March 12, 2014

Not Your Average Mean

Forecasting is all about what's likely to happen, and what's likely to happen is usually an average (aka "mean").  Life would be pretty sweet if there were only one way to calculate a mean.  But there's not -- and if you use the wrong one, your forecast will be wrong.

Sadly, I speak from experience:  I used the wrong average for several months, and couldn't figure out why my results were always overstated.

Which one to use depends upon the type of data you're averaging.  
  •  If you're measuring the same thing fluctuating over different periods (e.g., month-over-month growth rates of a stock), then your variables are dependent, and you need to use the geometric mean.   This is sometimes referred to as a compound average.
To calculate the geometric mean, multiply together the growth rates and take the root of the number of elements.  (Note:  Express them all as a percentage of the previous value -- hence, if a value decreased by half, represent it as .5, and if it doubled, represent it as 2.)
Geometric Mean
  • If you're measuring truly different elements (such as growth rates for a bunch of different stocks), then your variables are independent, and if the values are more-or-less evenly distributed, then you're best off using the arithmetic mean: the "classic" average that we all used in school.
The arithmetic mean is the sum of all values, divided by the number of values.
Arithmetic Mean
  • If you're measuring different elements (i.e., independent variables) but you have crazy outliers that can skew your results (which is most common when averaging rates), you should use the harmonic mean. 
This is the number of values divided by the the sum of the reciprocal of each value.  (<phew!>)
Geometric Mean

So, let's say that you're trying to figure out the average return on six stocks, where three decreased by 50% and three increased by 50%.   It would be (.5 + .5 + .5 + 1.5 + 1.5 + 1.5)/6 = 1.  (Right back to where you started)!

...However, let's say that you were trying to calculate the return on a single stock over six periods, where it decreased by 50% three times, and increased by 50% three times.  In that case, it would be (.5 * .5 * .5 * 1.5 * 1.5 * 1.5) ^ (1/6) =  .86.   ...In other words, an average drop of 14% per period.  

Finally, let's say you were taking the average of some internet traffic rates (which are measured in bits per second):  5 Mbps, 8 Mbps, 20 Mbps, 25 Mbps and 5000 Mbps.   Your (harmonic) average would be 5 / (1/5 + 1/8 + 1/20 + 1/25 + 1/5000)  = 12.04Mbps. 

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