Log of Average v. Average of Log I am constructing a dataset of monthly averages based on daily data. This dataset will be used for standard regression analysis. I anticipate wanting to transform the dependent variable, which has an approximately log-normal distribution. My question is whether it is more appropriate to transform the data before or after taking the monthly average.
 A: If you maintain the assumption that the daily dependent variable $Y_{ji}$ of month $i$ follows a log-normal distribution, this means that
$$\ln Y_{ji} \sim \mathbf N (\mu_{ji}, \sigma_{ji}^2)$$
Then, denoting $d_i$ the number of days of month $i$, we also have
$$ \frac {1}{d_i} \ln Y_{ji} \sim \mathbf  N\left(\frac {\mu_{ji}}{d_i}, \frac  {\sigma_{ji}^2}{d_i^2}\right)$$
If you also maintain the assumption that your sample is comprised of independent observations, the sum of independent normal random variables is certain to also follow a normal distribution and so
$$\sum_{j=1}^{d_i}\frac {1}{d_i} \ln Y_{ji} =\frac {1}{d_i} \sum_{j=1}^{d_i}\ln Y_{ji}  \sim N\left(\frac {1}{d_i} \sum_{j=1}^{d_i}\mu_{ji}, \frac {1}{d_i^2} \sum_{j=1}^{d_i} \sigma_{ji}^2\right)$$
In words, if a log-normality assumption is stated at the level of a sample of independent daily data, then the monthly average of the logs of the original daily variables (their geometric mean, as a comment mentioned) will also be normally distributed.
A: The question is what is log-normally distributed?
I'm assuming it's monthly series. In this case get the average, then log. 
If you thought that the daily series are log-normally distributed, then your average monthly series would be very close to normal distribution if there is no large autocorrelation.
