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Introduction
Meaning
Central Tendency
• Central Tendency or more simply average is a measure
of finding a representative single figure for a large set of data
for making meaningful comparisons with others sets or body
of data. Such an average is somewhere within range of data
and is therefore a measure of central tendency.
•Some of the techniques that can be used for measuring
Central Tendency are given below1.Arithmatic Mean
2. Median
3. Mode
4. Geometric Mean 5. Harmonic Mean 6. Weighted Mean
These tools can be used for statistical population or samples
depending upon type of data.
“An average is a single figure that represents the whole
group.”-Clark
Definitions
“An average is a typical value that represents all the
individual values in a series.”-Croxton and Cowden
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Purpose & functions
Central Tendency
1. Brief and Simple- It can present raw data which may be
unorganised and complex into a simple and systematic form
reducing the mass of data into single figure to draw conclusions.
2. Helpful in comparison- It helps in making comparison of different
sets of data. For e.g average age of male population in India may
be compared with similar statistic in UK to understand
demographic patterns.
3. Helpful in policy formulation – Once inferences are drawn by
use of average, social, economic, educational, industrial and other
policies can be formed to take corrective actions to improve
conditions.
4. Basis of statistical analysis- Average figure can put focus on
general patterns, interests, orientation of a large population and
give an inspiration for further study of drivers of such interest.
5. Representation of Universe- It can provide a single
representative figure for a large data set or for sub components
for understanding general position or state of affairs.
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Characteristics & types
Good
Average
Central Tendency or Average
Characteristics of good average are1. It should be simple to compute.
2. Should be easy to understand.
3. Should be uniquely defined.
4. Should be based on all observations without unduly
affected by extreme observations.
5. Should be capable for further algebraic treatment.
Types of Averages
Types of
average
Mathematical Averages
Arithmatic
Mean
Geometric
Mean
Harmonic
Mean
Positional Averages
Median
Mode
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Arithmatic Mean
Definition
Central Tendency
Arithmatic mean is most popular and widely known measure
of central tendency. It is defined as value arrived at by
adding all items of a series and then diving it by total
number items.
It may be divided into 2 types1. Simple arithmatic mean
2. Weighted arithmatic mean
In case of individual series, arithmatic mean can be computed
by applying any of 2 methods
a. Direct method
Individual
series
Formula
Where is arithmatic mean
∑x is values of items of a series
n is number of observations
contd….
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Arithmatic Mean
Short Cut
method
Central Tendency
When number of observations are large, arithmatic mean can
be calculated by short cut method. The deviations are taken
from an assumed mean. The formula to calculate it isWhere ‘d’ is deviations of items from assumed mean and ‘A’ is
assumed mean.
To calculate Arithmatic mean in discrete series, 3 methods
may be used
1. Direct method 2. Shortcut method
3. Step deviation method
--------------------------------------------------------------------------------The following formula is used for direct method-
Discrete
series
Where f is frequency, x is value of variables and N is total
number of observations.
Formula for Shortcut method
Where A is assumed mean, Efdx=Ef(X-A) and N is sum of
frequency
contd…. 5
Central Tendency
Formula used in this method is
Step
Deviation
method
Where C is the common factor.
For calculating arithmatic mean for continuous series,
3 methods are used
1. Direct
2. short cut
3. Step deviation
DirectContinuous
series
Short Cut-
Step Deviation-
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Arithmatic Mean (AM)
Properties
Merits &
demerits
Central Tendency
1. The sum of deviations of items from AM is always zero.
2. The sum of squared deviations of the items from AM is minimum
i.e. it is less than squared deviation of items from any other
value.
3. If each item of series is
4. The product of
Merits1. It is easy to calculate and simple to understand.
2. It is based on all observations.
3. It is rigidly defined and based on calculated value rather than the
positional value.
Demerits1. Its outcome is affected by extreme values.
2. It cannot be used inn an open series.
3. Sometimes gives confusing results like number of children born
to population may be presented in decimal points.
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Median
Central Tendency
Introduction
Median is another important measure of central tendency based
on the positional average. It is defined as middle value of
series when the series is arranged in either ascending or
descending order.
Definition
“The median is that value of the variable which divides the
group into two equal parts, one part comprising all value
greater and the other values less than the median.”-Conner
Calculation of Median M=size of {N+1}th item
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Individual
Series
Odd number series- If the number of items are odd, then the
medium is middle value after the items are arranged in
ascending or descending order.
Even number items- In case of even number of items, medium is
arithmetic median of 2 middle items after the items are
arranged in ascending or descending order of their
magnitude.
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Median
Discrete
Series
Central Tendency
Calculation of Median M=size of {N+1}th item
2
Continuous
Series
Mid value
Series
Merits
1. It is easy to understand and compute.
2. Median is not affected by extreme values.
3. Median is most suitable for finding qualitative aspects and
also provides most appropriate average in open ended
classes.
4. It is rigidly defined.
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Median & Partition values
Demerits
Central Tendency
1. It requires arranging of data but other averages do not need
this.
2. Not based on all the observations of the series so it is a
positional average.
3. It cannot be computed correctly if the numbers of items in the
series is even.
4. It is difficult to calculate if the items in a series are very small
or very large.
Just as Median divides the series into 2 equal parts, there are
other measures which divides the series into 4, 10 and 100
equal parts. These are called quartiles, deciles and
percentiles.
Partition
Values
Quartiles-Divides the series into 4 equal parts and are denoted
by Q1, Q2 and Q3 for any series.
Deciles- Divides the series into 10 equal parts and every series
have nine deciles denoted by D1 through D9.
Percentiles- Divides the series into 100 equal parts and every
series have ninety nine deciles denoted by P1 through P99.
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Mode
Introduction
Definition
Central Tendency
Mode is another important measure of central tendency and is
defines as value which appears most frequently in the series.
“The value of the variable which occurs most frequently in
distribution is called the mode.”- Kenny and Keeping
“Mode is the value which has greatest frequency density.”A.M.Tuttle.
In case of individual series mode can be computed by 2 methods
1. Inspection method-This involves inspecting the series to find
most frequently occurring value which is mode.
Individual
Series
2. By changing individual series into discrete series-When the
number of items is very large, individual series is first
converted into discrete series to find the value corresponding
to which there is highest frequency.
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Mode
Discrete
Series
Central Tendency
Under this Mode can be calculated using 2 methods1.Inspection method- Mode is determined by inspection only. By
the method is generally used where frequency increase upto
a point and decrease after reaching a maximum point.
2. Grouping method-
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