If G be the G.M. of the product of k, set of observations with G.M.'s $G_{1}$ , $G_{2}$ , $G_{3}$ , ..., $G_{k}$ respectively, then G equals

log G_{1} + log G_{2} + . . . + log G_{k}

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G_{1} G_{2} G_{3} . . . . . . . . . . G_{k}

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log G_{1} log G_{2} log G_{3} . . . log G_{k}

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k log (G_{1} G_{2} G_{3} G_{k})

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Solution

The correct option is B $G_{1} G_{2} G_{3} . . . . . . . . . . G_{k}$
If $x_{1}$ , $x_{2}$ , $x_{3}$ , ..., $x_{k}$ be the $k$ variables and $x = x_{1} x_{2} . . . x_{k}$ denote their product.

$∴$ $log x = log x_{1} + log x_{2} + . . . + log x_{k}$
$∴$ $\sum log x = \sum log x_{1} + \sum log x_{2} + . . . + \sum log x_{k}$
$∴$ $\frac{\sum log x}{n} = \frac{log x_{1}}{n} + \frac{\sum log x_{2}}{n} + . . . + \frac{\sum log x_{k}}{n}$
$∴$ $log G = log G_{1} + log G_{2} + . . . + log G_{k}$
$\Rightarrow G = G_{1} G_{2} G_{3} . . . G_{k}$

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