Interpret Logarithmic Values like Mean and Std

Question

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 1

Let $x_i$ your observations. Then the (observed) arithmetic mean $\hat\mu_{\log}$ of the logarithm of those is $$ \frac{\sum_{i = 1}^n\log(x_i)}{n} = \frac{\log\left(\prod_{i = 1}^n x_i\right)}{n} = \log\left(\prod_{i = 1}^n x_i^{1/n}\right) $$ which is the logarithm of the (observed) geometric mean of the original values (i.e. $\exp(\hat\mu_{\log})$ is the geometric mean of the $x_i$).

The (observed) standard deviation of these values is $$ \hat\sigma_{\log} = \sqrt{\frac{\sum_{i = 1}^n(\log(x_i) - \hat\mu_{\log})^2}{n-1}} $$ but the $^2$ makes it a bit more difficult to transform the way we transformed $\hat\mu_{\log}$ above. However, $\exp(\hat\sigma_{\log})$ 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 $\frac12$ and exactly half are $2$, the geometric mean $\exp(\hat\mu_{\log})$ is $1$, and the "geometric standard deviation", $\exp(\hat\sigma_{\log})$ 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 $\exp(0.55)\approx 1.73$, and a standard multiplicative deviation from that mean of $\exp(0.69)\approx 1.99$. So what would usually be an interval of "mean plus or minus a standard deviation" now becomes $$ [1.73/1.99, 1.73\cdot1.99] = [0.87, 3.46] $$

Answer 2

You can interpret a logarithmic scale as the "order of magnitude" of the data. Assuming decimal logarithms,

• magnitude $-2\leftrightarrow 0.01$,

• magnitude $-1\leftrightarrow 0.1$,

• magnitude $0\leftrightarrow 1$,

• magnitude $1\leftrightarrow 10$,

• magnitude $2\leftrightarrow 100$

• ...

and all intermediates.

So you are estimating the central trend and the dispersion of the orders of magnitude.

A grocer will have low mean and deviation. A car seller, high mean and low deviation. A general store, medium mean and high deviation.

Note that the given values are unrealistic.