Arithmetic, Geometric and Harmonic Means
The three classical Pythagorean means are the arithmetic mean(AM), the geometric mean(GM), and the harmonic mean(HM).
The harmonic mean has the least value compared to the geometric and arithmetic mean: min<HM<GM<AM<max
Arithmetic Mean
The arithmetic mean is calculated by adding all of the numbers and dividing it by the total number of observations in the dataset.
For example: Arithmetic Mean => 4 + 10 + 7 => 21/3 => 7
The arithmetic works well when the data is in an additive relationship between the numbers, often when the data is in a ‘linear’ relationship which when graphed the numbers either fall on or around a straight line. But not all datasets establish a linear relationship, sometimes you might expect a multiplicative or exponential relationship and in those cases, arithmetic mean is ill-suited and might be misleading to summarize the data.
Geometric Mean
The geometric mean works well when the data is in an multiplicative relationship or in cases where the data is compounded; hence you multiply the numbers rather than add all the numbers to rescale the product back to the range of the dataset. The data is seen as a scaling factor and does not contain null or negative values. Geometric mean is used when the data is not linear and specifically when a log transformation of data is taken.
Suppose you invested $500 initially which yielded 10% return the first year, 20% return the second year and 30% return the third year. After three years, you have $500 * 1.1 * 1.2 * 1.3 = $858.00.
Whereas if you taking arithmetic mean, it’s 10+20+30 = 20% return on average per year, so after three years you would have $500 * 1.4 * 1.4 * 1.4 = $1372. As we can see, arithmetic mean overestimates earnings by nearly $514 which is not right since we applied an additive operation to a multiplicative process.
Investors usually consider using geometric mean over arithmetic mean to measure the performance of an investment or portfolio.
Harmonic Mean
Harmonic mean is used when we want to average units such as speed, rates and ratios.
Say for example: I drove at an speed of 60km/hr to Seattle downtown and returned home at a speed of 30km/hr and the distance from my house to Seattle is 20 miles. What was my average speed for the whole trip?
Harmonic Mean => 2 /(1/A + 1/B) = 2AB/(A+B)
In our case, A = 60 and B = 30. Therefore, Harmonic Mean = 40km/hr.
If you take arithmetic mean of the two speeds, it would be 45km/hr which is not correct.
Hence, choosing the right mean for the right process is crucial.
Thanks for reading!
