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Unit 4: Central Tendency: Mean, Median and Mode and their Properties

that too in a restricted field. The harmonic mean of a series of values is the reciprocal of the arithmetic Notes
mean of the reciprocals of the individual values. Reciprocal tables are used with ease for this. The
Harmonic Mean is less than the geometric mean of the same observations. The formula to calculate
harmonic mean is:
N
1 1 1 1
H.M. = + + + ...
X 1 X 2 X 3 X n
or, reciprocal of —

1 + 1 + 1 ... 1
X 1 X 2 X 3 X n
N

( )∑Reciprocal of X
or, reciprocal of
N
where X, X , X , ... X are the observations or the values of the series.
1 2 n
Merits of H.M.

(1) Harmonic mean is calculated by taking into account all the items of the series.
(2) In series with wide dispersion, this method is useful.
(3) It gives less weight to large items and more weight to small ones (because reciprocals are used).
(4) The method is useful in calculating rate.
(5) While calculating harmonic mean, the values get weight automatically and there is no need to
assign weights specifically.
Demerits of H.M.

(1) It is difficult to calculate.
(2) Difficult to be understood by common man.
(3) Harmonic mean cannot be calculated if even one item of the series is missing.
(4) The value of harmonic mean obtained may be a value which is no item in the given series.
Geometric Mean

Geometric mean of ‘n’ numbers is defined as the n root of the product of ‘n’ numbers. Symbolically,
th
()
G.M. = n ( )X ( 1 )X ( 2 3 ) . X ... X n
where X X ... X are the various values of the series.
1 2 n
‘n’ is the number of items. To make the calculations of finding out n root simpler, logarithms are
th
used. G.M. by using logs is found thus:
∑log X
G.M. = Antilog of .
N
Where, measuring the ratios of change is required, the geometric mean are used. It is also most
suitable when large weights have to be given to small items and small weights to large items, which
is usually required to study economic and social phenomena.

Merits
(1) Based on all the values of the series.
(2) Capable of further alzebraic treatment.

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