MEASURES OF LOCATION AND DISPERSION MEASURES OF LOCATION AND DISPERSION It is impossible to remember all the values of data, In our d...
MEASURES OF LOCATION AND DISPERSION
MEASURES OF LOCATION AND DISPERSION It is impossible to remember
all the values of data, In our daily problems we try
to find a value which is neither very small nor very large but lie in the Center of
series of observations and may be used to represent the whole given data
such a value in Statistics is technically known as average, Averages help to
locate the center of data hence they are also called measures of central
tendency or measures of location.
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| MEASURES OF LOCATION AND DISPERSION|Types of Average |
These measures always lie in the Center of given data and remain unchanged
even if the order of values is changed.
There are two purposes of measures of central tendency (i) To find the
central value about which all the values are clustered (ii) To find balancing point
of the data. The most common and well known measure of central tendency is
only arithmetic mean, In Statistics five types of average are studied, which are
suitable in different situations depending upon the nature of data.
These types of average are.
1. Mathematical Average
Its types are:
a) Arithmetic Mean (X)
b) Geometric Mean (GM)
c) Harmonic Mean (H,M)
2: Average of Position or Average of Location
Its types are:
a) Median (R)
b) Mode R
Arithmetic mean, median and mode are known as averages of first order,
properties/Qualities/Criteria of Good Average:
i) It is well defined, It means there is no confusion in its definition. For
example sum of values divided by their number, is clear definition it will
be considered well defined, Demand of most repeated value of data is ill
defined statement, because there 'is puzzlement, if there is no most
repeated value or there are two, three, or more, most equally repeated
values then what will happen.
ii)It uses all the values. It means while calculating the average each value
of data is used in the calculation,
Iii)It is able for more action i.e it may be further solved or it is possible to
calculate some other quantities from the average, For example it is
possible to calculate total of values from the arithmetic mean,
iv)It is easy to understand and solve, It means its calculation requires no
much practice and a common man can understand it easily when
interpreted,
v)t is less affected by extreme values, The values whose magnitude is
much higher than central values are extremely large values and whose
magnitude is very small than central values are extremely small values,
Both extremely large and extremely small values ,are known as extreme
values or outlines,
Vi)If the experiment is repeated a large number of times it gives unwavering
and reliable results.
ARITHMETIC MEAN (X )
It is well-known average and suitable when there are no very small or very large
values in the data, It is highly affected by extremely large values, It can be
calculated for every kind of data easily except open-end frequency distribution
and missing values without assuming the ends or missing values, Arithmetic
mean gives misleading results if there are extreme values, Mathematically it is
obtained by dividing the, sum of values by their number and denoted by x.
Properties of Arithmetic Mean:
1)Sum of deviations (differences) of observations from their mean is zero.
2)Sum of squares of deviations of observations from their arithmetic mean
is minimum than sum of squares of deviations of observations from any
provisional mean.
3) Arithmetic mean is affected by change of origin (addition of some
constant to each value of the variable is change of origin) and scale
(multiplication by some constant to each value of the variable is change
of scale) i.e if Y -a+bX then Y -a +bX
Methods of Computing Arithmetic Mean:
There are three methods of computing arithmetic mean:
(a) Direct method
(b) Short cut method
(c) Coding method.
Methods of Calculating Arithmetic Mean
1. Short cut method
2. Direct method
3. Coding method
(a) Direct Method:
It is most common method of computing arithmetic mean.
Common people know only this method, If data is ungrouped it is computed by
dividing the sum of values by their number. It is usually denoted by X and read
as X bar. If the data is grouped into a frequency distribution then we first
calculate the mid points of the classes and arithmetic mean is obtained by
dividing the sum of products of mid points by their frequencies by the sum of
Frequencies.
(b) Short cut method:
Calculation of arithmetic mean from small values
without calculator is easy but from large values it becomes difficult. Now a day's
scientific calculators are very advance and cheaper. In the ancient times the
they were very costly and slow. There was a handle attached to operate them.
A short cut method was introduced to compute arithmetic mean at that time.
Now this method is not so important, The formula for grouped and grouped
Where D X — A, A is usually known as provisional mean or arbitrary
mean, It is any value (from the given data or outside the data) which is
subtracted from each value of X, If the data is grouped, a point whose
frequency is highest is selected as provisional mean but? if any other mid paint 10
suitable may be selected as .
(c) Coding method:
If the data is ungrouped and values are equidistant or if
the data is grouped and class interval is uniform, Arithmetic mean is computed
by changing the origin and scale of variable by transferring the values as
where h is size of distance between the values or size of class
interval, First we find mean of u, then mean of original values is obtained by
multiplying the mean of Il by h and adding This method of finding the
arithmetic mean is known as coding method. This method is used if the class interval is equal,
Weighted mean or weighted arithmetic mean denoted by X is -suitable
average when all the commodities are not equally important or sampling is done
with replacement, In such situations we assign numbers known as weights to
the commodities, which show relative importance of commodities in their group,
calculations of index numbers, birth and death rates in vital statistics.
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