You have already studied about arithmetic mean which belongs to the category of mathematical averages. Now, you will study about the two other mathematical averages viz, Geometric Mean and Harmonic Mean.

**Uses and Limitations Geometric Mean**

**Uses: **

1) For computing the averages of ratio and percentages,
geometric mean is the most suitable average.

2) As it has bias towards lower values, it is particularly
useful when a given phenomenon has a limit for lower values but no such limit
for upper values. For example, price
cannot be below zero.

3) In the construction of index numbers, geometric mean is
considered to be the best average. It is
especially used in developing Fisher’s Ideal Formula that satisfies time
reversal and factor reversal tests. (The
study of these concepts is beyond the scope of this course.)

4) When large weights are desired to be assigned to small
items and small weights are to be assigned to large items, it is a more
suitable average than arithmetic mean.

**Limitations:
**

1) Even if the single item of the given series is zero,
geometric mean will be zero. Hence, it cannot
be computed. For example, geometric mean
of the three items 0, 10, 100 will be: √0 × 10 × 100 = 0.

2) If any of the items is negative, geometric mean does not
exist.

3) The computational procedure is difficult especially when
the items are very large.

4) Its bias for lower values obstructs its use in the
situations where disparities are to be highlighted as the case of income
distributions.

**Properties of Harmonic Mean **

1) If each value of the narrate is replaced by harmonic mean,
the total of reciprocals of values of the narrate remains the same.

2) Harmonic mean is the reciprocal of the arithmetic mean of
the reciprocals of the individual observations.

3) Like arithmetic mean and geometric mean, it lends itself to
further algebraic treatment.

4) Amongst the three means (viz., arithmetic mean, harmonic
mean and Tendency

geometric mean), harmonic mean is the least i.e., AM ≥ GM ≥
HM.

**Uses and Limitations of Harmonic Mean**

**Uses: **

1) For the rates and ratios involving speed, time and
distance, harmonic mean is used to find out the average speed.

2) For the rates and ratios involving price and quantity (both
amount of money spent and the units per rupee are given), harmonic mean is
used. In general, if reciprocals of items are used in obtaining their combined effect,
harmonic mean is to be used for averaging them.

3) In a given data set if there are a few large values, the
reciprocals will tone down the effect of large numbers. In such cases harmonic
mean is to be used.

4) When it is desired to assign greater weight to smaller
values and smaller weight to larger values of a variate, its use is
recommended.

**Limitations: **

1) It is difficult to compute and understand.

2) It cannot be computed when one or more items are zeros. In
fact in such a case HM will be always zero whatever may be the value of other
items.

3) To assign the largest weight to the smallest item, it is
not always a desirable feature and has a limited scope in the analysis of
economic data.

**Harmonic Mean Versus Arithmetic Mean **

In order to derive averages of the rates and ratios (that
involve speed, time and distance or price, quantity and amount of money spent,
etc.) making a choice between the harmonic mean and arithmetic mean is not very
easy. In some situations harmonic mean seems to be more proper, whereas in
other situations harmonic mean is found more suitable to derive the correct
answer. Such a choice mainly depends on the nature of the data. Based on it,
some general guidelines for a judicious choice can be prescribed.