MODULE:3 MEASURES OF CENTRAL TENDENCY AND LOCATION

Learning Module in
Elementary Statistics and
Probability

Module 3: Measures
Of Central Tendency
And Location

Subject Teacher: Mr. Jocel D. Bigalbal

MEASURES OF CENTRAL TENDENCY AND LOCATION

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Objectives:
At the end of this module, the students are expected to:

A. Employ summation notation and apply operations involving the summation.

B. Compute and interpret the different measures of central tendency and location.
C. Compare the different measures of central tendency and location.
D. Recognize the advantages and disadvantages of each measure of average.
E. Apply their knowledge to real life situations.

3.2 Measures of Central Tendency

In Statistics, an average is a measure of central tendency. It is a single number that can

represent a set of data. There are three kinds of averages, namely: mean, median and mode.

Definition: A measure of central tendency is any single value that is used to identify the
“center” of the data or typical value. It is often referred to as the average.

3.2.1 The Arithmetic Mean

The most average and sometimes simply referred to as the mean.

The sum of all values of the observations divided by the number of observations.
The mean for a finite population with N elements, denoted by the Greek letter µ (mu).
The sample mean, used to estimate the population mean µ, is computed as

x =∑ xni , where n is the number of observations in the sample.

i=1

Examples:
1. The number of employees at 5 different gift shops are 4, 8, 10, 12, and 6. Find the mean
number of employees for the 5 stores.
Solution:

4+8+10+12+6 40

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x∑
= xi = =
5
= 8 The mean number of employees for the 5 stores is 8.
i=1 5 5

2. Scores in the Math 120 first long quiz for a sample of 10 students are as follows: 84, 75, 90,
98, 88, 79, 95, 86, 93, and 89.
Solution:

µ=10 84+75+90+98+88+79+95+86+93+89 877

∑x i = = = 87.7
i=110 10 10

The average score is

87.7.
Approximating The Mean From A Frequency Distribution
This is possible only when the class mark can be assumed to be representative of all the
values in that class. If the assumption holds, the following equation mat be used to approximate
the mean from a frequency distribution.

∑f x i i
i=1

x=
n

Where f i = the frequency of the ith class

x i = the class mark of the ith class
k = total number of classes
n = total number of observations
Example: Scores of 110 students in an Achievement Test
Score Frequency (fi) Class Mark (xi) fixi
50-54 10 52 520
55-59 3 57 171
60-64 8 62 496
65-69 13 67 871
70-74 17 72 1224
75-79 19 77 1463
80-84 22 82 1804

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85-89 13 87 1131
90-94 4 92 368
95-99 1 97 97
TOTAL 110 8145

k
Then, ∑x =f x i
8145
n
i
=
110
= 74.0
i=1

The average score of 110 students in achievement test is 74.0.

3.2.2 The Median

The positional middle of an array.

In an array, one-half of the values precede the median and one-half follow it.

The first step in calculating the median, denoted by Md, is to arrange the data in an array.
Let X(i) be the ith observation in the array, i = 1, 2, …N
( N +1) ( N +1)
If N is odd, the median position equals , and the value of the observation in
2 2
the array is taken as the median, i.e.
Md = x
N +1
2

If N is even, the mean of the two middle values in the array is the median, i.e.

x +x
Md =
2
Example: Find the median of the given data set: 75, 75, 67, 71, 72.
Solution:
X1 X2 X3 X4 X5
67 71 72 75 75

Md = x = x3 =72
The average (median) is 72.

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Approximating The Median From A Frequency Distribution

This is possible only if it can be assumed that the values of the observations falling in the
median class are evenly spaced throughout the class. (The median class is the class containing
the median.)

*Median class: Starting from the top, locate the class with <CF greater than or equal to n/2 for
the first time.

Md = LCBmd + c
Where
LCBmd = the lower class boundary of the median class
c = class size of the median class
n = the total number of observations
<CFmd-1 = less than cumulative frequency of the class preceding the median class
fmd = frequency of the median class

Example: (Refer to the example on the scores of 110 students in an achievement test)
Score Frequency (fi) <CF
50-54 10 10
55-59 3 13
60-64 8 21
65-69 13 34
Median Class
70-74 17 51
75-79 19 70
80-84 22 92
85-89 13 105
90-94 4 109
95-99 1 110
TOTAL 110

Md = 74.5 + 5 =75.6

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The average (median) score of 110 students in the achievement test is 75.6.

3.2.3 The Mode

It is the observed value that occurs most frequently.

It locates the point where the observation values occur with the greatest density
It does not always exist, and if it does, it may not be unique. A data set is said to be
unimodal if there is only one mode, bimodal if there are two modes, trimodal if there are three
modes, and so on.
It is not affected by extreme values.
It can be used for qualitative as well as quantitative data.

Example: Identify the mode(s) of the following data sets.

Data Set 1.
2 5 2 3 5 2 1
Solution:
Mo = 2 because it has the most number of occurence. UNIMODAL
Data Set 2.
2 5 5 2 2 5 1
3 5 4 2 5 3 2
Solution:
Mo = 2, 5 because both occured 5 times. BIMODAL
Data Set 3.
1 2 3 3 2 1 2
3 1 4 4 5 5 5
Solution:
Mo = 1, 2, 3, and 5 because each have 3 occurrences. TRIMODAL
Approximating The Mode From A Frequency Distribution

Mo = LCBmd + c
Where
The modal class is the class with the highest frequency.

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LCBmo = the lower class boundary of the modal class

c = class size of the modal class
fmo = frequency of the modal class
f1 = frequency of the class preceding the modal class
f2 = frequency of the class following the modal class

Example:(Refer to the example on the scores of 110 students in an achievement test)

Score Frequency (fi)
50-54 10
55-59 3
60-64 8
65-69 13
70-74 17 Modal class
75-79 19
80-84 22
85-89 13
90-94 4
95-99 1
TOTAL 110

Mo = 79.5 + 5 = 80.8
Most students got a score of 80.8.

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Prepared by:
Mr. Jocel D. Bigalbal

Name: Score:
Course: Date:
Quiz
1. The grades of a student on seven examinations were 85, 96, 72, 89, 95, 82 and 85. Find the
student’s average grade.

2. The salaries of 4 employees were P12,000, P10,000, P15,000, and P18,000. What is the
average salary?

3. Out of 100 numbers, 20, 4, 40, 6, 35, 2, and 5. Find the mean.

4. Find the average weight of 50 male college students of Cavite West Point College.
Weight (in lbs) Number of students
118-126 3
127-135 7
136-144 11
145-153 14

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154-162 7
163-171 5
172-180 3
TOTAL 50

5. Find the median of the set of numbers:

5 3 7 3 8 2 1

6. Find the median of the set of numbers:

11 25 18 79 12 13

7. The reaction times of an individual to certain stimuli were measured by a psychologist to be

0.23, 0.52, 0.36, 0.25, 0.52, 0.26, 0.25, 0.39, and 0.22 seconds. Determine the mode of the
given reaction times.

8. The numbers of incorrect answers on a true-false test for 15 students were recorded as
follows: 2, 1,3, 0, 1, 3, 6, 0, 3, 0, 5, 2, 1, 4, 2. Find the median and mode.

Name: Score:
Course: Date:
Activity
1. The distribution of the number of mistakes made by 200 students taking German in a
multiple-choice quiz on vocabulary is as follows:
NUMBER OF MISTAKES NUMBER OF STUDENTS
6-10 12
11-15 73
16-20 52
21-25 39
26-30 24
TOTAL 200

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Find the mean, median and mode. If the instructor would like to claim that the students have
learned a lot in the course, which of the values should he use to support his claim?

2. A consumer testing service obtains the following mileage per gallon during 5 test runs
performed with each of three compact cars:
CAR A: 27.9 30.4 30.6 31.4 31.7
CAR B: 31.2 28.7 31.3 28.7 31.3
CAR C: 28.6 29.1 28.5 32.1 29.7
A. If the manufacturer of CAR A wants to advertise that their car performed best in the test,
which of the averages (mean, median, mode) could they use to substantiate their claim?
B. If the manufacturer of CAR B wants to advertise that their car performed best in the test,
which of the averages (mean, median, mode) could they use to substantiate their claim?
C. If the manufacturer of CAR C wants to advertise that their car performed best in the test,
which of the averages (mean, median, mode) could they use to substantiate their claim?

3. Find the mean, median, and mode of the data set given below:
Frequency Distribution of Grades in College Algebra
GRADE NUMBER OF STUDENTS
90-100 9
80-89 30
70-79 35
60-69 8
50-59 9
40-49 2
30-39 3
20-29 1

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10-19 2
0-9 1
TOTAL 100

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