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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.

1. It is rigidly defined.
2. It is based on all the observations.
3. 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).
4. It is not affected much by fluctuations in sampling.
5. 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.
6. It is useful for calculating average ratios and percentages and in determining rates of increase or decrease.

1. Because of its abstract mathematical character, the geometric mean is not easy to understand and interpret and hence it is not widely used.
2. If any one of the observations is zero the geometric mean becomes zero no matter how large the other values may be.
3. 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:

1. The geometric mean is used to find the rate of population growth and the rate of interest.
2. It is also used in the construction of index numbers.
3. 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%
4. 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:

1. 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.
2. 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.
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It is rigidly defined. It is based on all the observations. It is suitable for further mathematical treatment.

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