# I can't think how to approach this statistics problem?

A set of numbers is transformed by taking the log base 10 of each number. The mean of the transformed data is 1.65. What is the geometric mean of the untransformed data?

 Approach used so far:
i) So basically a * b * c ...*tn = 10^(1.65*n) where a,b,c and so on are the values in the data set taken.
ii) I have to find (a * b * c ... *tn^1/2 = 10^(1.65/2*n)


Lets define the N numbers in your set with $$x_i$$, where $$i \in \{1,...,N \}$$.
Since $$\sum_{i=1}^{N} \log_{10}x_i$$ = $$\log_{10}(\prod_{i=1}^{N} x_i)$$ and the mean of the transformed data is 1.65, it follows that $$$$\frac{\log_{10}(\prod_{i=1}^{N} x_i)}{N} = 1.65 \Leftrightarrow \prod_{i=1}^{N} x_i = 10^{N \cdot 1.65}$$$$ Taking by the power of $$\frac{1}{N}$$ on both sides yields you the geometric mean of the untransformed data: $$$$\left(\prod_{i=1}^{N} x_i\right)^{1/N} = 10^{1.65}$$$$