The document discusses the analysis, interpretation, and application of test data, focusing on measures of central tendency (mean, median, and mode) and measures of variability (range, variance, and standard deviation). It includes statistical concepts like skewness, quartiles, deciles, and percentiles, which help summarize and interpret data distributions. Through the explanation of examples and formulas, the document elaborates on how these measures provide insights into the central location and spread of data.

1. Analysis,
Interpretation,
and Use of Test
Data
Lesson 8

2. Desired Significant Learning Outcomes
Analyze, interpret, and use test data applying:
a. Measures of central tendency
b. Measures of variability
c. Measures of position
d. Measures of covariability

3. Measures of
Central
Tendency
-means the central location or
point of convergence of a set
of values. The value is the
average of the set of scores.
MCT gives a single value that
represents a given set of
scores.

4. 3 Measures of Central Tendency or Measures
of Central Location
1.Mean
2.Median
3.Mode

5. MEAN
 Arithmetic Mean
FORMULA:
The sum of
all the scores
The number of
scores in the
set
Mean

6. Scores of 100 College Students in a Final
Examination
53 30 21 42 33 41 42 45 32 58
36 51 42 49 64 46 57 35 45 51
57 38 49 54 61 36 53 48 52 49
41 58 42 43 49 51 42 50 62 60
33 43 37 57 35 33 50 42 62 49
75 66 78 52 58 45 53 40 60 33
46 45 79 33 46 43 47 37 33 64
37 36 36 46 41 43 42 47 56 62
50 53 49 39 52 52 50 37 53 40
34 43 43 57 48 43 42 42 65 35

7. The MEAN is the sum of all the scores from 53
down to the last score, which is 35
FORMULA:
= 53 +36+57+………… = 35
100

8. Frequency Distribution of Grouped Test Scores
Class
Interval
Midpoint
(X)
f X1f Cumulative
Frequency
(cf)
Cumulative
Percentage
75-80
70-74
65-69
60-64
55-59
50-54
45-49
40-44
35-39
30-34
25-29
20-24
77
72
67
62
57
52
47
42
37
32
27
22
3
0
2
8
8
17
18
21
13
9
0
1
231
0
134
496
456
884
846
882
481
288
0
22
100
97
97
95
87
79
62
44
23
10
1
1
100
97
97
95
87
79
62
44
23
10
1
1
TOTAL N 100 ΣX1f=4720

9. FORMULA:
= 4720 = 47.2
100
1 f

10. MEDIAN
Is the value that divides the ranked score into
halves, or the middle value of the ranked
scores

11. Frequency Distribution of Grouped Test Scores
Class
Interval
Midpoint
(X)
f X1f Cumulative
Frequency
(cf)
Cumulative
Percentage
75-80
70-74
65-69
60-64
55-59
50-54
45-49
40-44
35-39
30-34
25-29
20-24
77
72
67
62
57
52
47
42
37
32
27
22
3
0
2
8
8
17
18
21
13
9
0
1
231
0
134
496
456
884
846
882
481
288
0
22
100
97
97
95
87
79
62
44
23
10
1
1
100
97
97
95
87
79
62
44
23
10
1
1
TOTAL N 100 ΣX1f=4720

12. Formula
Mdn= Lower limit of + size of the (n/2)- cumulative frequency
below the medial class
median class class interval
frequency of the median class

13. Applying the formula:
1. You need a column for cumulative frequency.
2. Determine n/2, which is one-half of number of scores of
examinees.
3. Find the class interval of the 50th score. In this case where there
are 100 scores, the 50th score is in the class interval of 45-49.
this class interval of 45-49 becomes the median of the class.
4. Find the exact limits of the median class. In this case, class 44.5-
49.5. The lower limit then is 44.5

14. Frequency Distribution of Grouped Test Scores
Class
Interval
Midpoint
(X)
f X1f Cumulative
Frequency
(cf)
Cumulative
Percentage
75-80
70-74
65-69
60-64
55-59
50-54
45-49
40-44
35-39
30-34
25-29
20-24
77
72
67
62
57
52
47
42
37
32
27
22
3
0
2
8
8
17
18
21
13
9
0
1
231
0
134
496
456
884
846
882
481
288
0
22
100
97
97
95
87
79
62
44
23
10
1
1
100
97
97
95
87
79
62
44
23
10
1
1
TOTAL N 100 ΣX1f=4720

15. Substituting:
5(100/2-44)
Median= 44.5 + 18
= 44.5 + 5(6)
18
= 46.17

16. MODE
- Is 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.
- In a graphical representation of the frequency distribution,
the mode is the value in the horizontal axis at which the
curve is at its highest point. When all the scores in a

17. Activity: Find the Mean,Mdn and Mo
Class
Interval
Midpoint
(X)
f X1f Cumulative
Frequency
(cf)
Cumulative
Percentage
80-86
73-79
66-72
59-65
52-58
45-51
38-44
31-37
24-30
17-23
10-16
3-9
8
2
9
7
5
5
18
19
9
8
5
5
TOTAL N 100 ΣX1f=

18. Create the table of frequency distribution of the
scores in Mathematics of the 100 BSE
Students.Calculate the Mean Md, Mo
53 61 66 80 36 42 36 77 52 85
59 69 65 81 39 82 35 68 54 86
62 59 54 77 32 88 48 74 50 35
70 54 52 76 34 77 41 75 60 37
75 52 51 74 33 76 42 64 66 34
62 51 59 59 42 73 52 61 53 34
89 33 56 68 41 71 53 69 42 84
44 58 48 67 36 70 64 68 37 86
45 57 48 62 33 80 65 63 38 87
61 54 44 47 51 69 63 57 40 83

19. How do measures of central tendency
determine skewness?

20. Symmetrical Distribution
• Inthis distribution , the Mean, Median and Mode have the
same value. The value of the Median is between the
mean and the mode.

21. Positively Skewed
• When the distribution becomes positively skewed,
there are variations in their values. The mode stays
at the peak of the curve and its value will be the
smallest. The mean will be pulled out from the peak
of the distribtion toward the direction of the few high
scores..The mean gets the largest value. The
Median is between the mode and the Mean.

22. Negatively Skewed
• The Mode remains at the peak of the
curve, but it will have the largest value.
The mean will have the smallest value as
influenced by the extremely low scores,
and the Median still lies between the
Mode and the Mean

23. MEASURES OF DISPERSION

24. What do you notice?
• Different distributions are symmetrical,
• may have the same values or average
• Scores in A. range between 0-400
• Scores in B range between 125- 275
Scores in C 150-250

25. Measures of variability
• gives us the estimate to determine
how the scores are compresses,
which contributes to the “flatness” or
“peakedness”of the distribution

26. • Indices of Variability
a. Range
b. Variance and Standard deviation

27. a. RANGE
-it is the difference between the highest (XH ) and the lowest
Score (X L) in a distribution.
-it is the simplest measure of variability but also considered
as the least accurate measure of dispersion because its
value is determined by just two scores in a group. It does
not take into consideration the spread of all scores.
-Its value depend on the highest and lowest scores.
-value could be drastically changed by a single value.

28. b. Variance and Standard Deviation
• the most widely used measure of variability and is
considered as the most accurate to represent the
deviations of individual scores from the mean values in
the distribution.

29. CLASS A CLASS B CLASS C
22 16 12
18 15 12
16 15 12
14 14 12
12 12 12
11 11 12
9 11 12
7 9 12
6 9 12
5 8 12

30. Mean
A. ∑ X = 120 B. ∑ X = 120 C. ∑ X = 120
X = 120/10 X=120/10 X=120/10
X= 12
= 12 =12 = 12

31. CLASS A CLASS B CLASS C
22 16 12
18 15 12
16 15 12
14 14 12
12 12 12
11 11 12
9 11 12
7 9 12
6 9 12
5 8 12

32. Recall that ∑(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 rool is the Standard
deviation.

33. population
variance
no. of
scores in the
distribution
population
mean
Unbiased
estimate

34. CLASS A CLASS B
X (X- ) (X- )2 X (X- ) (X- )2
22 22-12 100 16 16-12 16
18 18-12 36 15 15-12 9
16 16-12 16 15 15-12 9
14 14-12 4 14 14-12 4
12 12-12 0 12 12-12 0
11 11-12 1 11 11-12 1
9 9-12 9 11 11-12 1
7 7-12 25 9 9-12 9
6 6-12 36 9 9-12 9
5 5-12 49 8 8-12 16
=12 (X- )2 = 276 = 12 (X- )2 =74

35. Computation
SA
2 = 276 = 30.67
10-1
SB
2 = 74 = 8.22
10-1

36. RAW SCORE FORMULA

37. Note:
• Standard deviation is a measure of
dispersion, it means that a large SD
indicates greater score variability and
if SD is small, the scores are closely
clustered around the mean.

38. MEASURES OF POSITION.
A. Quartile
b. Decile
c. Percentile

39. QUARTILE
• Quartlies are the three values that divide a set of scores
into four equal parts, with one-fourth of the data values in
each part. This means about 25% of the data falls at or
below the first quartile (Q1); 50% of the data falls at or
below the 2nd quartile (Q2), and 75% falls at or below the
3rd Quartile (Q3). Q2 is also the median. Q1 is the
median of the first half of the values, and Q3 the median
of the 2nd Half of the values. Thus, the upper quartile
represents on average the mark of the top half of the
class, while the lower quartile represents that of the
bottom half of the class.

40. QUARTILES
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
abovementioned measures.

41. Example: Given the following scores, find the 1st
Quartile, 3rd Quartile, Quartile Deviation.
90, 85, 85, 86, 100, 105, 109, 110, 88, 105, 100, 112
Steps:
1. Arrange the scores in the decreasing order
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 Q1 and Q3 , respectively.
4. Find Q using the formula. Q= Q3-Q1
2

42. Applying these steps in the above example,
we have:
112
110
109 Q3 = 109+105 =107
105 2
105
100
100
90
88 Q1 = 88+ 86 =87
86 2
85
85

43. • Consequently, applying the formula:
Q3- Q1 gives the quartile deviation
2
Q = 107-87 = 10
2

44. 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, (D), 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 number of data values, decile
is not appropriate to use.

45. 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.

46. PERCENTILE
• 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. For example, 75% as a
percentage scores means you get 75 items correct out of
100 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. Percentiles are commonly used in
national assessments or university entrance