Utilization of assesmant data

1. UTILIZATION OF
ASSESSMENT DATA
D. MEASURES OF CENTRAL TENDENCY
Michael Dell A. Tuazon, LPT, MEDM

2. Measures of Central Tendency
In our everyday life, we are more often than not, exposed to statements involving
the concept of “average” such as:
 the average price of car,
 the average amount of rainfall in the Philippines or
 the average salary of the nurses abroad.
Somehow, informed individuals generally make decisions based on their
understanding of this important idea.

3. What is an “average”?
Values on the scale where most of the other scores are
clustered
The most typical score
Can be classified as the mean, median and mode

4. How is an “average” computed?
Can be computed or merely inspected from a set of data;
Either in their original form (called raw data) or
From that have been organized into frequency distribution (called
grouped data)

5. Measure of Central Tendency
Mean
The balancer of the distribution
If the entire distribution is likened to a “see-saw”, then the mean
serves as the “fulcrum”
NOTE:
Can be classified as an arithmetic mean, weighted mean,
harmonic mean, etc

6. Mean
Arithmetic Mean is obtained by adding the scores in a distribution and dividing it
by the number of scores, that is:
𝑀𝑒𝑎𝑛 =
𝑠𝑢𝑚 𝑜𝑓 𝑠𝑐𝑜𝑟𝑒𝑠
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑐𝑜𝑟𝑒𝑠

7. In research, the rule is for us to round-up heavily, which means that we
only have to retain the digits that are meaningful.
Example: Compute the average score in science of the 12 students who
obtained the following scores: 22, 24, 17, 21, 19, 18, 21, 30, 30, 17, 15,
and 20.
Mean

8. Example: The following data represent the net take home pay of
five rank and file employees of a certain company. Determine the
average net take home pay of the employees.
NET TAKE HOME PAY: P4,750; P4,535; P4,380; P3,895; and P9,307.
Mean

9. How do we estimate the mean of the data if the scores are not
available?
The mean of a grouped data can be computed by:
Finding mean of a frequency distribution.
Mean
Where:
f = class frequency;
X = class midpoint; and
N is the total number of cases in the frequency distribution

10. Example: Find the mean of the data as shown in the frequency distribution below.
Mean

11. Solution:
Step 1: Expand the table by constructing the columns for X (midpoints)
Step 2: Find fX, (the products of corresponding midpoints and class frequencies).
Mean

12. The following are the grades of a college student
in each subject. Compute the mean score.
Subject Score Number of
Units
Math 1.1 3
Science 1.2 3
Rizal 1.7 2
English 1.3 3
P.E 2.0 1
Weighted Mean
ത𝑋 =
σ 𝑤𝑋
σ 𝑤
=
3 1.1 +3 1.2 +2 1.7 +3 1.3 +1 2
12
=
16.2
12
=1.35 is the mean score of the college
student in each subject.
Example:

15. The score where half of the total number of scores is found below it and the other
half above it.
The middle score when all the scores have been ranked.
No common symbol for the median. (but we use Md for median)
An appropriate measure of central tendency when the variable is measured in at
least the ordinal scale.
Median (Raw Data)

16. Suppose you are tasked to put the following words in a
backdrop, which letters will you start for it to be symmetrical?
Common application of median
HAPPY
SUMMER
VACATION
Median (Raw Data)

17. 1. Arrange the data or scores in ascending order (from lowest to highest);
2. If n is odd, there will be a middle score. This middle score is the median.
If n is even, there will be two middle scores and the median is taken as the
arithmetic average of the two middle scores.
The following are the steps in finding the median based on raw data:
Median (Raw Data)
NOTE:
Because the median depends only on the number of cases, it is more preferred than the
mean whenever extreme values occur in a data set.

18. Solution: Arranging the scores in ascending order and identifying the
middle score, the median score is 21.
Example: The scores of nine students in a science test are:
22, 24, 17, 21, 19, 18, 21, 30, 30 . Find the median score.
Median (Raw Data)

19. Solution: Because the number of cases is 6 (even), there will be two middle scores. The age in
ascending order are listed below. From the arranged data, the two middle scores are 27 and 30.
Hence, the median is 28.5.
Example: The age (in years) of six teachers are listed below:
42, 23, 24, 30, 27, and 34. Find the median age of the six teachers.
Median (Raw Data)

20. When n is large, locating the position of the median by ocular inspection may not be
easy. However, based on the definition given and the parity (oddness or evenness) of
n, we can use the following formulas for locating the middle score (s).
Median (Raw Data)

21. Median (Raw Data)
Thus, if n=157 (odd)
score79th
2
1157





 
thThe median is the
th
n





 
2
1
oddisnifformula
Thus, if n=346 (even)
score1
2
n
and
2
evenisnifformula thth
n












The median is the 173
2
346





 1741
2
346







Hence, the median is the average of the 173rd and 174th scores.

22. The formula can be used in finding the median of a frequency distribution
Median (Grouped Data)
where,
LL = true lower limit or lower class boundary of the median class;
Fb = the sum of all frequencies below the median class (or the <cf directly below the
median class)
f = frequency corresponding to the median class; and
c = class size.

23. Find the median of the following frequency distribution.
Median (Grouped Data)

24. Solution: We note that = 50%(30) = 15. Looking at the <cf, 15 is between 7 and 17.
Hence the median class is 80 – 84.
With reference to the median class, we have ;
Fb = 7; f = 10; and c = 5. Therefore, the value of the median is given by
Median (Grouped Data)

26. The value or the score that occurs most frequently in a collection of
scores.
Represented by the tallest column on a histogram or the highest peak in
a frequency polygon.
Appropriate when the variable is measured in the nominal scale.
Mode (Raw Data)

27. Solution: We first arrange the scores in ascending order and take note of the frequency of
occurrence of each score.
Example: Find the modal age of the 10 Grade III pupils whose ages are listed: Age x
(in years): 10.25, 9.0, 10.25, 9.5, 9.0, 10, 9.0, 9.25, 10.75, 10.
Mode (Raw Data)

28. 1, 2, 3, 4, 5
No Mode
1, 2, 3, 4, 4
Mode = 4
Find the mode of the following data set.
1, 2, 3, 3, 4, 4
Modes = 3 and 4
2, 2, 3, 3, 4, 4
No Mode
2, 2, 2, 2, 2
Mode = 2
Mode (Raw Data)

29. For data that are summarized in a frequency distribution, an
approximate value of the mode called the “crude mode” is
defined as the midpoint of the class interval with the highest
class frequency (called the modal class), and thus is also easily
obtained by inspection.
Mode (Grouped Data)

30. where,
LL = true lower limit of the modal class;
d1 = absolute difference between the frequencies of the
modal class and the lower class interval (interval just below it);
d2 = absolute difference between the frequencies of the
modal class and the higher class interval (interval just above it);
c = the class size.
A more accurate value of the mode (exact mode) for grouped data is also obtained by linear
interpolation. The resulting formula is given by
Mode (Grouped Data)

31. Solution: The modal class of the frequency distribution shown below is 85 – 89 since it has the
largest value of f.
Example: Find the exact mode based on the same frequency distribution given;
Modal Class
Mode (Grouped Data)

32. Thus, with reference to the modal class, the
frequency of the lower class interval is 7 while the
frequency of the higher class interval is 8. The
values needed to compute the exact mode are:
LL = 84.5
d1 = |12 - 7| = 5
d2 = |12 – 8| = 4
c = 5.
Modal
Class
Mode (Grouped Data)

35. When the distribution of a set of data is
symmetric, the three measures of central
tendency have the same values
Relationship of the Three Measures of Central Tendency

36. For skewed distributions…
Relationship of the Three Measures of Central Tendency

37. Summary of Measures of Central Tendency