Page 126 - DCOM203_DMGT204_QUANTITATIVE_TECHNIQUES_I

Unit 6: Measures of Central Tendency

Self Assessment Notes

State whether the following statements are true or false:
25. The geometric mean of a series of n positive observations is defined as the nth root of their
product.

26. In case of arithmetic mean, the sum of deviations of logarithms of values from the log GM
is equal to zero.

6.7 Harmonic Mean

The harmonic mean of n observations, none of which is zero, is defined as the reciprocal of the
arithmetic mean of their reciprocals.

6.7.1 Calculation of Harmonic Mean

Individual Series
If there are n observations X , X , ...... X , their harmonic mean is defined as
1 2 n
n n
HM
1 1 1 n
 1
X 1 X 2 X n X
1
i
i
Example: Obtain harmonic mean of 15, 18, 23, 25 and 30.
Solution:
5 5
HM = = = 20.92 Ans.
1 1 1 1 1 0.239
+ + + +
15 18 23 25 30
Ungrouped Frequency Distribution

For ungrouped data, i.e., each X , X , ...... X , occur with respective frequency f , f ...... f , where
1 2 n 1 2 n
f = N is total frequency, the arithmetic mean of the reciprocals of observations is given by
i
1 n f i
N i 1 X i .

N
Thus, HM
f
i
X i
Example: Calculate harmonic mean of the following data:

X : 10 11 12 13 14
f : 5 8 10 9 6

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