Question

The geometric mean of $10$ observations on a certain variable was calculated as $16.2$. It was later discovered that one of the observations was wrongly recorded as $12.9$; infact it was $21.9$. The correct geometric mean is:

• A. ${\left(\frac{(16.2{)}^9\times 21.9}{21.9}\right)}^{1/10}$
• B. ${\left(\frac{(16.2{)}^{10}\times 21.9}{21.9}\right)}^{1/10}$
• C. ${\left(\frac{(16.2{)}^{10}\times 21.9}{12.9}\right)}^{1/10}$
• D. ${\left(\frac{(16.2{)}^{11}\times 21.9}{21.9}\right)}^{1/11}$

Answer 1

The correct option is A ${\left(\frac{(16.2{)}^9\times 21.9}{21.9}\right)}^{1/10}$

Geometric mean of $n$ numbers $=({\prod }_{i=1}^n{x}_i{)}^{1/n}$

Here, $({\prod }_{i=1}^{10}{x}_i{)}^{1/10}=16.2$

$\Rightarrow \left({\prod }_{i=1}^{10}{x}_i\right)=(16.2{)}^{10}$

Now suppose $x_{10}$ was wrongly recorded, so we rewrite above relation as $\left({\prod }_{i=1}^9{x}_i\right)\times x_{10}=(16.2{)}^{10}$

$\Rightarrow \left({\prod }_{i=1}^9{x}_i\right)=\frac{(16.2{)}^{10}}{x_{10}}$

Now, the correct value is $21.9$, so multiply both sides by $21.9$ and also put value of $x_{10}=12.9$ in above equation

$\left({\prod }_{i=1}^9{x}_i\right)\times 21.9=\frac{(16.2{)}^{10}}{12.9}\times 21.9$

$=$ Correct Geometric mean=$\left(\right({\prod }_{i=1}^9{x}_i)\times 21.9{)}^{1/10}$

$={(\frac{(16.2{)}^{10}}{12.9}\times 21.9)}^{1/10}$

$={\left(\frac{(16.2{)}^{10}\times 21.9}{12.9}\right)}^{1/10}$

Hence, $\left(C\right)$ is correct.