measures of central tendency,types of averages,mean,median,mode,geometric mean,harmonic mean

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MEASURES OF CENTRAL TENDENCY
MEANING OF MEASURES OF CENTRAL TENDENCY:
A measure of central tendency is a single value, which
describes a set or group of data by identifying the central position within set or group of
data.
Some times Measures of central tendency are also
called as measures of central location or central value. Here the central value or location is
called average.
Definitions:
According to Croxton & Cowden:
An average value is a single value within the range of the data
that is used to represent all the values in the series.
According to Clark:
An average is a figure (number) that represents the whole group.
According to M.R.Speigal:
An average is a value, which is representative of a set of data.
Properties of Good Measures of Central Tendency:
The ideal measures of central tendency are as follows, which are
1. It should be based on all observations in given data.
2. Its definition should be in the form of a mathematical formula.
3. It should be easy to calculate
4. It should be simple to understand
5. It should be rigidly (accurately) defined.
6. It should be capable further algebraic treatments.
7. It can be found by graphical method also.
8. It should have sampling stability.
Functions (or) objectives of averages:
The main objectives of the measures of central tendency are as
follows. Which are:
1. Representative of the group
2. Brief description

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3. Comparison
4. Formulation of Economic policies
5. Other statistical analysis.
1. Representative of the group:
The average is the single value, which represent whole group of data.
So this single value will help to identifying entire data with in short period of time
2. Brief description:
The average gives a small or brief description of the whole
data in systematic manner.
3. Comparison:
The measure of central value is helpful in comparison with other
groups.
4. Formulation of economic policies :
The measure of central tendency helps to develop the business in
case of economical activities as well as state and central governments are also widely
using the averages to formulate the economic policies.
5. Other statistical analysis:
The use of averages becomes compulsory for other statistical analysis,
such as index numbers, analysis of time series etc.
Types of averages (or) measures of central tendency:
The measures of central tendency (or) the averages are classified in to
the following types .which are
Types of Averages (or) Different
Measures of Central Tendency:
• Arithmetic Mean
• Geometirc Mean
• Harmonic Mean
• Median
• Mode

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1. Arithmetic Mean:
The most widely & popular used measure for representing the
entire data by on value is arithmetic mean. It is a scientific method. Here each & every
item taken in to account.
Arithmetic Mean (A.M) means adding all items then the number of
items to be divided. The result is known as A.M (or) X bar ( ).
Sum of All the Items (or) Observations.
A.M= =
Number of items (or) Observations.
X1+X2+X3+…………….Xn
A.M= X =
n
∑ X
A.M= X =
n
Where ∑ X =sum of all observation in the given data
n = number of observation in the given data
Merits of Arithmetic Mean:
1. It is simple to understand and easy to calculate
2. In the calculation of mean each & every item or observation is taken in to account.
3. It is good to comparison
4. It is possible to calculate even some of the details of the data are locking or
unknown.
5. It doesn’t require arranging the data in the order i.e., ascending or descending order.
Demerits of Arithmetic Mean:
1. It can’t find by the simple observation of the given series or data.
2. If any item of the series is ignored, then the accuracy of the mean will be affected.
3. It is not possible to find graphically.
4. It can’t be calculated exactly in case of open-end classed.
Types of Arithmetic Mean:
There are two types of arithmetic mean, which are

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1. Simple (or) Unweighted Arithmetic Mean
2. Weighted Arithmetic Mean
1. Simple (or) Unweighted Arithmetic Mean:
The simple A.M can be calculated for 3 types of data .which are
1. Calculation of A.M for personal data or individual data
2. Calculation of A.M for discrete data
3. Calculation of A.M for continuous data
Calculation of A.M for personal data or Individual data:
If the given data is individual data then A.M can be
calculated as follows.
 Add all the observations(x )in the given data, it gives ∑x
 Count the number of observation, it gives n
 Apply the following formula
X1+X2+X3+…………….Xn
A.M= X =
n
∑ X
A.M= X =
n
Calculation of A.M for discrete data:
If the given data is discrete data then A.M can be calculated as
follows.
 Calculate the sum of frequencies ,it gives N. i,e., N=∑f
 Multiply each frequency value(f) with the corresponding observation value(x),it gives
fx
 Calculate the sum of all fx values ,it gives ∑fx
 Apply the following formula
f1 x1+f2 x2+f3 x3+…………….fn xn
A.M= X =
N
∑f x
A.M= X =
N
Where ∑f x= the sum of multiplication of f and x
∑f = the sum of frequencies

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Calculation of A.M for continuous data:
If the given data is continuous data then A.M can be calculated as
follows.
 Calculate the sum of frequencies ,it gives N. i,e., N=∑f
 Find the mid values(x) in a separate column for the given class intervals (C.I).the mid
values can be calculated by using the following formula.
Upper limit +Lower limit
Mid value(x) =
2
 Multiply each frequency value(f) with the corresponding mid value(x),it gives fx
 Calculate the sum of all fx values ,it gives ∑fx
 Apply the following formula
f1 x1+f2 x2+f3 x3+…………….fn xn
A.M= X =
N
∑f x
A.M=X =
N
Where ∑f x= the sum of multiplication of f and x
∑f = the sum of frequencies
2. Weighted Arithmetic Mean:
The weighted A.M calculated as follows
 Multiply weights(w) by the variables(x) and add up the wx values, it gives ∑wx
 Calculate the sum of weights ,it gives ∑w
 Apply the following formula
∑wx
XW =
∑w
Where XW = weighted arithmetic mean
W=weights
X =variables
2. Geometric Mean :

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Geometric mean is obtained by multiplying the values of the items &
extracting the root of the product corresponding to the number of items.
G.M is defined as the nth root of the products of n items or values. If
there are two items we can take the square root, if there are three items we can take cube
root so on…….if there are n items then we can take the nth root so on….
G.M=n√x1 x2 x3 ……..xn
G.M=( x1 x2 x3 ……..xn)1/n
Take log on both sides
Log (G.M) =Log (x1 x2 x3 ……..xn)1/n
Log (G.M) = (1/n) [Log (x1 x2 x3 ……..xn)]
Log (G.M) = (1/n) [log x1+log x2+log x3+…….log xn]
Log (G.M) = (1/n) [∑logx]
Merits of Geometric Mean:
 It is rigidly defined
 It is based on all observations
 As compared with mean, geometric mean is affected to a lesser by extreme
observations
 It is not much affected much by fluctuations of sampling
 It is useful for construction of index numbers
 It can be calculated with mathematical precision provided all the values are positive
Demerits of Geometric Mean:
 It can’t be used when any observation is zero or negative value
 It is not easily understand
 It is very difficult to calculate
 It can’t be computed if any value is missing
Calculation of G.M:
The geometric mean can be calculated for 3 types of data .which are
1. Calculation of G.M for personal data or individual data
2. Calculation of G.M for discrete data
3. Calculation of G.M for continuous data
Calculation of G.M for personal data or Individual data:
G.M= Antilog [(∑logx)/n]

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If the given data is individual data then G.M can be
calculated as follows.
 Find out the logarithm of each value (x),it gives log(x) values
 Add all the values of log(x),it gives ∑log(x)
 Count the number of observation, it gives n
 Apply the following formula
Calculation of G.M for discrete data:
If the given data is discrete data then G.M can be calculated
as follows.
 Calculate the sum of frequencies ,it gives N. I,e., N=∑f
 Find out the logarithm of each value, it gives log(x)
 Multiply each log(x) value with its corresponding frequency value(f),it gives flog(x)
 Add all the flog(x) values, it gives ∑ flog(x)
 Apply the formula
Where ∑f log(x) = the sum of multiplication of f and log(x)
N=∑f = the sum of frequencies
Calculation of G.M for continuous data:
If the given data is continuous data then G.M can be calculated as
follows.
 Calculate the sum of frequencies ,it gives N. i,e., N=∑f
 Find the mid values(x) in a separate column for the given class intervals (C.I).the mid
values can be calculated by using the following formula.
Upper limit +Lower limit
Mid value(x) =
2
 Find out the logarithm of each mid value(x), it gives log(x)
 Multiply each log(x) value with its corresponding frequency value(f),it gives flog(x)
 Add all the flog(x) values, it gives ∑ flog(x)
 Apply the formula
G.M= Antilog [(∑logx)/n]
G.M= Antilog [(∑flogx)/N]
G.M= Antilog [(∑flogx)/N]

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Contact Number :8688683936
Where ∑f log(x) = the sum of multiplication of f and log(x)
N=∑f = the sum of frequencies
3. Harmonic Mean :
The harmonic mean is based on the reciprocal of numbers averaged. It is defined as the
reciprocal of the A.M of the reciprocal of the individual observations.
If X1,X2,X3,……………,Xn are n observations then the harmonic mean of these
observations will be as follows.
n
H.M=
(1/ X1)+ (1/ X2)+ (1/ X3)+……. (1/ Xn)
Where n= number of observations
Merits of Harmonic Mean:
 H.M is rigidly defined
 It is based on all the observations
 It is suitable for further mathematical treatment
 It is not affected by fluctuations of sampling
 It gives greater importance to small items and also it is useful only when small items
have to be given a greater weight
Demerits of Harmonic Mean:
 It is not easy to understand
 It is difficult to calculate
 It gives greater importance to small items
Calculation of H.M:
The harmonic mean can be calculated for 3 types of data .which are
1. Calculation of H.M for personal data or individual data
2. Calculation of H.M for discrete data
3. Calculation of H.M for continuous data
Calculation of H.M for personal data or Individual data:
H.M=n/[∑(1/x)]

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If the given data is individual data then
H.M can be calculated as follows.
 Find the reciprocal of each observation ,it gives (1/x)
 Add all the reciprocals ,it gives ∑(1/x)
 Count the number of observations, it gives n
 Apply the formula
Calculation of H.M for discrete data:
If the given data is discrete data then H.M can be calculated as
follows.
 Calculate the sum of frequencies, It gives N. i,e., N=∑f
 Find the reciprocal of each observation ,it gives (1/x)
 Multiply the reciprocal (1/x) of each observation by its corresponding frequency ,it
gives f(1/x)
 Add all the f(1/x) values , it gives ∑f(1/x)
 Apply the formula
Where N=∑f = the sum of frequencies
Calculation of H.M for continuous data:
If the given data is continuous data then H.M can be calculated
as follows.
 Calculate the sum of frequencies, it gives N. i,e., N=∑f
 Find the mid values(x) in a separate column for the given class intervals (C.I).the mid
values can be calculated by using the following formula.
Upper limit +Lower limit
Mid value(x) =
2
 Find the reciprocal of each mid value ,it gives (1/x)
 Multiply the reciprocal (1/x) of each mid value by its corresponding frequency ,it
gives f(1/x)
 Add all the f(1/x) values , it gives ∑f(1/x)
 Apply the formula
H.M=n/ [∑ (1/x)]
H.M=N/ [∑f (1/x)]
H.M=N/ [∑f (1/x)]

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Where N=∑f = the sum of frequencies
4. Median :
Median is the value which divides the given series into two equal parts, when
series arranged in ascending order (or) descending order.
The number of items less than the median value and the number of items more
than the median value will be equal.
Merits of Median:
 It is easy to understand and easy to calculate
 It can be calculated by graphically
 Median can be calculated in case of open-end classes
 It can be located by inspection, after arranging the data in to ascending or
descending order.
 Median is also use to calculate median deviation & standard deviation.
Demerits of Median:
 It may not show correct values, if the series was not arranged in order.
 All the items of series are not taken into account
 It is not based on all observations. So it is called positional measure also.
Calculation of Median:
The median can be calculated for 3 types of data .which are
1. Calculation of Median for personal data or individual data
2. Calculation of Median for discrete data
3. Calculation of Median for continuous data
Calculation of Median for personal data or Individual data:
If the given data is individual data then Median can be
calculated as follows.
 Arrange the observations in ascending /descending order
 Locate middle value
 If the number of observations(n) are odd, then median is the middle value
Or
If the number observations are odd, then Median =[(n+1)/2 ]thterm
 If the number of observations(n) are even, then median is the average of
middle two values

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Or
If the numbers of observations (n) are even, then
[(n/2)th term +((n/2)+1)th term]
Median =
2
Calculation of Median for discrete data:
If the given data is discrete data then Median can be calculated as
follows.
 Arrange the data in ascending /descending order
 Calculate the sum of frequencies, it gives N. i,e., N=∑f
 Calculate the cumulative frequency by adding the frequency values one by one
 Locate the median value based on (N/2)th term value in cumulative frequency ,if it is
not exist in cumulative frequency, then find the greater value of (N/2)th term
 apply the formula
Median = (N/2)th term
Where N= sum of frequencies
Calculation of Median for continuous data:
If the given data is continuous data then Median can be calculated as follows.
 Calculate the sum of frequencies, it gives N. i,e., N=∑f
 Calculate the cumulative frequency by adding the frequency values one by one
 Find (N/2)th term
 Find (N/2)th term value in cumulative frequency ,if it is not exist in cumulative
frequency, then find the greater value of (N/2)th term
 Identify L,f,m values in the table
 Apply the formula
[(N/2)-m]
Median=L+ *C
f
Mode:
Mode may be defined as the value that occurs most frequently in a statistical
distribution. It is denoted by Z.
Merits of Mode:
 It is easy to understand
 It is not affected by the extreme values
 It can be calculated for open-end class intervals

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 It can be calculated by the graphical method also
 It is usually an actual value as it occurs most frequently in the series.
Demerits of Mode:
 it is not based on all observations
 it is not capable for further mathematical treatment
 As compared with mean, mode is affected much by fluctuations of sampling.
 Some time mode can’t be identify clearly
Calculation of Mode:
The mode can be calculated for 3 types of data .which are
1. Calculation of Mode for personal data or individual data
2. Calculation of Mode for discrete data
3. Calculation of Mode for continuous data
Calculation of Mode for personal data or Individual data:
If the given data is individual data then Mode can be
calculated as follows.
 Maximum repeated value in the given data is called mode
Calculation of Mode for discrete data:
If the given data is discrete data then Mode can be calculated as
follows.
 Mode can be obtained by inspection. The value of the variable having maximum
frequency is known as modal value.
Calculation of Mode for continuous data:
If the given data is discrete data then Mode can be calculated using the
following formula
f - f1
Mode (Z) =L+ *c
2f-f1-f2
Where L=lower limit of modal class
C=class interval
f= frequency of modal class

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f1=frequency of preceding the modal class
f2=frequency of succeeding the modal class