Most people are familiar with a traditional mean, or an arithmetic mean. We commonly call this the "average," or as my students often say, "you know, where you add all the numbers then divide by however many there are." But the mean is just one of many "averages" in the realm of math, with the more common arithmetic mean being one of three Pythagorean means, the other two being the geometric mean and the harmonic mean.
Without getting too math-y, here is the basic different between them.
- The arithmetic mean is found by adding your raw data, then dividing by the number of data pieces in your set.
- The geometric mean indicates the typical value of a set of numbers by using the nth root of their products.
- The harmonic mean is the reciprocal (think flipped) of the average (arithmetic mean) of the reciprocals (again, flipped) of each piece of data.
There is an appropriate time to use each one of these means in data analysis, depending on type of data in question. However, in all cases of data sets > 0, regardless of skew, the harmonic mean tends strongly toward the smaller numbers in a data set, so it mitigates the impact of large outliers and aggravates the impact of small outliers. As such, the harmonic mean will always be the lowest of the averages. Or, if you are more visual...
There is no shortage of theological extrapolation that can be done on the differences and similarities of the three Pythagorean means. If you really want to read more (or perhaps just be put to sleep...!), check this out. (Note: I am not in line with that author theologically, but did find the concept of his writing quite fascinating.) Or this. And for more about Pythagoras and his philosophy, check this out. (e.g. "abstain from beans, do not touch a white cock (ha! as in rooster, though...), men and women are equal, and property is communal," just to name a few tenets.) For those of you so inclined, I will leave this extrapolation to each of you for pondering, and I invite you to comment if you so choose.
