Statistics – Measures of Central Tendency – Geometric Mean & Harmonic Mean

Geometric Mean –

Geometric Mean of a set of n observations is the nth root of their product. It is denoted by G.M.

If we have two observations, then their G.M. will be the square root of their product.

If we have three observations, then their G.M. will be the cube root of their product. And so on……

GM has three types of series, which are

Individual Series –

Discrete Series –

Continuous Series –

For example – G.M. of 3 and 12

Product of 3 and 12 = 36

No. of observation = n = 2

Then G.M. = square root of 36 = 6

In case of n>2, calculations are done by the Logarithms –

Properties of Geometric Mean

1. Geometric Mean is less than the Arithmetic Mean. [ G.M. < A.M. ]
2. The G.M. of the ratios of the corresponding observations in a series is equal to the ratios of their G.M.
3. The product of all observations will remain the same if each observation is replaced by the G.M.
4. It is mostly used in case of Index No. Ratio & Proportion etc.

Weighted Geometric Mean –

Example –

Combined Geometric Mean –

Merits of G.M.

1. It is based on all observations.
2. It is rigidly defined.
3. It gives less weight to the bigger or larger observation.
4. It is never greater than A.M.
5. Further mathematical operations can be done.
6. It is the best measure of ratios of change.
7. It is least affected by the extreme observations.

Demerits of G.M.

1. It is difficult to calculate.
2. It is not suitable for open end distributions.
3. It any of the value is negative, then GM becomes indeterminate.
4. If any of the value is zero, the G.M. will also become zero.
5. It can’t be calculated unless all of the observations are known.
6. It may be a value which does’t exist in the series.

Harmonic Mean –

Harmonic Mean is the reciprocal of Arithmetic Mean of the reciprocal of all observations in a series. It is denoted by H.M. –

H.M. has three types of series like A.M. which are

Individual Series –

Example –

Discrete Series –

Example –

Continuous Series –

Example –

Weighted Harmonic Mean –

Example –

Merits of H.M.

1. It is rigidly defined.
2. It is based on all the observations.
3. It is a mathematical average.
4. Further mathematical operations can be done in case of H.M. data.
5. It is applicable in case of quantitative data.

Demerits of H.M.

1. It is difficult to understand & calculate.
2. It’s value can’t be obtained if any one of the observation is zero.
3. It gives larger weight to the smallest observations.

Relationship between A.M., G.M., and H.M. –

1. For any two positive numbers GM = Squared root of AM*HM
2. If all observations are the same then, AM = GM = HM,
3. If not then, AM> GM> HM.