abbracciopj

2022-06-20

Interpret Logarithmic Values like Mean and Std
let's say i have a series of data containing prices on the log scale. How would i interpret a arithmetic mean of 0.55 and a std of 0.69 (both metrics are computed with the log prices).
Is there an intuitive explanation in terms of percentage change or anything similar?

### Answer & Explanation

timmeraared

Let ${x}_{i}$ your observations. Then the (observed) arithmetic mean ${\stackrel{^}{\mu }}_{\mathrm{log}}$ of the logarithm of those is
$\frac{\sum _{i=1}^{n}\mathrm{log}\left({x}_{i}\right)}{n}=\frac{\mathrm{log}\left(\prod _{i=1}^{n}{x}_{i}\right)}{n}=\mathrm{log}\left(\prod _{i=1}^{n}{x}_{i}^{1/n}\right)$
which is the logarithm of the (observed) geometric mean of the original values (i.e. $\mathrm{exp}\left({\stackrel{^}{\mu }}_{\mathrm{log}}\right)$ is the geometric mean of the ${x}_{i}$).
The (observed) standard deviation of these values is
${\stackrel{^}{\sigma }}_{\mathrm{log}}=\sqrt{\frac{\sum _{i=1}^{n}\left(\mathrm{log}\left({x}_{i}\right)-{\stackrel{^}{\mu }}_{\mathrm{log}}{\right)}^{2}}{n-1}}$
but the ${}^{2}$ makes it a bit more difficult to transform the way we transformed ${\stackrel{^}{\mu }}_{\mathrm{log}}$ above. However, $\mathrm{exp}\left({\stackrel{^}{\sigma }}_{\mathrm{log}}\right)$ does have an interpretation as a measure of deviation from the (geometric) mean, not in terms of difference but in terms of ratio.
When you have a lot of observations where exactly half of them are −1 and exactly half of them are 1, then the arithmetic mean is 0, and the standard deviation is a little larger than 1 (it tends to 1 as the number of observations grows), which is the observed deviation from the arithmetic mean in all cases.
In exactly the same way, for a long list of observations where exactly half are $\frac{1}{2}$ and exactly half are 2, the geometric mean $\mathrm{exp}\left({\stackrel{^}{\mu }}_{\mathrm{log}}\right)$ is 1, and the "geometric standard deviation", $\mathrm{exp}\left({\stackrel{^}{\sigma }}_{\mathrm{log}}\right)$ is a little larger than 2 (and tends to 2 as the number of observations grows), since all the observations deviate from the geometric mean by a factor of 2.
In your case, this means that you have observed a geometric mean of $\mathrm{exp}\left(0.55\right)\approx 1.73$, and a standard multiplicative deviation from that mean of $\mathrm{exp}\left(0.69\right)\approx 1.99$. So what would usually be an interval of "mean plus or minus a standard deviation" now becomes
$\left[1.73/1.99,1.73\cdot 1.99\right]=\left[0.87,3.46\right]$

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Recalculate according to your conditions!