The document defines various measures of central tendency including mean, median, and mode for both ungrouped and grouped data. It also defines key terms related to frequency distributions such as lower class limit, upper class limit, class boundaries, class marks, class width, and cumulative frequency. An example is provided to illustrate the construction of a grouped frequency distribution table involving 7 classes with a class width of 7 using data on exam scores of 40 students.

1. STATISTICS
MEASURES OF CENTRAL
TENDENCY

2. DEFINITIONS OF TERMS
• MEAN (Ungrouped Data)
 is the average of all the elements of a set of data and is
denoted by
𝑋 =
𝑥1 + 𝑥2 + 𝑥3 + 𝑥4 … 𝑥𝑛−1 + 𝑥𝑛
𝑁
=
𝑥
𝑁
Whereas,
𝑋= Mean
𝑥= Sum of all values in a data
𝑁=Total number of elements in a given data

3. MEDIAN (UNGROUPED)
 Is usually denoted by 𝑥. By definition, median
is the value at the middle when all the
elements in a given set of data are
arranged in ascending order.
NOTE: If n is odd number, select the middle
data value
𝑥=
𝑛+1
2
If n is even number, find the mean of the
two middle values.

4. MEDIAN OF A GROUPED DATA
𝑥 = 𝐶𝑙𝑏 + 𝑖
𝑁
2
−< 𝑐𝑓𝑏
𝑓𝑐
𝑥=Median
𝐶𝑙𝑏= Lower class boundary
𝑓𝑐= Median Class Frequency
𝑐𝑓𝑏= Cumulative frequency before the median class
MEDIAN CLASS
𝑀 =
𝑁
2
𝑖= Class interval
n= Total number of observation

5. MODE (UNGROUPED)
 defined as the element in a set of data that has the
greatest number of frequencies. It is donated by 𝑋.
The value that occurs most often in a data
set.
𝑋= Most frequent
Different types of mode:
Unimodal
Bimodal
Multimodal
No mode

6. MODE (GROUPED)
 𝑥 = 𝐶𝐵𝑙 +
∆𝑓𝑏
∆𝑓𝑏+∆𝑓𝑎
𝑖
Whereas:
𝑥 = 𝑀𝑜𝑑𝑒
𝐶𝐵𝑙 = 𝐿𝑜𝑤𝑒𝑟 𝑐𝑙𝑎𝑠𝑠 𝑏𝑜𝑢𝑛𝑑𝑎𝑟𝑦
∆𝑓𝑏 = 𝑀𝑜𝑑𝑎𝑙 𝑐𝑙𝑎𝑠𝑠 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑚𝑖𝑛𝑢𝑠 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑏𝑒𝑓𝑜𝑟𝑒
∆𝑓𝑎 = 𝑀𝑜𝑑𝑎𝑙 𝑐𝑙𝑎𝑠𝑠 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑚𝑖𝑛𝑢𝑠 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑎𝑓𝑡𝑒𝑟
𝑖 = 𝐶𝑙𝑎𝑠𝑠 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙

7. DEFINITIONS OF TERMS
• FREQUENCY DISTRIBUTION
 A frequency distribution is a way of presenting and
organizing the data connected in tabular form using
classes and frequencies.
The most convenient method of organizing the data
is to construct a frequency distribution in order to
describe situations, draw conclusions, or make
inferences about events, the researcher must organize
the data in some meaningful way.

8. GROUPED FREQUENCY DISTRIBUTION
Lower class limit
 the smallest data value that can be included in
the class.
Upper class limit
 the largest data value that can be included in
the class.
Class boundaries
 are used to separate classes so that there are
no gaps in the frequency distribution.
Class marks
 the midpoint of the classes (average)
𝑙𝑜𝑤𝑒𝑟 𝑙𝑖𝑚𝑖𝑡 + 𝑢𝑝𝑝𝑒𝑟 𝑙𝑖𝑚𝑖𝑡

9. GROUPED FREQUENCY DISTRIBUTION
Class width
 the difference between two consecutive lower class
limits
Cumulative frequency
 classes are increasing order is the sum of
the frequencies for that class and all
previous class.

10. BASIC TYPES OF FREQUENCY
DISTRIBUTION
UNGROUPED FREQUENCY
DISTRIBUTION
Classifies a given data
set (usually n≤30) under
a specific category or class.
Frequencies of each data is
treated as individual data
points or a discrete data
GROUPED FREQUENCY
DISTRIBUTION
Having an interval or a
ratio –level data, and
beyond the sample size
of 30.
Frequencies of each data
points are clustered in a
specific class interval.

11. EXAMPLES
UNGROUPED FREQUENCY
DISTRIBUTION
AGE TALLY FREQUENCY
19
18
17
16
15
14
GROUPED FREQUENCY
DISTRIBUTION
CLASS TALLY FREQUENCY
95-99
90-94
85-89
80-84
75-79
70-74

12. Example1
Scores of 15 students in Mathematics I quiz consist of 25
items. The highest score is 25 and the lowest score is 10.
Here are the scores: 25, 20, 18, 18,17, 15, 15, 15, 14,
14, 13, 12, 12, 10, 10. Find the mean in the following
scores. X (Scores)
25 14
20 14
18 13
18 12
17 12
15 10
15 10
15
𝑋 =
𝑥
𝑁
𝑋 =
228
15
𝑋 =15.2

13. UNGROUPED FREQUENCY DISTRIBUTION
• Twenty-five positive cases
of COVID-19 were given a
blood test to determine
their blood type. The data
set is as follows:
•A B B AB O
•O O B AB B
•B B O A O
•A O O O AB
•AB A O B A
CLASS TALLY FREQUENC
Y
A
B
O
AB
5
7
9
4
Total (N) =
25

14. STEPS IN CONSTRUCTING FDT
1.Arrange the data in ascending order.
2.Determine the range.
3.Decide the class size or the number of classes.
4.Divide the range by the class size to identify
the class interval.
5.Construct a frequency distribution table.
6. Tally and count the observations under each
classes.

15. EXAMPLE #2 GROUPED FREQUENCY
DISTRIBUTION
Step.1
Arrange the score from lowest to
highest.
The following are the scores obtained by 40
students of BSIT in 100 item Mathematics quiz.
40 41 42 44 45 46 47 48
49 50 51 52 53 54 55 56
57 58 59 60 61 61 62 63
64 64 65 65 65 66 66 70
75 75 76 76 82 84 92 98

16. EXAMPLE #2 GROUPED FREQUENCY
DISTRIBUTION
Step.2
Determine the range.
The following are the scores obtained by 40
students of Grade 10 in 100 item MMW quiz.
Range= Highest score-
Lowest score
Range=98-40
R= 58
40 41 42 44 45 46 47 48
49 50 51 52 53 54 55 56
57 58 59 60 61 61 62 63
64 64 65 65 65 66 66 70
75 75 76 76 82 84 92 98

17. EXAMPLE #2 GROUPED FREQUENCY
DISTRIBUTION
The following are the scores obtained by 40
students of Grade 10 in 100 item MMW quiz.
Sturge rule:
Step.3
Determine the class size
-A rule for determining
the desirable number of groups
into which a distribution of
observations should be
classified.
K=1+ 3.322logN
K=1+ 3.322log(40)
K=6.322 round
up
K= 7
40 41 42 44 45 46 47 48
49 50 51 52 53 54 55 56
57 58 59 60 61 61 62 63
64 64 65 65 65 66 66 70
75 75 76 76 82 84 92 98

18. EXAMPLE #2 GROUPED FREQUENCY
DISTRIBUTION
The following are the scores obtained by 40
students of Grade 10 in 100 item MMW quiz.
𝑪𝒘 = 𝒊 =
𝑹
𝑲
Step.4
Find the class width.
R=58,
K=7
𝑪𝒘 =
𝟓𝟖
𝟕
𝑪𝒘 =8.286 round
up
𝑪𝒘 = 9
40 41 42 44 45 46 47 48
49 50 51 52 53 54 55 56
57 58 59 60 61 61 62 63
64 64 65 65 65 66 66 70
75 75 76 76 82 84 92 98

19. The following are the scores obtained by 40
students of Grade 10 in 100 item MMW quiz.
Step.5
Select as starting point either the lowest score or the lower class limits. Add
the class width to the starting point to get the second lower class limits. Then enter
the upper class limits.
Class
Interval
Tally
Frequenc
y (f)
Class
mark (x)
94-102 I 1 98
85-93 I 1 89
76-84 IIII 4 80
67-75 III 3 71
58-66 IIIII-IIIII-
IIII
14 62
49-57 IIIII-IIII 9 53
40- 48 IIIII-III 8 44
𝑖 = 9
Class width (𝑪𝒘 = 𝒊) =9
number of classes (K)=7
40 41 42 44 45 46 47 48
49 50 51 52 53 54 55 56
57 58 59 60 61 61 62 63
64 64 65 65 65 66 66 70
75 75 76 76 82 84 92 98

20. EXAMPLE #1
Scores of 40 students of BS-Criminology in MMW
exam. Construct a frequency distribution table using
the class size of 6.

21. EXAMPLE #1
Scores of 40 students of BS-Criminology in MMW
exam. Construct a frequency distribution table using
the class size of 6.

22. EXAMPLE #1
Scores of 40 students of BS-Criminology in
MMW exam. Construct a frequency
distribution table using the class size of 6.
R= HS-LS K=6 𝒊 =
𝑹
𝑲
R= 50-9 𝐍𝐨𝐭𝐞: 𝐂𝐒 = 𝒊 𝒊 =
𝟒𝟏
𝟔
R= 41 𝒊 = 𝟔. 𝟖𝟑 = 𝟕

23. EXAMPLE #1
Scores Tally Frequen
cy
Class Boundaries
(CB)
Class
Mark
(CM)
𝐿𝐶𝐿 + 𝑈𝐶𝐿
2
Less than
Cumulati
ve
Frequenc
y (<Cf)
(f+Cf)
LCB
(LCL-
0.5)
UCB
(UCL+0.
5)
44-50
37-43
30-36
23-29
16-22
9-15
i= 7

24. EXAMPLE #1
Scores Tally Frequen
cy
Class Boundaries
(CB)
Class
Mark
(CM)
𝐿𝐶𝐿 + 𝑈𝐶𝐿
2
Less than
Cumulati
ve
Frequenc
y (<Cf)
(f+Cf
LCB
(LCL-
0.5)
UCB
(UCL+0.
5)
44-50 IIII 4
37-43 IIII-I 6
30-36 IIII-II 7
23-29 IIII-IIII-I 11
16-22 IIII-IIII-I 11
9-15 1 1
i= 7 n=40

25. EXAMPLE #1
Scores Tally Frequen
cy
Class Boundaries
(CB)
Class
Mark
(CM)
𝐿𝐶𝐿 + 𝑈𝐶𝐿
2
Less than
Cumulati
ve
Frequenc
y (<Cf)
(f+Cf
LCB
(LCL-
0.5)
UCB
(UCL+0.
5)
44-50 IIII 4 43.5
37-43 IIII-I 6 36.5
30-36 IIII-II 7 29.5
23-29 IIII-IIII-I 11 22.5
16-22 IIII-IIII-I 11 15.5
9-15 1 1 8.5
i= 7 n=40

26. EXAMPLE #1
Scores Tally Frequen
cy
Class Boundaries
(CB)
Class
Mark
(CM)
𝐿𝐶𝐿 + 𝑈𝐶𝐿
2
Less than
Cumulati
ve
Frequenc
y (<Cf)
(f+Cf
LCB
(LCL-
0.5)
UCB
(UCL+0.
5)
44-50 IIII 4 43.5 50.5
37-43 IIII-I 6 36.5 43.5
30-36 IIII-II 7 29.5 36.5
23-29 IIII-IIII-I 11 22.5 29.5
16-22 IIII-IIII-I 11 15.5 22.5
9-15 1 1 8.5 15.5
i= 7 n=40

27. EXAMPLE #1
Scores Tally Frequen
cy
Class Boundaries
(CB)
Class
Mark
(CM)
𝐿𝐶𝐿 + 𝑈𝐶𝐿
2
Less than
Cumulati
ve
Frequenc
y (<Cf)
(f+Cf)
LCB
(LCL-
0.5)
UCB
(UCL+0.
5)
44-50 IIII 4 43.5 55.5 47
37-43 IIII-I 6 36.5 43.5 40
30-36 IIII-II 7 29.5 36.5 33
23-29 IIII-IIII-I 11 22.5 23.5 26
16-22 IIII-IIII-I 11 15.5 22.5 19
9-15 1 1 8.5 15.5 12
i= 7 n=40

28. EXAMPLE #1
Scores Tally Frequen
cy
Class Boundaries
(CB)
Class
Mark
(CM)
𝐿𝐶𝐿 + 𝑈𝐶𝐿
2
Cumulati
ve
Frequenc
y (Cf)
(f+Cf)
LCB
(LCL-
0.5)
UCB
(UCL+0.
5)
44-50 IIII 4 43.5 55.5 47 40
37-43 IIII-I 6 36.5 43.5 40 36
30-36 IIII-II 7 29.5 36.5 34 30
23-29 IIII-IIII-I 11 22.5 23.5 26 23
16-22 IIII-IIII-I 11 15.5 22.5 19 12
9-15 1 1 8.5 15.5 12 1
i= 7 n=40

29. EVALUATION

30. Construct the problem using frequency
distribution.
Given 50 multiple choice items in their final
test in English, the score of the students are
the following:

31. EXAMPLE #3 GROUPED FREQUENCY
DISTRIBUTION
Step.1
Arrange the score from lowest to
highest.
Construct a frequency distribution table given
the set of data.
Ages of people going to Boracay
14 29 43
17 32 44
21 34 47
21 35 52
22 35 53
25 36 54
26 37 55
27 39 60
28 41 60
28 42 63

32. Answer the following:
1. Construct a frequency distribution table
given the set of data using 6 classes.