Interpret Logarithmic Values like Mean and Stdlet's say i have a series of data containing...
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
Beginner2022-06-21Added 22 answers
Let your observations. Then the (observed) arithmetic mean of the logarithm of those is
which is the logarithm of the (observed) geometric mean of the original values (i.e. is the geometric mean of the ). The (observed) standard deviation of these values is
but the makes it a bit more difficult to transform the way we transformed above. However, 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 and exactly half are 2, the geometric mean is 1, and the "geometric standard deviation", 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 , and a standard multiplicative deviation from that mean of . So what would usually be an interval of "mean plus or minus a standard deviation" now becomes
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