Is it possible to scale back the average of a logarithmic number sequence to the average of the original number sequence? If so, how?

Question

Assume you have a number sequence: 55,15,10,18,2. Then its average is 20 and the standard deviation: is 18.3.

If this sequence is logarithmized its values are: 4,2.3,2.7,2.9,0.7. With a mean of 2.5 and a sd of 1.1.

If one only has the logarithmized mean (2.5) and sd (1.1), can it be scaled back to 20 and 18.3?

Answer 1

No - it is not possible to recover the original mean of $20$ and standard deviation about $18.3$.

A couple of examples which have the same means of the logarithms about $2.52$ and standard deviations of the logarithms about $1.07$ as your original data:

• either $106.555,7.266,7.266,7.266,7.266$ with mean about $27.1$ and s.d. about $39.7$
• or $1.4505,21.27,21.27,21.27,21.27$ with mean about $17.3$ and s.d. about $7.9$