Geometric Mean

Geometric Mean - Definition:

It is defined as the nth root of the products of the n set of observations.

Formulas to be used:

The geometric mean G, of n observations y_i : i = 1to n is provided as:

G = (y1, yn)^1/n

Geometric Mean for ungrouped data:

The geometric mean for an ungrouped data can be calculated using the following formula:

G = Antilog (1 / n ∑ log y_i) where i = 1 to n

Geometric Mean for frequency distribution:

The geometric mean for the frequency distribution can be estimated using the given formula:

G = Antilog (1 / N ∑ f_i log y_i) where N = ∑ f_i, i = 1 to n

Geometric Mean for Continuous or Grouped data:

The geometric mean for the continuous or grouped data can be calculated using the below formula:

G = Antilog (1 / N ∑ f_i log y_i) where N = ∑ f_i, i = 1 to n and x takes the value corresponding to the mid-point of the class intervals.

Geometric Mean of the Combined Group:

Let n1 and n2 be the sizes and G1 and G2 be the geometric means of 2 series respectively. Then the geometric mean of the combined group is given by:

Log G = (n₁ log G₁ + n₂ log G₂)/( n₁ + n₂ )

(or)

G = Antilog [(n₁ log G₁ + n₂ log G₂)/ (n₁ + n₂)]

Merits of Geometric Mean:

The following are the merits of the geometric mean:

• It is strictly defined
• It is grounded upon all the values.
• It is worthy for further mathematical treatment.
• It is not affected by sampling fluctuations.
• It gives relatively more weight to small items. 

Uses of Geometric Mean:

• Geometric Mean is used to calculate the rate of interest and population growth.
• Geometric Mean is used in the construction of index numbers.