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.