Applications of Arithmetic, Geometric, Harmonic, and Quadratic Means
Date: 11/22/2006 at 05:51:29 From: Irfan Subject: Application of Harmonic Mean I'm wondering when to apply the arithmetic, geometric and harmonic means for a certain data set. I have data that is widely spread. What technique will give me the correct results?
Date: 11/22/2006 at 13:46:41 From: Doctor Peterson Subject: Re: Application of Harmonic Mean Hi Irfan - The choice depends not so much on the spread of the data, but on their meaning, and specifically how the numbers naturally combine. The basic idea is that a mean is a number that can be used in place of each number in a set, for which the NET EFFECT will be the same as that of the original set of numbers. What determines which mean to use is the way in which the numbers act together to produce that net effect. For example, if you are looking for a mean amount of rainfall, you note that the total amount of rain, which affects crop growth, etc., is found by ADDING the daily numbers; so if you add them up and divide by the number of days, the resulting ARITHMETIC mean is the amount of rain you could have had on EACH of those days, to get the same total. If you have several successive price markups, say by 5% and then by 6%, and want to know the mean markup, you note that the net effect is to first MULTIPLY by 1.05 and then by 1.06, equivalent to a single markup of 1.05*1.06 = 1.113; taking the square root of this, if you had TWO markups of 5.499% each, you would get the same result. This is the GEOMETRIC mean. In general, you use it where the product is an appropriate "total"; another example is when you combine several enlargements of a picture. If you want the mean speed of a car that goes the same distance (not time!) at each of several speeds, then the net effect of all the driving (the total time taken) is found by dividing the common distance by each speed to get the time for that leg of the trip, and then adding up those times. The constant speed that would take the same total time for the whole trip is the HARMONIC mean of the speeds. This amounts to the reciprocal of the arithmetic mean of the RECIPROCALS of the individual speeds. In general, we use the harmonic mean when the numbers naturally combine via their reciprocals. Another example is combining resistances in a parallel electrical circuit. The quadratic mean (which you didn't mention--also called "root-mean- square") is used in situations where it is the SQUARE of the values that matters; for example, electrical current squared is proportional to power, so where it is the total power (energy) that matters, you need the quadratic mean. In summary, you use the ... arithmetic mean when numbers just add up ... geometric mean when numbers multiply together ... harmonic mean when the reciprocals of the numbers add up ... quadratic mean when the squares of the numbers add up to produce the net effect you are interested in. Here is another explanation: Arithmetic vs. Geometric Mean http://mathforum.org/library/drmath/view/52804.html If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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