Measures of Central Tendency: Mean, Median, and Mode for Grouped data

Measures of
Central Tendency
for Grouped Data

Measures 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 single value that is used to
describe the “center” of the data.
There are three commonly used measures of
central tendency. These are the following:
 MEAN
 MEDIAN

Recall : Measures of Central
Tendency
Measures of Central Tendency for Grouped Data

Grouped Data are the data or scores that are
arranged in a frequency distribution
Frequency distribution is the arrangement of
scores according to category of classes
including the frequency.
Frequency is the number of observations
falling in a category.
Measures of Central Tendency for Grouped
Data

The formula in solving the mean for grouped data
is called the midpoint method. The formula is:
where:
the arithmetic mean
= the midpoint or class mark of each class
f = the frequency in each class
n = sum of the frequencies
Mean for Grouped Data

1. Find the midpoint or class mark () of each class by
using the formula: (= lower limit and = upper limit)
2. Multiply the frequency and the corresponding
class mark in each class interval
3. Find the sum of the results in step 2. ()
4. Solve the mean using the formula:
Steps in solving the Mean for Grouped
Data

The scores of 40 students in a Science class are
tabulated below. Find the mean.
Scores 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

Step 1. Find the class mark for each interval.
Scores f x
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
12
17
22
27
32
37
47
42
52

Step 2. Multiply each frequency with the its
corresponding class mark.
Scores f x fx
10 – 14 5 12
15 – 19 2 17
20 – 24 3 22
25 – 29 5 27
30 – 34 2 32
35 – 39 9 37
40 - 44 6 42
45 – 49 3 47
50 - 54 5 52
60
34
66
135
64
333
252
141
260

Step 4. Find n, , and solve for x using the formula.
Scores f x fx
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
𝒙=
𝜮 𝒇𝒙
𝒏
¿
𝟏 𝟑𝟒𝟓
𝟒𝟎
𝒙=𝟑𝟑.𝟔𝟐𝟓
63
n = 40 1 345

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 the students who got scores above
33.63 performed well.

Properties of the Mean
 It measures stability. Mean is the most stable
among the measures of central tendency
because every score contributes to the value of
the mean.
 The sum of each score’s distance from the mean
is zero.
 It may easily be affected by the extreme scores.
 It can be applied to interval level of
measurement.
 It may not be an actual score in the distribution.

When to Use the Mean
 If sampling stability is desired.
 When other measures such as standard
deviation, coefficient of variation and skewness
are to be computed.

 The Median is what divides the scores in the
distribution into two equal parts.
Fifty percent (50%) lies below the median value
and 50% lies above the median value.
 It is also known as the middle score or the 50th
percentile.
Recall : The Median

The formula in solving the median for grouped data is:
where:
the median
= lower boundary of the median class
f = frequency of the median class
cf = cumulative frequency above the median class if the
scores
are arranged from lowest to highest
= class size (number of scores in an interval)
Median for Grouped Data

Steps in solving Median for Grouped Data
1. Complete the table for <cf.
2. To Identify the Median Class, get of the
scores in the distribution.
3. Determine the Lower boundary (),
cumulative frequency above the median class
(cf ),
the frequency of the median class (f )
and the class size (i )
4. Solve for the median using the formula

tabulated below. Find the median.
Scores 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

Scores f >cf
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
Step 1. Complete the table for <cf.
5
7
10
15
17
26
32
35
40
N = 40

Scores f >cf
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
Step 2. Identify the median class.
5
7
10
15
17
26
32
35
40
N = 40
To Identify the median class, get
the value of first.
Median Class = = 20
Locate 20 in >cf column.
If the cf = 20 cannot be find,
choose the higher cf, in this case,
26. The location of cf = 26 is where
the median class is.

Scores f >cf
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
Step 3. Determine the Lower boundary (), cumulative frequency above the median class
(cf ), the frequency of the median class (f ), and the class size (i )
5
7
10
15
17
26
32
35
40
N = 40
The Lower Limit of the
Median class is 35, therefore,
= 34.5
cf = 17
f = 9
i = 5

Scores f >cf
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
Step 4. Solve for the Median using the formula.
5
7
10
15
17
26
32
35
40
N = 40
Given the following:
= 34.5 cf = 17 f = 9 i = 5
~
𝒙 = 𝑳 𝑩 +(
𝑵
𝟐
− 𝒄𝒇
𝒇 )𝒊
¿ 𝟑𝟒 . 𝟓+( 𝟐𝟎 − 𝟏𝟕
𝟗 )𝟓
¿ 𝟑𝟒 . 𝟓 + ( 𝟑
𝟗 )𝟓
¿ 𝟑𝟒 . 𝟓 +𝟏 . 𝟔𝟕
~
𝒙 =𝟑𝟔 . 𝟏𝟕

When to use the Median
 It may not be an actual observation in the data
set.
 It can be applied in ordinal level.
 It is not affected by extreme values because
median is a positional measure.
Properties of the Median
 The exact midpoint of the score
distribution is desired.
 There are extreme scores in the
distribution.

 The Mode in a frequency distribution is within the class
interval with the highest frequency. The class interval
with the highest frequency is known as the modal class.
A crude mode may be determined by taking the class
interval with the highest frequency. However, the rough
approximation may be improved by considering the
frequencies adjoining the modal frequency.
Recall : The Mode

The formula in solving the mode for grouped data is:
where:
the mode
lower boundary of the modal class
the difference between the frequency of the modal class and the
frequency above the modal class when the scores are arranged from
lowest to
highest
the difference between the frequency of the modal class and the
frequency below the modal class when the scores are arranged from
lowest to
highest
class size (number of scores in an interval)
Mode for Grouped Data

tabulated below. Find the mode.
Scores 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

Step 1. Identify the modal class.
Scores 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
The modal class is the
interval containing the
highest frequency.

Step 2. Identify the lower boundary (), ,
Scores 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
In the modal class we can
identify the following:
Lower Limit of the Modal class =
35
9 – 2 = 7
9 – 6 = 3

Step 3. Solve for the mode using the formula.
Scores 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
Given the following:
9 – 2 = 7 9 – 6 = 3

Learning Activity.
Compute for the mean, median, and mode of the table below:
Weight (in lbs.) Number of students
77 – 83 3
84 – 90 2
91 – 97 4
98 – 104 3
105 – 111 7
112 – 118 12
119 – 125 11
126 – 132 9
133 – 139 8
140 – 146 5
147 – 153 2
n =

Measures of Central Tendency: Mean, Median, and Mode for Grouped data

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