 22 1 1
Total (N) 100 X¡ f =4720

10. What are
measures of
central
tendency?
- Mean
In the Traditional way, it cannot be argued that you can see at a
glance how the scores are distributed among the range of values in a
condensed manner. You can even estimate the average of the scores
by looking at the frequency in each class interval. In the absence of
statistical program the mean can be computer by the following
formula:
𝑋 = 𝑋𝑖𝑓
N
Where 𝑋𝑖 = midpoint of the class interval
𝑓 = frequency of each class interval
N = total frequency
Thus, the mean on the test scores in Table 7.1 is calculated as follows:
𝑋 = 𝑋𝑖𝑓 =4720 = 47.2
N 100

11. What are
measures of
central
tendency?
- Median
Median
• The value that divides the ranked score into
halves, or the middle value of the ranked
scores.
• If the number of scores is odd, then there is
only one middle value that gives the median .
However, if the number of scores in the set is
an even number, then there are two middle
values.

12. What are
measures of
central
tendency?
- Median
Table 8.2 Frequency Distribution of Grouped Test Scores
This formula will help you determine the median:
Mdn = lower limit + size of the class interval of median class
[ (
𝑛
2
) – cumulative frequency below the median class ]
frequency of the median class
Class Interval Midpoint
(X)
f X¡ F Cumulative
Frequency(cf)
Cumulative
Percentage
75-80
70-74
65-69
60-64
55-59
50-54
77
72
67
62
57
52
3
0
2
8
8
17
231
0
134
496
456
884
100
97
97
95
87
79
100
97
97
95
87
79
45-49 47 18 846 62 62
40-44
35-39
30-34
25-29
20-24
42
37
32
27
22
21
13
9
0
1
882
481
288
0
22
44
23
10
1
1
44
23
10
1
1
Total (N) ∑ of X¡F= 4720

13. What are
measures of
central
tendency?
- Mode
Mode
• The easiest measure of central tendency to
obtain.
• It is the score or value with the highest
frequency in the set of scores.
• If the scores are arranged in a frequency
distribution, the mode is estimated as the
midpoint of the class interval which has the
highest frequency.
• This class interval with the highest frequency
is also called the modal class.

14. When are
mean,
median, and
mode
appropriately
used?
• To appreciate comparison of the three
measures of central tendency, a brief
background of level of measurement is
important.
• The level of measurement helps you decide
how to interpret the data as measures of
these attributes, and this serves as the guide
in determining in part the kind of descriptive
statistics to apply in analyzing the test data.

15. Scale of
Measurement
There are
four levels of
measurement that
apply to the
treatment of test
data: nominal,
ordinal, interval,
and ratio.
1.) NOMINAL
 The number is used for labeling or identification
purposes only.
Example: student’s identification number or
section number.
2.) ORDINAL
 It is used when the values can be ranked in some
order of characteristics.
 The numeric values used is indicate the difference
in traits under consideration.
Example: Academic awards are made on the
basis of an order of performance: first honor,
second honor, third honor, and so on.

16. Scale of
Measurement
3.) INTERVAL
 Which has the properties of both nominal and
ordinal scales.
 Attained when the values can describe the
magnitude of the differences between groups or
when the intervals between the numbers are equal.
 “Equal interval” means that the distance between
the things represented by 3 and 4 is as the same
distance represented by 4 and 5.
Example: the most common example of interval
scale is temperature readings.
(the difference between the temperatures 30° and
40° is the same as that between 90° and 100°. )

17. Scale of
Measurement
4.) RATIO
 It the highest level of measurement.
 As such, it carries the properties of the
nominal, ordinal, and interval scales.
 Its additional advantage is the presence of a
true zero point, where zero indicates the
total absence of the trait being measured.

18. How do
measures of
central
tendency
determine
skewness?
 SYMMETRICAL DISTRIBUTION
- the mean, median, and mode have the same value, and
the value of the median is between the mean and the mode.
Figure 8.1. Mean, Median and Mode in a Symmetrical Distribution

19. How do
measures of
central
tendency
determine
skewness?
 POSITIVELY-SKEWED DISTRIBUTION
- the mode stays at the peak of the curve and its value will be
smallest.
- the mean will be pulled out from the peak of the distribution
toward the direction of the few high scores.
thus, the mean gets the largest value. The median is between
the mode and the mean.
Figure 8.2 Mean, Median, and Mode in a Positively- Skewed
Distribution

20. How do
measures of
central
tendency
determine
skewness?
 NEGATIVELY-SKEWED DISTRIBUTION
the mode remains at the peak of the curve, but it will have the
largest value.
- the mean will have the smallest values as influenced by the
extremely low scores, and the median still lies between the
mode and the mean.
Figure 8.3. Mean, Median, and Mode in a Negatively-Skewed
Distribution

21. What are
measures of
dispersion?
- Which indicates “variability”, “ spread”, or “scatter”.
- Measures of variability give us the estimate too determine how
the scores are compressed, which contributes to the “flatness”
or “peakedness” of the distribution.
- There are several indices of variability, and the most commonly
used in the area of assessment are the RANGE, VARIANCE,
AND STANDARD
Figure 8.4. Measures of Variability of Sets of Test Scores

22. Indices of
Variability
 RANGE
- It is the difference between the highest scores and the lowest scores in
a distribution.
The simplest measures of variability but also considered as least
accurate measure of dispersion.
Determine the range for the following scores: 9, 9, 9, 12, 12, 13, 15,
15, 17, 17, 18, 18, 20, 20, 20.
Range = Highest Score (HS) – Lowest Score (LS)
= 20 – 9
= 11
Now, replace a high score in one of the scores, say, the last score and
make it 50. The range becomes:
Range = HS – LS
= 50 – 9
= 41

23. Indices of
Variability
 VARIANCE AND STANDARD DEVIATION
- The most widely used measure of variability and considered as the
most accurate to represent the deviations of individual scores from the
mean values in the distribution.
Examine the following test score distributions:
ΣX = 120 ΣX = 120 ΣX = 120
x
̅ = 120 x
̅ = 120 x
̅ = 120
10 10 10
= 12 = 12 = 12
Class A Class B Class C
22
18
16
14
12
11
9
7
6
5
16
15
15
14
12
11
11
9
9
8
12
12
12
12
12
12
12
12
12
12

24. VARIANCE
AND
STANDARD
DEVIATION
You will note that while the distribution contain different scores, they have
the same mean. If we ask how each mean represents the score in their
respective distribution, there will be no doubt with the mean of distribution C
because each score in the distribution is 12. How about distributions A and B?
For these two distributions, the mean of 12 is a better estimate of the scores
in distribution B than in distribution A. We can see that no score in B is more
than 4 points away from the mean of 12. However, in distribution A, half of the
12 scores is 4 points or more away from the mean. We can also say that there
is less variability of scores in B than A. However , we cannot just determine
which distribution is dispersed or not by merely looking at the numbers
especially when there are many scores. We need a reliable index of variability,
such as variance or standard deviation, that takes into consideration all the
scores.
Recall that Σ (X - x
̅ ) is the sum of the deviation scores from the mean,
which is equal to zero. As such, we square each deviation score, then sum up
all the squared deviation scores, and divide it by the number of cases. This
yields the variance. Getting its square root is the standard deviation.

25. VARIANCE
AND
STANDARD
DEVIATION
The measure is generally defined by the formula:
σ² = Σ ( X - )²
N
Where σ² = population variance
μ = population mean
X = score in the distribution
finding the square gives us this formula for the standard deviation.
That is,
σ =
Σ ( X – μ )²
N
Where = population standard deviation
μ = population mean
X = score in distribution

26. VARIANCE
AND
STANDARD
DEVIATION
If we are dealing with the sample data and wish to calculate an estimate
of s, the following formula is used for such statistic:
s =
Σ ( 𝐗 – x̅ )2
/
𝐍
𝐍
Where s = standard deviation
X = raw score
x
̅ = mean score
N = number of scores in the distribution
The value of 276 and 74 are the sum of the squared deviations of scores
in Class A and Class B, respectively. If these are divided by number of scores
in each class, this gives the variance (S2):
𝑆𝐴
2
= 276 = 30.67 𝑆𝐵
2
= 74 = 8.22
10 – 1 10 – 1

27. VARIANCE
AND
STANDARD
DEVIATION
The value above are both in squared units, while our original units of scores are
not in squared units. When we find their square roots, we obtain values that are on
the same scale of units as the original set of scores. These too give the respective
standard deviation (S) of each class and computed as follows:
SA = 𝑆𝐴
2 SB = 𝑆𝐵
2
= 30.67 = 8.22
= 5. 538 = 2. 867
Using the scores in Class A and Class B in the above dataset, we can apply the
formula:
Class A Class B
X ( X - x
̅ ) ( X - x
̅ )2
22 22-12 100
18 18-12 36
16 16-12 16
14 14-12 4
12 12-12 0
11 11-12 1
9 9-12 9
7 7-12 25
6 6-12 36
5 5-12 49
X ( X - x
̅ ) ( X - x
̅ )2
16 22-12 16
15 15-12 9
15 15-12 9
14 14-12 4
12 12-12 0
11 11-12 1
11 11-12 1
9 9-12 9
9 9-12 9
8 8-12 16
x
̅ = 12 Σ (X - x
̅ )2 = 276 x
̅ = 12 Σ (X - x
̅ )2 = 74

28. VARIANCE
AND
STANDARD
DEVIATION
In addition, since the standard deviation is a measure of
dispersion, it means that a large standard deviation indicates greater
score variability than lower standard deviation. If the standard
deviation is small, the scores are closely clustered around the mean,
or the graph of the distribution is compressed even if it is symmetrical
or skewed.
Figure 8.5. Homogenous Test Score Distributions in Different Skewness

29. What are the
measures of
position?
 While measures of central tendency and measures of dispersion
are used often in assessment, there are other methods of
describing data distributions such as using measures of position
or location.
 What are these measures? QUARTILE, DECILE, and
PERCENTILE.
QUARTILE
 The quartiles are the three values that divide a set scores into
four equal parts, with one-fourth of the data values in each part.
Quartiles are also used as a measure of the spread of data
in the interquartile range (IQR), which is simply the difference
between the third and first quartiles (Q3 – Q1). Half of this gives
the semi-interquartile range or quartile deviation (Q). The
following example illustrates the above mentioned measures.

30. QUARTILE
Example: Given the following scores, find the 1st quartile, 3rd
quartile, and quartile deviation.
90, 85, 85, 86, 100, IOS, 109.110.88, 105, 100, 112
Steps:
1. Arrange the scores in the decreasing order. Steps:
2. From the bottom, find the points below which 25% of the
score value and 75 % of the score values fall.
3. Find the average of the two scores in each of these
points to determine QI and Q3, respectively.
4. Find Q using the formula:
Q = Q3 – Q1
2

31. QUARTILE
Applying these steps in the above example, we have:
Note that in the above example, the upper and lower 50% contains
even center values, so the median in each half is the average of the
two center values.
Consequently, applying the formula:
𝑄3 – 𝑄1
2
gives the quartile deviation.
That is,
Q = 107 – 87 = 10
2

32. DECILE
 It divides the distribution into 10 equal parts. There
are 9 deciles such that 10% of the distribution are
equal or less than decile 1,(D1), 20% of the scores
are equal or less than decile 2 (D2); and so on. A
student whose mark is between the first and
second deciles is in decile 2, and one whose mark
is above the ninth decile belongs to decile 10. If
there are small numbers of data values, decile is
not appropriate to use.

33. PERCENTILE
 It divides the distribution into one hundred equal
parts. In the same manner, for percentiles, there are
99 percentiles such that 1% of the scores are less
than the first percentile, 2% of the scores are less
than the second percentile, and so on. For example,
if you scored 95 in a 100-item test, and your
percentile rank is 99th, then this means that 99% of
those who took the test performed lower than you.
This also means that you belong to the top 1% of
those who took the test. In many cases, percentiles
are wrongly interpreted as percentage score.

34. PERCENTILE
 For example, 75% as a percentage score means you
get 75 items correct out of a hundred items, which is
a mark of grade reflecting performance level. But
percentile is a measure of position such that 75th
percentile as your mark means that 75% of the
students who took the test got lower score than you,
or your score is located at the upper 25% of the class
who took the same test. For very large data set,
percentile is appropriate to use for accuracy. This is
one reason why percentiles are commonly used in
national assessments or university entrance
examinations with large dataset or scores in
thousands.

35. What is
coefficient of
variation as a
measure of
relative
dispersion?
 We need a measure of relative dispersion which
dimensionless or “unit free”.
 This measure of relative dispersion is also known as
coefficient of variation.
 This is simply the ratio of the standard deviation of a
distribution and the mean of the distribution.
cv= σ (100)
μ
• The above formula indicates that coefficient of
variation is a percentage value.

36. What is
coefficient of
variation as a
measure of
relative
dispersion?
• Thus, if the mean score of the students in mathematics is 40 with a
standard deviation of 10, then the coefficient of variation is
computed as:
cv= 10 (100)
40
= 0.25
= 25%
• Suppose the mean score of students in science is 18 with standard
deviation of 5. The coefficient of variation is:
cv= 5 (100)
18
= 0.227
= 28%
• From the computed coefficient of variations as measure of relative
dispersion, we can clearly see that the scores in mathematics are
more homogenous than the scores in science.

37. • Example question: Two versions of a test are given to students. One
test has pre-set answers and a second test has randomized answers.
Find the coefficient of variation.
Step 1: Divide the standard deviation by the mean for the first sample:
11.2 / 50.1 = 0.22355
Step 2: Multiply Step 1 by 100:
0.22355 * 100 = 22.355%
Step 3: Divide the standard deviation by the mean for the second
sample:
12.9 / 45.8 = 0.28166
Step 4: Multiply Step 3 by 100:
0.28166 * 100 = 28.266%
How to Find
aCoefficient
ofVariation?
Regular Test Randomized Answers
Mean 50.1 45.8
SD 11.2 12.9

38. • The standard deviation is viewed as the
most useful measure of variability because
in many distributions of scores, not only in
assessment but in research as well, it
approximates the percentage of scores that
lie within one, two or three standard
deviations from the mean.
How is
standard
deviation
applied in a
normal
distribution?

39.  Special kind of symmetrical distribution that
is most frequently used to compare scores.
 Are drawn as a smooth curve, one curve
stands out, which is the bell-shaped curve.
 It is also called Gaussian distribution,
named after Carl Friedrich Gauss.
 The standard deviation is used to
determine the percentage of scores that fall
within a certain number of standard
deviations from the mean.
The Normal
Distribution

40. Figure 8.6. The Normal Curve
 In assessment, the area in the curve refers to the number of scores
that fall within a specific standard deviation from the mean score, in
other words, each portion under the curve contains a fixed
percentage of cases as follows:
- 68% of the scores fall between one standard deviations below
and above the mean
- 95% of the scores fall between two standard deviations below
and above the mean
- 99.77% of the scores fall between three standard deviations
below and above the mean.
The Normal
Distribution

41. • The following figure further illustrates the theoretical model:
Figure 8.7. The Areas under the Normal Curve
From the above figure, we can state the properties of the normal
distribution:
1. The mean, median, and mode are all equal.
2. The curve is symmetrical. As such, the value in a specific area on the left
is equal to the value of its corresponding area on the right.
3. The curve changes from concave to convex and approaches the X-axis,
but the tails do not touch the horizontal axis.
4. The total area on the curve is equal to 1.
The Normal
Distribution

42. WHAT ARE
STANDARD
SCORES?
 According to study.com, Standard Score is a set of scores that
have the same mean and standard deviation so they can be
compared.
 The most useful is the z-score, which is often used to express a
raw score in relation to the mean and standard deviation.
 This relationship is expressed in the following formula
Z=
𝑋− 𝑋
S
 Recall that X-X is a deviation score.
 With this difference, we are able to know whether your test
score, say, X is above or below average score.
 The standard deviation helps you locate the relative position of
the score in a distribution.
 A z-score is called a Standard Score, simply because it is a
deviation score expressed in standard deviation units.

43. WHAT ARE
STANDARD
SCORES?
 If raw score are expressed as z-scores, we can see their relative
position in their respective distribution. Moreover, if the raw scores
are already converted into standard scores, we can now compare
the two scores even when these scores comes from different
distributions or when scores are measuring two different things.
Figure 8.8. A Comparison of Score Distributions with Different Means
and Standard Deviation

44. WHAT ARE
STANDARD
SCORES?
In the figure, a score of 86 in English indicates better performance
than a score of 90 in Physics. Let us suppose that the standard
deviations in English and Physics are 3 and 2, respectively.
ZE= 86 – 80 ZP= 90 – 95
3 2
= 2 = -2.5
 From the above, if 86 and 90 are your scores in two subjects, you
can in English say that, compared with the rest of your class, you
performed better in English than in Physics. That is because in
English, your performance is 2 standard deviations above the
mean. While 90 is numerically higher, this score is more than half
standard deviation below the average performance of the class
where you belong, while 86 is above the mean and even 2
standard deviations above it.

45. WHAT ARE
STANDARD
SCORES?
Figure 8.9. Different Raw Scores in one Z-score Distribution
 Note that in Figure 8.8, the shaded area in two graphs indicate the
proportion of scores below yours. This is the same as saying that
this proportion is the number of students in your class who scored
lower than you.
 Examining the area under the normal curve, we can say that about
98% in your class scored below you in English, while the 1.2%
scored below you in Physics.
 We assume here that the scores are normally distributed.

46. T-Score
 As you can see in the computation of the z-score, it can
give you a negative number, which simply means the
score is below the mean. However, communicating
negative z-score as below as the mean may not be
understandable to others.
 One option is to convert a z-score into a T-score, which
is a transformed standard score.
 There is scaling in which a mean of 0 in a z-score is
transformed into a mean of 50, and the standard
deviation in z-score is multiplied by 10. The
corresponding equation is:
T-score = 50+10z

47. T-Score
For example, a z-score of -2 is equivalent to a T-score of
30. That is:
T-score = 50+10(-2)
= 50-20
= 30
Looking back at the English score of 86, which
resulted in a z-score of 2 as shown in figure 8.7, T-score
equivalent is ;
T-score = 50+10(2)
=50+20
= 70

48. STANINE
SCORES
 Another standard score is a stanine, shortened from
standard nine.
 With nine in its name, the scores are on a nine-point
scale.
 In a z-score distribution, the mean is 0, and the standard
deviation is 1.
 In this scale, the mean is 5 and the standard deviation is
2.
 Each stanine is one-half standard deviation-wide.
 Like the T-Score, stanine score can be calculated from the
z-score by multiplying the z-score by 2 and adding 5.
Stanine=2z + 5

49. STANINE
SCORES
 Going back to our example on a score of 86 in English that is
equivalent to a z-score of 2, its stanine equivalent is
Stanine=2(2) + 5 = 9
Example:
Scores in stanine scale have some limitations. Since they are in
9-point scale and expressed as a whole number they are not
precise.
Z-Score T-Score Stanine
2.1 71 9
2.0 70 9
1.9 69 9

50. STANINE
SCORES
On the assumptions that stanine scores are normally
distributed, the percentage of cases in each band or
range of scores in the scale as follows:
Stanine Score Percentage of Scores
1 Lowest 4%
2 Next Slow 7%
3 Next Slow 12%
4 Next Slow 17%
5 Middle 20%
6 Next High 17%
7 Next High 12%
8 Next High 7%
9 Highest 4%

51. STANINE
SCORES
With the above percentage distribution of scores in each stanine,
you can directly convert a set of raw scores into stanine scores.
Simply arrange the raw scores from the highest to lowest, and with
the percentage of scores in each stanine, you can directly assign
the appropriate stanine score in each raw score.
Figure 8.10. The Normal Distribution and the Standard Scores

52. What are
Measure of
Covariability?
• Measures of covariability tell us to a certain extent the
relationship between two tests or two factors. A score one gets
may not only be due to a single factor but with other factors
directly or indirectly observable, which are also related to one
another. This section will be limited to introducing two scores that
are hypothesized to be related to one another.
• Correlation between two variables - is finding the degree of
relationship between two scores.
• The statistical measure is the correlation coefficient, an index that
ranges from -1.0 to +1.0. the value -1.0 indicates a negative
perfect correlation, 0.00 no correlation at all, and 1.00 a perfect
positive correlation.
• Correlation coefficients did not result in exact values of 0.00, 1.0
and -1.0; instead, the correlation values are either closer to 1.0 or
-1.0.

53. What are
Measure of
Covariability?
• The various types of relationships are illustrated in the
following scatter-plot diagrams:
Figure 8.11. Various Types of Relationships

54. What are
Measure of
Covariability?
• Some examples of interpreting Correlation Coefficients are as
follows:
• To further analyze the relationship between two scores, a
statistical test to determine the correlation coefficient of two
variables is needed. In actual practice, we deal with test scores,
so the dataset is at interval and ratio level measurements. Hence,
the commonly used statistical analysis is the Pearson Product-
Moment correlation
Coefficient Direction Strength
r = -0.90 negative strong
r = 85 positive strong
r = 68 positive moderate
r = -30 negative weak
r = 0.05 positive very weak/negligible

55. • Differentversionsof
Pearson Product-
MomentCorrelation
1.A
Deviation-
score
Formula
𝑟 =
(x − 𝑋 )(𝑌 − 𝑌)
(𝑋 − 𝑋)2(𝑌 − 𝑌)2
𝑁 =
𝑥𝑦
𝑥2 𝑦2
• where r = correlation coefficient
X = raw score in variable X
𝑋 = mean score in the first variable
Y = raw score in the second variable
𝑌 = mean score in the second variable
• Note that in 𝑥𝑦, x and y are derivation score from 𝑋 and 𝑌,
respectively.

56. • Different versionsof
Pearson Product-
MomentCorrelation
2. Raw
Score
Formula
𝑟 =
𝑁 (𝑋𝑌 − 𝑋 𝑌
𝑁 𝑋2 − ( 𝑋)2 𝑁 𝑌2 − ( 𝑌)2
• The raw score formula is very easy because a simple
scientific calculator can provide the values for
𝑋, 𝑋2 , 𝑌 , 𝑌2.This formula has been introduced
earlier in lesson 6, when you were thought on how to
compute the reliability coefficient of scores. This is the
same Pearson r, but this time, it is used to establish
relationship between two sets of data.

57. • Computation
for
Correlation
for Raw
Score Data
Table 8.3. Computation for Correlation for Raw Score Data
X Reading Y Problem
Solving
x2 y2 XY
4 6 16 36 24
9 11 81 121 99
4 5 16 25 20
11 10 121 100 110
12 8 144 64 96
5 5 25 25 25
7 10 49 100 70
5 8 25 64 40
6 9 36 81 54
2 5 4 25 10
X = 65 Y = 77 x2 = 517 y2 = 461 XY = 548

58. • Computation
for
Correlation
for Raw
Score Data
So that:
Correlation coefficient (r) = 𝑁 (𝑋𝑌− 𝑋 𝑌
𝑁 𝑋2−( 𝑋)2 𝑁 𝑌2−( 𝑌)2
=
10 548 −(65)(77)
10 517 − 65 2 [10 641 −(77)2]
=
5480−5005
(5170−4225)(6410−5929)
=
475
(945)(481)
=
475
454545
=
475
674.19
r= 0.7045
 The above mathematical processes gave a correlation coefficient of
0.705 between performance scores in Reading and Problem
Solving. This coefficient indicates a moderate correlation between
performance in reading and problem solving.

59. • Precautions
to be
observed
with regard
to the
computed r
as correlation
coefficient:
- It should not be interpreted as percent.
o Thus, 0.705 should not be interpreted as 70%. If we want to
extend the meaning of 0.705, then compute for r2, which
becomes the coefficient of determination. This coefficient
explains the percentage of the variance in one variable that is
associated with the other variable. With the reference to the two
variables indicated in table 8.3, and the computed r of 0.70
(rounded off to the nearest hundredths), it results to an r2 of
0.49, which can be taken as 49%. This can be interpreted as
49% of the variance in Problem-Solving test scores is associated
with the Reading scores. Thus, r2 helps explain the variance
observed in the problems score. The total variance is equal to 1,
in percent, 100%. If 49% of the variance observed in problem-
solving scores was attributable to reading scores, then the other
51% of the variance in problem-solving test scores is due to
other factors. This concept is concretized in figure 8.12.

60. • Precautions
to be
observed
with regard
to the
computed r
as correlation
coefficient:
- While the correlation coefficient shows the relationship
between two variables, it should not be interpreted as
causation.
o Considering our example, we could not say that the scores in
reading test causes 49% of the variance of the problem-solving
test scores. Relationship is different from causation.
Figure 8.12. Covariation of Performance in Reading and Problem-Solving

61. APPLY:
1. Refer to this figure below as the frequency polygons
representing entrance test scores of three groups of
students in different field of specialization.

62. APPLY:
a. What is the mean score of Education students?
b. What is the mean score of engineering students?
c. What is the mean score of Business students?
d. Which group of students had the most dispersed scores
in the test? Why do you say so?
e. What distribution is skewed? Why do you say so?

63. APPLY:
2. Determine the type of distribution depicted by
the following measures:
a. 𝑥 = 80.45 𝑀𝑑𝑛 = 80.78 𝑀𝑜𝑑𝑒 = 80.25
b. 𝑥 = 120 𝑀𝑑𝑛 = 130 𝑀𝑜𝑑𝑒 = 150
c. 𝑥 = 89.78 𝑀𝑑𝑛 = 82.16 𝑀𝑜𝑑𝑒 = 82.10
Note:
The higher the mode from mean – negative
The higher the mean from mode – positive

64. APPLY:
3. The following is a frequency distribution of
scores of 10 persons.
a.What is the mean of distribution?
b.What is the median?
c. What is the mode?
d.What is the range?
e.What is the standard deviation?
X f
30 1
28 2
20 3
17 0
15 1
13 2
10 1

65. APPLY:
4. The following is the frequency distribution of year-end
examination marks in a certain secondary school.
Class Interval f Midpoint F*m cf
60-65 2 63 126 100
55-59 5 57 285 98
50-54 6 52 312 93
45-49 8 47 376 87
40-44 11 42 462 79
35-39 10 37 370 68
30-34 11 32 352 58
25-29 20 27 540 47
20-24 17 22 374 27
15-19 6 17 102 10
10-14 4 12 48 4
100 3347

66. APPLY:
a.Compute the mean, median and mode of the
frequency distribution.
b. Find :
1. Third quartile or the 75th percentile (𝑃75)
2. First quartile or the 25th percentile
3. Semi – interquartile range

67. APPLY:
5. A common exit examination is given to 400 students in a
university. The scores are normally distributed and the mean
is 78 with a standard deviation of 6. Daniel had a score of 72
and Jane score 84. What are the corresponding z-score of
Daniel and Jane? How many students would be expected to
score between the score of Daniel and Jane? Explain your
answer.
6. James obtained a score of 40 in his Mathematics test and
34 in his Reading test. The class mean score in Mathematics
is 45 with a standard deviation of 4 while in Reading; the
mean score is 50 with a standard deviation of 7. On which test
did James do better compared of the rest of the class?
Explain your work.

68. APPLY:
Following are sets of score on two variables: X for reading
comprehension and Y for Reasoning skills administered to sample of
students.
X: 11 9 15 7 5 9 8 4 8 11
Y: 13 8 14 9 8 7 7 5 10 12
X Y X2 Y2 XY
11 13 121 169 143
9 8 81 64 72
15 14 225 196 210
7 9 49 81 63
5 8 25 64 40
9 7 81 49 63
8 7 64 49 56
4 5 16 25 20
8 10 64 100 80
11 12 121 144 132
= 87 = 93 = 847 = 941 XY = 879

69. APPLY:
a. Compute the Pearson Product-Moment Correlation for
the above data.
Pearson r?
𝒓 =
𝑵 𝒙𝒚 − 𝒙 𝒚
𝑵 𝑿𝟐 − ( 𝑿)𝟐 [𝑵 𝒀𝟐 − ( 𝒀)𝟐]
b. Describe the direction and strength of the relationship
between readings.
c. The Coefficient Determination. Interpret the results.

70. EVALUATION:
In each item, choose the letter which you think can
best represent the answer to the given problem
situation. Give a statement to justify your choice.
1. Which distribution is negatively skewed in the figure shown?
a. Distribution X
b. Distribution Y
c. Distribution Z
d. Distribution W

71. EVALUATION:
2. What is the preferred measure of central tendency in a test
distribution where there are small number of scores that
are extremely high or low?
a. Median
b. Mean
c. Variance
d. Mode
3. The following scores were obtained from a short spelling
test:
10, 8, 11, 13, 12, 13, 8, 16, 11, 8, 7, 9.
What is the modal score?
a. 8
b. 10
c. 11
d. 13

72. EVALUATION:
4. What does it mean when a student got a score at the 70th
percentile on a test?
a. The performance of the student is above average
b. The student answered 70 percent of the items correctly
c. The student got at the least 70 percent of the correct
answers
d. The student’s score is equal or above 70 percent of the
other students in the class.
5. Which best describes a normal distribution?
a. Positively skewed
b. Negatively skewed
c. Symmetric
d. Bimodal

73. EVALUATION:
6. What does a large standard deviation indicate?
a. Scores are not normally distributed
b. Scores are not widely spread, and the median is unreliable
measure of central tendency.
c. Scores are widely distributed where the mean may not be a
reliable measure of central tendency.
d. Scores are not widely distributed, and the mean is
recommended as more reliable measure of central
tendency
7. In a normal distribution, approximately what percentage of
scores are expected to fall within three-standard deviation
from the mean?
a. 34%
b. 68%
c. 95%
d. 99%

74. EVALUATION:
8. Which of the following is interpreted as the percentage of
scores in a reference group that falls below a particular
raw score?
a. Standard scores
b. Percentile rank
c. Reference group
d. T-score
9. For the data illustrated in the scatter plot below, what is the
reasonable product-moment correlation coefficient?
a. 1.0
b. -1.0
c. 0.90
d. -0.85

75. EVALUATION:
10.A Pearson test statistic yields a correlation coefficient
(r) of 0.90. if X represents scores on vocabulary and
Y, the reading comprehension test scores, which of
the following best explains r=0.90?
a. The degree of association between X and Y is 81%.
b. The strength of relationship between vocabulary and
reading comprehension is 90%
c. There is almost perfect positive relationship between
vocabulary test scores and reading comprehension.
d. 81% of the variance observed in Y can be attributed to
the variance observed in X.

76. LESSON 8 :
ANALYSIS,
INTERPRETATIO
N,ANDUSEOF
TEST DATA