The geometric mean of defined as the Nth root of a product of N numbers or values. If there are two items we take the square root of their product to calculate the geometric mean.
Advantages/Merits of Geometric Mean:
- It is rigidly defined.
- It is based on all the observations.
- It is suitable for further mathematical treatment. If m and n are the sizes, G1 and G2 are the geometric means of two series respectively, then we can find the geometric mean G using the below formula.
- log(G) = m*log(G1)+n*log(G2)/(m+n).
- It is not affected much by fluctuations in sampling.
- It gives comparatively more weight to small items compared to the arithmetic mean. In fact, the geometric mean is always less than or equal to the arithmetic mean.
- It is useful for calculating average ratios and percentages and in determining rates of increase or decrease.
Disadvantages of Geometric Mean:
- Because of its abstract mathematical character, the geometric mean is not easy to understand and interpret and hence it is not widely used.
- If any one of the observations is zero the geometric mean becomes zero no matter how large the other values may be.
- If some of the observations are negative, it may sometimes not be possible to compute the Nth root of the product of the data values.
Uses of Geometric Mean:
- The geometric mean is used to find the rate of population growth and the rate of interest.
- It is also used in the construction of index numbers.
- It is used to find the average percentage increase in sales, production, or other economic or business series.
- For example, if the prices of a commodity increased by 5, 10, and 18 percent over a span of three years then the average annual increase is not 11% as given by the arithmetic mean.
- The annual average growth rate is instead equal to the geometric mean which is equal to 10.9%
- This average is most suitable when large weights have to be given to small items and small weights have to be given to large items.
Properties of Geometric Mean:
- The product of the values of the data will remain unchanged when the value of the geometric mean is substituted for each individual value.
- For example, if we consider the three values 5, 25, and 125 then their geometric mean is 25 and we have that, 5 x 25 x 125 = 15625 = 25 x 25x 25.
- The value of the geometric mean balances the ratio deviations of the observations from the geometric mean.
- The product of ratio deviations of values lower than the geometric mean is equal to the product of ratio deviations of values greater than the geometric mean.
- Considering the same numbers as above we see that, 25/5 x 25/25 = 5 = 125/25.