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SKewness and arithmetic and geometric means (reworking of Arthur Barnett's example

18-04-2017 21:38

In this reworking of Arthur Barnett's example, values that exceed the original average 0.5 value are the counterparts of the values that fall below it, e.g. 0.4 in the second iteration of the left skewed example becomes 0.625 in the second iteration of the right skewed example so that the geometric mean of the two values, 0.4 and 0.625 is 0.5. Note that this makes a big difference in the symmetric example, where the arithmetic mean shows higher and higher values with increasing variance but the geometric mean remains constant at 0.50. In the variant of the symmetric example, this distribution would be considered skewed to the right since the upper values differ from the median value by much more than the lower values do, however if one looks at the logs of the values there is symmetry. For example: ln(2.5)-ln(0.5)=ln(0.5)-ln(0.1)=ln(5).

#Jevons #Formulaeffect

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In this reworking, as the variance increases where there are lower and higher values around the median, it is the geometric mean that always reflects the median value, while the arithmetic mean drifts upward.