# Calculate mean/stddev on log and real scales

1. Feb 19, 2015

### gummz

1. The problem statement, all variables and given/known data
The logarithm of an object changes between days according to a normal distribution with median 0,00065 and std dev 0,0038. Calculate the mean and std dev after 249 days for the real and logarithmic scales.

2. Relevant equations
Standard standard deviation as far as I am aware. This type of problem is unbeknownst to me and my textbook.

3. The attempt at a solution
I know the formula for the real scale, but I don't know what they mean by a logarithmic scale. Do they mean to just take log() of the std dev for the real scale?

2. Feb 19, 2015

### Ray Vickson

If $X$ is the variable on the real scale and $Y$ is on the logarithmic scale, they just mean that $Y = \ln X$ (assuming natural logs). After that, the question is ambiguous, because it is poorly worded. One interpretation would be that $Y_i$ = change of log on day $i$, so that the final log after $N$ days would be $W = \sum_{i=1}^N Y_i$. Then, of course, it matters if the $Y_i$ are independent or not, and you did not say whether that is the case. Assuming it IS the case, computing the mean and standard deviation of $W$ is straightforward because you are told the mean and standard deviation of each $Y_i$ separately. In the non-logarithmic scale, the final after $N$ days would be $U = X_1 X_2 \cdots X_N = \prod_{i=1}^N e^{Y_i}$, because $\ln U = W = \sum Y_i = \sum \ln(X_i).$. Remember: that is just my interpretation of a not-well-stated question.

As for the rest: Google 'lognormal distribution'.