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Harmonic and Other Averaging
The applet below illustrates a property of means. The number bars are draggable up and down, while the exponent r can be changed by dragging the horizontal slider. Drag the number bars until one reads: 60, while the other reads: 30. Now slide the "exponent" cross-hair sideways to the left, or to the right, until the exponent reads: -1 or +1, respectively. When the exponent is +1, the mean is an arithmetic one. When the exponent is -1, the mean is harmonic. When the exponent is +2, the mean is a summation of squares. For finding the mean of a sum of squares (r = +2), try sliding the number bars until one of them reads 1 and the other reads 7. For adventure, try making the size of the set "3" within the little input box and then tap your keyboard's "Enter" key while your mouse cursor is still within it. If you are looking for the harmonic mean of three numbers (while the exponent is -1), then a couple of values to start playing around with, are: 90, 72, and 20. Have fun with this!
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Mean Formula
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(A1r + A2r + ... + Anr) / n = Br
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General Case....
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(3011 + 6021) / 2 = 451
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Arithmetic Mean of 60 and 30
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(301-1 + 602-1) / 2 = 40-1
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Harmonic Mean of 60 and 30
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(112 + 722) / 2 = 52
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Squared Mean of 1 and 7
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(201-1 + 722-1 + 903-1) / 3 = 40-1
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Harmonic Mean of 20, 72, and 90
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Next Lesson: Commentary and conjecture on the commonality of Golden Polynomials.
References:
The Harmonic Mean
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http://vinyasi.info/vinyasi/older-1st-version/book/harmonic_mean.shtml
Monday, 16 August 2004 15:30:00 MST
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