9 Measures of Central Tendency
9 Measures of Central Tendency
9 Measures of Central Tendency
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ACKNOWLEDGMENT
Inspiration and hard work always play a key role in the success of
any venture. At the level of practice, it is often difficult to get knowledge
without guidance. “Project is like a bridge between theory and practice.”
With this will I felt pleasure to undertake this project.
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CONTENT
7 Conclusion 22
8 Bibliography 23
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INTRODUCTION
CENTRAL TENDENCY
In statistics, a central tendency (or measure of central tendency) is a central or typical
value for a probability distribution. It may also be called a center or location of the distribution.
Colloquially, measures of central tendency are often called averages. The term central
tendency dates from the late 1920s.
The most common measures of central tendency are the arithmetic mean, the median and
the mode. A central tendency can be calculated for either a finite set of values or for a theoretical
distribution, such as the normal distribution. Occasionally authors use central tendency to denote
"the tendency of quantitative data to cluster around some central value."
While others might stifle a yawn at the mere word, Six Sigma practitioners who have
passed the Lean Six Sigma Green Belt course knows that without statistics, no Six Sigma project
will succeed. Statistics is simply part of the data-driven Six Sigma approach. At the Six Sigma
Green Belt level, we need to have the basic understanding of many statistical methods. Six
Sigma Black Belts and Green Belts typically use a statistical program such as MINITAB to
perform calculations. However, it is always good to understand the underlying principles of a
measure, such as the measures of central tendency, instead of just conjuring the results at a click
of a button without knowing exactly what you are calculating. Statistical tools are used in the
Measure and Analyze phases of the DMAIC process, as you will learn from a reputable free
Lean Six Sigma courses. The most basic statistics that a Six Sigma practitioner will look at are
measures of central tendency. Measures of central tendency are one of the first things that Six
Sigma teams look at after collecting data.
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MEASURES OF CENTRAL TENDENCY
MEAN
The mean, which is also known as the average, is the most popular and well known of the
measures of central tendency. It can be used with both discrete and continuous data, although it
is used more often with continuous data. The mean is equal to the sum of all the values in the
data set divided by the number of values in the data set. Therefore, the formula for the mean
would be read: sum of the observations divided by number of observations.
MEDIAN
Of the measures of central tendency, it is the median that is also known as the positional
average. Positional averages are based on the position of the given observation in a series,
arranged in an ascending or descending order. The median is that value which divides the group
into two equal parts, one part comprising all values greater, and the other, all values less than
median. In other words, the median of a sequence of numbers is the value that lies in the middle
of the sequence when sorted from smallest to largest or vice versa. The median is an example of
one of the measures of central tendency.
MODE
PROCEDURE OF CALCULATION
In this article, we’ll look at some examples to describe each of the measures of central
tendency. Let’s look at some examples of the mean in the figure below. The numbers ranging
from 1 to 8 have been summed and the sum has been divided by the number of observations,
which is 8. Hence, the answer is 4.5. The average or mean is 4.5.
In the second example, the waiting time in the hospital has been impacting turnaround
time for basic blood analysis. The data for turnaround time was collected for 10 such tests on a
given day to calculate the average turnaround time. The unit of measurement for data collection
was minutes. To calculate the mean, we summed the turnaround time for ten blood analysis tests
and then; the sum was divided by the number of observations to arrive at the average turnaround
time for basic blood analysis. The answer here is 61.3 minutes. This is an example of the most
well known of the measures of central tendency.
The formula for calculating the median is read as: ‘n’ plus 1 divided by 2. The median
will be the value lying at the position; which has been suggested by the solution to the formula.
Have a look at the two examples below. The numbers ranging from 1 to 8 have been counted
according to the formula. The answer is 4.5. The median is not 4.5. The value that lies at the
4.5th position is the median value. The question is: how will you calculate the value which lies at
the 4.5th position? The formula to do this would be 4th value plus half of the 5th value minus 4th
value.
For the second example, let’s use the same example that we used for calculating the
average. The data for turnaround time was organized in ascending order. To calculate the
median, we will have to count the number of data points and compose the formula. The answer
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here is 5.5. The value that lies at the 5.5th position is the median value. Hence, the
median is 61 minutes.
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average would be unrepresentative. Similarly, to study wage level in cotton mill industry
of India, separate averages should be calculated for the male and female workers.
Average Formula
In working with an average, there is one central formula that is used to answer questions
pertaining to an average. This formula can be manipulated in many different ways, enabling test
writers to create different iterations on mean problems.
The following is the formal mathematical formula for the arithmetic mean (a fancy name for the
average).
Discrete series means where frequencies of a variable are given but the variable is
without class intervals.
Here each frequency is multiplied by the variable, taking the total and dividing total by
total number of frequencies, we get X.
Symbolically,
X = ∑fx/N
Where f = frequency,
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Example 1. Calculate A.M. from the following data
Solution:
Here Assumed Mean is taken and taking deviations of variable from it. We obtain X by
using the following formula.
dx = (X-A);
(Note :-This formula is often used when the variables are large in size or infractions and direct
formula is not easy to use.)
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Example 2. Calculate the Arithmetic Mean using short-cut method:
Solu
tion: How to calculate arithmetic mean in continuous series:
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Overtime hour worked by 100 employees in a year are given below. Calculate overtime
hour worked per employee.
No. of employee: 10 22 30 20 18
-2
0-10 10 5 -20 -200 -20
20-30 30 25 0 0 0 0
30-40 20 35 10 200 1 10
40-50 18 45 20 360 2 20
N = 100
=1
4
hours
Step-Deviation method:
hour
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CALCULATION OF MEDIAN
Median is defined as the middle value of a set of observation after these observation are
arranged either is increasing order or in decreasing order. Thus, it refers to the value which lies at
the middle position of the series. Therefore, median is termed as positional average. The place of
median in a series is such that half of the observations are smaller and half are larger than this
value.
CALCULATION OF MEDIAN:
1. Individual series:
30 35 28 16 18 26 10 12 15 22
Solution:
10 12 15 16 18 22 26 28 30 35
Here N=10
2. Discrete series:
X: 5 10 15 20 25 30 60 40 45 50 55 35
F: 3 7 10 15 18 25 7 35 20 12 8 40
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Solution: calculation of median
X F CF
5 3 3
10 7 10
15 10 20
20 15 35
25 18 53
30 25 78
35 40 118
40 35 153
45 20 173
50 12 185
55 8 193
60 7 200
N=200
Median= item
Looking at CF column we will find that no CF is equals to 100.5. so, the next higher
= 118. The value of the variable (X) corresponding to CF 118 is 35. Thus, median=35
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Calculate median from the following data:
No.of 7 18 25 30 20 10
person
Solution:
Calculation of median:
C.I. F CF
10-20 7 7
20-30 18 25
30-40 25 50
40-50 30 80
50-60 20 100
60-70 10 110
N=110
To find that the exact value of median we have to apply the following formula:
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=
CALCULATION OF MODE
Mode is that in a series which occurs maximum number of times. In otherworld’s, mode
represents the value which most frequently occurs in a set of observations.
A manufacturer of ready garments is interested to know the size of the garment that fits most
people rather than then mean size of the garments. Similarly a shoe maker is interested to know
the size that fits majority. Like-wise a cigarette company wants to know the preference of a
particular brand of cigarette. In all these cases, mode provides the correct answer rather than
mean and median.
CALCULATION OF MODE
Individual series:
The value which is repeated maximum number of times is the modal value.
Example: 1
The marks secured by 10 students (out of 50) are; 22,25,22,35,35,40,42 and 35.find the modal
mark
Solution:
In the above given series, the modal mark is 35 as it maximum number of times (4time)
Example: 2
Daily wages of 8 workers in rupees are given below find the modal wage.
Solution:
Since all items occur only once, here mode is ill-defined and cannot be calculated in the above
process. However, it can be computed indirectly through median and mean by applying the
formula;
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Mode (Z) = 3 Median – 2 mean.
2. Discrete series
A. By inspection:
The value of the variable whose frequency is maximum is called mode in discrete
frequency distribution series. As such mode can be determined by inspection.
Example 3
Frequency (f) 12 15 17 18 32 16 10
Solution:
Here, the maximum frequency is 32. So the size of the item corresponding to it is 50.
B. By Grouping method:
We have seen that in discrete series mode can be determined by the inspection.
Therefore, an error of judgment may occur if;
i. The difference between the maximum frequency and the frequency preceding or
succeeding it is very small.
ii. A series has two or more same maximum frequencies.(A series having two , three and
more same maximum frequency is called a bi-modal, trimodal and multi-modal series
respectively.)
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iii. Where the maximum frequency occurs either in the beginning or at the end of the
series. In the above situations modal can be determined by preparing grouping table
and analysis table.
ANALYSIS TABLE:
Example: 4
Marks:
10 20 30 40 50 60 70 80 90 100
No. of students:
2 5 8 9 12 14 14 15 11 13
Solution:
Here the maximum frequency is 15 but the difference between this and the
frequencies on either side of it is very small. Therefore, mistake may arise if we take
80 as the modal value.
Thus, we have to prepare the grouping table and analysis table to assess the correct
modal value.
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Grouping table
2
10 7
20 5 13 15
30 8 17 22 29
40 9 21
50 12 26 35 40
60 14 28 43
70 14 29 39
80 15 26 40
90 11 24
100 13
Analysis Table
10 20 30 40 50 60 70 80 90 100
1 1
2 1 1
3 1 1
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4 1 1 1
5 1 1 1
6 1 1 1
total 1 3 5 4 1
3. CONTINUOUS SERIES
No.of 7 18 25 30 20 10 12
person
GROUPING TABLE:
class F1 F2 F3 F4 F5 F6
10-20 7 25
20-30 18 43 50 73
30-40 25 55 75
40-50 30 50
50-60 20 30 60 42
60-70 10 22
70-80 12
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ANALYSIS TABLE:
F1 1
F2 1 1
F3 1 1
F4 1 1 1
F5 1 1 1
F6 1 1 1
total 1 3 6 3 1
Here the class 40-50 is having more frequency that is 5 so we will take the 40-50 class for
calculating mode.
F 1−F 0
Mode ¿ L+ ×i
2 F 1−F 0−F 2
30−25
¿ 40+ ×10=¿ 36.67
2 ×30−25−20
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Conclusion
A measure of central tendency is a measure that tells us where the middle of a
bunch of data lies.
The three most common measures of central tendency are the mean, the
median, and the mode.
Mean: Mean is the most common measure of central tendency. It is simply
the sum of the numbers divided by the number of numbers in a set of data. This is
also known as average.
Median: Median is the number present in the middle when the numbers in a
set of data are arranged in ascending or descending order. If the number of
numbers in a data set is even, then the median is the mean of the two middle
numbers.
Mode: Mode is the value that occurs most frequently in a set of data.
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BIBLIOGRAPHY
WWW.Google.com
www.wikipedia measures of central tendency
Business mathematics and statistics text book
Text book of application statistics
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