BUSINESS STATISTICS NOTES
B.COM 2ND AND 3RD SEM NEW SYLLABUS (CBCS PATTERN)
MEASURE OF CENTRAL TENEDENCY (AVERAGE)
Meaning of Average
One of the most important
objectives of statistical analysis is to get one single value that describes
the characteristics of the entire mass of data. Such a value is called the
central value or an “average” or the expected value of the variables.
In the words of Croxton and
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.”
In the words of Clark,” Average
is an attempt to find one single figure to describe whole of figures.”
From the above explanation we can
say that average is a single value that represents a group of values. It
depicts the characteristic of the whole group. The value of average lies
between the maximum and minimum values of the series. That is why it is also
called measure of central tendency.
Objectives of averaging
The objective of study of
averages is listed below:
a) To get single value that
describes the characteristic of the entire group.
b) To facilitate comparison of
data either at a point of time or over a period of time.
Requisites of a good average
The following are the important properties
which a good average should satisfy
1. It should be easy to understand.
2. It should be simple to compute.
3. It should be based on all the items.
4. It should not be affected by extreme values.
5. It should be rigidly defined.
6. It should be capable of further algebraic
treatment.
Types of average
Average is divided into three
main categories:
a) Mean which is further
classified as: Arithmetic mean, Weighted Mean, Geometric Mean and Harmonic
Mean.
b) Median and
c) Mode
Arithmetic Mean: Meaning, Properties, Merits and Demerits
It is a value
obtained by adding together all the items and by dividing the total by the
number of items. It is also called average. It is the most popular and widely
used measure for representing the entire data by one value.
Arithmetic mean may
be either:
(i)
Simple arithmetic
mean, or
(ii)
Weighted arithmetic
mean.
Properties of
arithmetic mean:
1.
The sum of deviations
of the items from the arithmetic mean is always zero i.e. ∑(X–X) =0.
2. The Sum of the
squared deviations of the items from A.M. is minimum, which is less than the
sum of the squared deviations of the items from any other values.
3. If each item in
the series is replaced by the mean, then the sum of these substitutions will be
equal to the sum of the individual items.
Merits of A.M.:
(i)
It is simple to understand
and easy to calculate.
(ii)
It is affected by the
value of every item in the series.
(iii)
It is rigidly
defined.
(iv)
It is capable of
further algebraic treatment.
(v)
It is calculated
value and not based on the position in the series.
Demerits of A.M.:
(i)
It is affected by
extreme items i.e., very small and very large items.
(ii)
It can hardly be
located by inspection.
(iii)
In some cases A.M.
does not represent the actual item. For example, average patients admitted in a
hospital is 10.7 per day.
(iv)
A.M. is not suitable
in extremely asymmetrical distributions.
Geometric Mean (GM): Meaning, Uses, Merits and Demerits
It is defined as nth root of the product of n items or values. i.e., G.M. = ^{n}√ (x1. x2. x3 ……xn)
Merits of G.M.:
(i)
It is not affected by
the extreme items in the series.
(ii)
It is rigidly defined
and its value is a precise figure.
(iii)
It is capable of
further algebraic treatment.
(iv)
It is Useful in
calculating index number.
Demerits of G.M.:
(i)
It is
difficult to understand and to compute.
(ii)
It cannot be computed when one of the values
is 0 or negative.
Uses of G.M.:
(i)
It is used to find average of the rates of
changes.
(ii)
It is Useful in measuring growth of
population.
(iii)
It is considered to be the best average for
the construction of index numbers.
Harmonic Mean (HM): Meaning, Uses, Merits and Demerits
It is defined as the reciprocal of the arithmetic mean of the reciprocal of the individual observations.


N 
H.M. 
= 
(1/x1 +
1/x2 + 1/x3 + ........ +1/xn) 
(i)
Like AM and GM, it is
also based on all observations.
(ii)
It is most
appropriate average under conditions of wide variations among the items of a
series since it gives larger weight to smaller items.
(iii) It is capable of further algebraic treatment.
(iv) It is extremely useful while averaging certain
types of rates and ratios.
Demerits of H.M.:
(i)
It is
difficult to understand and to compute.
(ii)
It cannot be computed when one of the values
is 0 or negative.
(iii)
It is necessary to know all the items of a
series before it can be calculated.
(iv) It is
usually a value which may not be a member of the given set of numbers.
Uses of H.M.: If there are two measurements
taken together to measure a variable, HM can be used. For example, tonne
mileage, speed per hour. In the above example tonne mileage, tonne is one measurement
and mileage is another measurement. HM is used to calculate average speed.
Median: Meaning, Merits and Demerits
Median may be defined as the size
(actual or estimated) to that item which falls in the middle of a series
arranged either in the ascending order or the descending order of their
magnitude. It lies in the centre of a series and divides the series into two
equal parts. Median is also known as an average of position.
Merits of Median:
(i)
It is simple to
understand and easy to calculate, particularly is individual and discrete
series.
(ii)
It is not affected by
the extreme items in the series.
(iii)
It can be determined
graphically.
(iv)
For openended classes, median can be
calculated.
(v)
It can be located by
inspection, after arranging the data in order of magnitude.
Demerits of Median:
(i)
It does not consider all variables because it
is a positional average.
(ii)
The value of median is affected more by
sampling fluctuations
(iii)
It is not capable of further algebraic
treatment. Like mean, combined median
cannot be calculated.
(iv) It cannot be computed precisely when it lies
between two items.
Mode: Meaning, Merits and Demerits
Mode is that value a dataset, which is
repeated most often in the database. In other words, mode is the value, which
is predominant in the series or is at the position of greatest density. Mode
may or may not exist in a series, or if it exists, it may not be unique, or its
position may be somewhat uncertain.
Merits of Mode:
(i)
Mode is the most representative value of
distribution, it is useful to calculate model wage.
(ii)
It is not affected by
the extreme items in the series.
(iii)
It can be determined
graphically.
(iv)
For openended classes, Mode can be calculated.
(v)
It can be located by
inspection.
Demerits of Mode:
(i)
It is not based on all observations.
(ii)
Mode cannot be calculated when frequency
distribution is illdefined
(iii)
It is not capable of further algebraic
treatment. Like mean, combined mode cannot
be calculated.
(iv) It is not
rigidly defined measure because several formulae to calculate mode is used.
Relationship between mean, median and mode
In a normal distribution Mean = Median = Mode. In an asymmetrical distribution median is always in the middle but mean and mode will interchange their positions or values.
Mode = 3 Median  2 Mean.
Or 3Median = 2Mean + Mode
Relation between arithmetic mean, geometric mean and harmonic mean
1. AM is greater than equal to GM and GM is greater than equal to HM
A.M.
> G.M. > H.M.
2.
GM is the square root of product of AM and HM.
GM
= √ (ARITHMETIC MEAN * HARMONIC MEAN)
Which Average is to be used?
No one average is suitable for
all circumstances. All average has its unique feature. The following points
must be taken into consideration in selection of an appropriate average:
1. The purpose which the average
is designed to serve.
2. Types of data available. Are
they badly skewed – avoid mean, gappy around the middle – avoid median and
unequal class interval – avoid mode.
3. Whether or not further
computations are possible?
4. The typical value required in
the particular problem.
Uses of various types of average
1) Arithmetic Mean: AM is
considered to be best average but in the below mentioned situations AM cannot
be used:
a) In highly skewed distributions.
b) In distribution with openend interval.
c) Irregular difference between the range of
data.
d) The arithmetic mean should not be used to
average ratios and rates of change.
e) AM will be misleading when there are very
large and very small items.
2) Median: The median
is generally the best average in openendgrouped distributions.
3) Mode: The mode is
best suited where there is large frequency. Also it can be used where
qualitative data is given.
4) Geometric Mean: Uses of
G.M.:
a)
It is used to find average of the rates of
changes.
b)
It is Useful in measuring growth of
population.
c)
It is considered to be the best average for
the construction of index numbers.
5) Harmonic Mean:
When there are two measurements taken together to measure a variable,
HM can be used. For example, tonne mileage, speed per hour. In the above
example tonne mileage, tonne is one measurement and mileage is another
measurement. HM is used to calculate average speed.
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