Central tendency m,m,m 1.2

Measures of Central Tendency:
Mean, Median, Mode
CasperWendy

Measures of Central Tendency
 It is also defined as a single value that is used to
describe the “center” of the data.
 Measure 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.
 There are three commonly used measures of central
tendency. These are the following:
 MEAN
 MEDIAN
 MODE

MEAN
 It is also referred as the “arithmetic average”
 It is the most commonly used measure of the center of data
 Computation of Sample Mean
_
X = Σ x = x1 + x2 + x3 + … xn
̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅
N
_
X = Σ x
̅ ̅ ̅ ̅
N
 Computation of the Mean for Ungrouped Data
n
_
X = Σ f x
̅ ̅ ̅ ̅
n

MEAN
Example:
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 = Σ x
̅ ̅ ̅ ̅
n
= 228
̅ ̅ ̅ ̅
15
= 15.2

MEAN
Analysis:
The average performance of 15 students who
participated in mathematics quiz consisting of
25 items is 15.20. The implication of this is that
student who got scores below 15.2 did not
perform well in the said examination. Students
who got scores higher than 15.2 performed
well in the examination compared to the
performance of the whole class.
_
X = 15.2

MEAN
Example:
Find the Grade Point Average (GPA) of Paolo Adade for the first
semester of the school year 2013-2014. Use the table below:
Subjects Grade (Xi) Units (wi) (Xi) (wi)
BM 112 1.25 3 3.75
BM 101 1.00 3 3.00
AC 103 1.25 6 7.50
EC 111 1.00 3 3.00
MG 101 1.50 3 4.50
MK 101 1.25 3 3.75
FM 111 1.50 3 4.50
PE 2 1.00 2 2.00
Σ(wi) = 26 Σ(Xi)wi)=32.00

MEAN
= 32
̅ ̅ ̅
_
X = Σ(Xi)wi)
̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅
Σ(wi)
26
= 1.23
The Grade Point Average of
Paolo Adade for the
first semester SY 2013-2014
Is 1.23.

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

MEAN
_
X = Σ f xm
̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅
The only one formula in solving the mean for grouped data is
called midpoint method. The formula is:
n
_
Where X = mean value
xm = midpoint of each class or category
f = frequency in each class or category
Σ f xm = summation of the product of f xm

MEAN
_
X = Σ f xm
̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅
Steps in Solving Mean for Grouped Data
1. Find the midpoint or class mark (Xm) of each class or category
using the formula Xm = LL + LU .
2
n
2. Multiply the frequency and the corresponding class mark f xm.
3. Find the sum of the results in step 2.
4. Solve the mean using the formula

MEAN
Example:
Scores of 40 students in a science class consist of 60 items and they
are tabulated below.
X f Xm fXm
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
n = 40 Σ f Xm = 1 345
_
X = Σ f xm
n
= 1 345
40
= 33.63

MEAN
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 those students who got scores
above 33.63 performed well.

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

MEAN
When to Use the Mean
• Sampling stability is desired.
• Other measures are to be computed such as
standard deviation, coefficient of variation and
skewness.

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

MEDIAN
1. Arrange the scores (from lowest to highest or highest to
lowest).
Median of Ungrouped Data
2. Determine the middle most score in a distribution if n is an
odd number and get the average of the two middle most
scores if n is an even number.
Example 1: Find the median score of 7 students in an English class.
x (score)
19
17
16
15
10
5
2

MEDIAN
x̃ = 16 + 15
2
x̃ = 15.5
Example: Find the median score of 8 students in an English
class.
x (score)
30
19
17
16
15
10
5
2

MEDIAN
Formula:
n_
x̃ = LB + _2 ̅ cfp_ X c.i
fm
X̃ = median value _n_
MC = median class is a category containing the 2
LB = lower boundary of the median class (MC)
cfp = cumulative frequency before the median class if the
scores are arranged from lowest to highest value
fm = frequency of the median class
c.i = size of the class interval
Median of Grouped Data

MEDIAN
1. Complete the table for cf<.
Steps in Solving Median for Grouped Data
_n_
2. Get 2 of the scores in the distribution so that you can
identify MC.
3. Determine LB, cfp, fm, and c.i.
4. Solve the median using the formula.

MEDIAN
Example: Scores of 40 students in a science class consist of 60 items
and they are tabulated below. The highest score is 54 and the
lowest score is 10.
X f cf<
10 – 14 5 5
15 – 19 2 7
20 – 24 3 10
25 – 29 5 15
30 – 34 2 17 (cfp)
35 – 39 9 (fm) 26
40 – 44 6 32
45 – 49 3 35
50 – 54 5 40
n = 40

MEDIAN
Solution:
_n_ _40_
2 = 2 = 20 _n_
The category containing 2 is 35 – 39.
LL of the MC = 35
Ln = 34.5
cfp = 17
fm = 9
c.i = 5
_n_
x̃ = LB + _2 ̅ cfp_ X c.i
fm
_20 – 17_ X 5
= 34.5 + 9
= 34.5 + 15/9
x̃ = 36.17

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

MODE
Trimodal is a distribution of scores that consists of three
modes or multimodal is a distribution of scores that
consists of more than two modes.
The mode or the modal score is a score or scores that occurred
most in the distribution.
Bimodal is a distribution of scores that consists of two modes.
It is classified as unimodal, bimodal, trimodal or mulitimodal.
Unimodal is a distribution of scores that consists of only one mode.

MODE
Example: Scores of 10 students in Section A, Section B and Section C.
Scores of Section A Scores of Section B Scores of Section C
25 25 25
24 24 25
24 24 25
20 20 22
20 18 21
20 18 21
16 17 21
12 10 18
10 9 18
7 7 18

MODE
The modes for Section C are 18, 21, and 25. There are three
modes for Section C, therefore, it is called a trimodal or
multimodal distribution.
The score that appeared most in Section A is 20, hence, the mode
of Section A is 20. There is only one mode, therefore, score
distribution is called unimodal.
The modes of Section B are 18 and 24, since both 18 and 24
appeared twice. There are two modes in Section B, hence,
the distribution is a bimodal distribution.

MODE
Mode for Grouped Data
In solving the mode value in grouped data, use the formula:
___d1___
X̂ = LB + d1 + d2 x c.i
LB = lower boundary of the modal class
Modal Class (MC) = is a category containing the highest frequency
d1 = difference between the frequency of the modal class and the
frequency above it, when the scores are arranged from lowest
to highest.
d2 = difference between the frequency of the modal class and the
frequency below it, when the scores are arranged from
lowest to highest.
c.i = size of the class interval

MODE
Example: Scores of 40 students in a science class consist of 60 items and
they are tabulated below.
x 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
n = 40

MODE
Modal Class = 35 – 39
LL of MC = 35
LB = 34.5
d1 = 9 – 2 = 7
d2 = 9 – 6 = 3
c.i = 5
___d1___
X̂ = LB + d1 + d2 x c.i
___7___
= 34.5 + 7 + 3 x 5
= 34. 5 + 35/10
X̂ = 38
The mode of the score distribution that consists of 40 students
is 38, because 38 occurred several times.

MODE
• It can be used when the data are qualitative as well as quantitative.
• It may not exist.
Properties of the Mode
• It may not be unique.
• When the “typical” value is desired.
When to Use the Mode
• When the data set is measured on a nominal scale.
• It is affected by extreme values.

Central tendency m,m,m 1.2

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Central tendency m,m,m 1.2