Geometric Mean

The set of n positive observations, the G.M. is defined as the n-th root of the product of the observations . If a variable x assumes n values x₁ , x₂ , x₃ , ---------- x_n (all observation should be positive) , then the G.M. of x is given by

G = ( x₁ . x₂ . x₃ . ---------- x_n)^1/n (for Raw data)

G = ( x₁^f₁ . x₂^f₂ . x₃ ^f₃ . ---------- x_n^f_n)^1/n (for ungrouped data)

G = ( x₁^f₁ . x₂^f₂ . x₃ ^f₃ . ---------- x_n^f_n)^1/n (for grouped data)

This formula is same as ungrouped data in this x_i = mid - point .

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Properties of G.M

(1) If z = x.y , then G.M of z = (G.M. of x) . (G.M. of y)

(2) If z =

x/y

, then G.M of z =

G.M of x/G.M of y

(3) Logarithm of G for set of observations is the A.M of the logarithm of the observations .

log(GM) =

logx₁ + log x₂ + ---------- logx_n/ n

or

G.M = antilog (

∑logx_i/n

)

G.M = antilog (

∑f_ilogx_i/N

)

x_i = Mid points of class intervals .

(4) If all the observation of a data are equal , say k , (k>0) , then the G.M of the data will also be k .

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Weighted Geometric Mean

When all the n individual observations in a set of data are not equally important . It is denoted as G.M_w

GM_w = antilog (

w₁logx₁ + w₂logx₂ + --------- + w_ilogx_i /w₁ + w₂ + --------- + w_i

)

or

GM_w = antilog (

∑w_ilogx_i /∑ w_i

)

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Combined Geometric Mean (GM_c)

GM_c = antilog (

n₁logGM₁ + n₂logGM₂ + --------- + n_klogGM_k /n₁ + n₂ + --------- + n_k

)

or

GM_c = antilog (

∑n_ilogGM_i /∑ n_i

)

Note :- G.M is specifically useful in averaging ratios , percentages and rates of change in one period over the other .