Question: On what occasions should geometric averages be used, rather than arithmetic averages?!

Question: On what occasions should geometric averages be used, rather than arithmetic averages?!

Question: On what occasions should geometric averages be used, rather than arithmetic averages?!

(7.6 Day Note, perhaps the topic is changed to what occasion use algebra (geometry,... ) are the average values appropriate? more appropriate)

People on many occasions (student performance statistics, socio-economic statistics and scientific experiments , etc...) ) for the statistical calculation of the average (the sum of the n samples is first calculated by the number of samples n ). This mean is called an algebraic (or arithmetic) average. In fact, mathematics also recommended a geometric mean (it is the product of the sample value and then open the N -th square), and even the harmonic mean (the inverse of the sample value of the algebraic average).
In an age of no calculators or computers, it is natural that algebraic averages are the easiest to calculate, and people are accustomed to using algebraic averages. But now that computers are so popular, is it customary to use algebraic averages, or does it satisfy some kind of theoretical requirement?
On what occasions should the geometric mean be counted instead of the arithmetic mean?

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Commented 1:zhaoxing [2010-7-5 16:59:41]

has been puzzled by this problem for a long time, the book of statistics has seen more than 10 20 kinds, but none of them have been clear about the problem.
Some say that geometric averages apply to time series data, and some say it applies to growth or rate of change calculations, but none of them say why.
From the algorithm, the personal sense of geometric average may be more " smooth " some, in the case of extreme values in the sample when the robustness of better.

Commented 2:lix [2010-7-5 18:05:19]

This question Zhang teacher is the expert, I calculate answer Zhang teacher's class question. I understand that the average is a simplified description of a distribution. This simplification is sure to lose some information. Then when to use the average, depending on what you are going to do with the average. For example, to use the average income of a certain resident to calculate its average happiness index? Since tens of millions of individuals are not necessarily more than 100,000 happier than the revenue of the revenues (subsequent calculations are non-linear), it is possible to consider using geometric averages to depress the overstating effect of tens of millions of individuals on the average happiness index. In general, consider whether the distribution itself is close to the normal distribution or the logarithm of the distribution is closer to the normal distribution. But there is no such thing as a fixed principle.

The following is the answer of the author in the first 2 days (2010.7.6): Understanding of the different averages

Thank Zhaoxing Zhaoshing teacher and Li Xiaowen Teacher's attention and the publication of the understanding. I would now like to add the following information:

1. algebraic mean is often used in many theoretical analyses, it is easy to calculate and easy to understand, and there is nothing wrong with it. After all, I do statistics, I am the boss.

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4. If you have these points of attention, why do you also calculate the geometric mean? Note that some of the variables themselves naturally have the impossibility of 0, can not take negative nature, seize this feature, may make it easier to see through its law. The kinetic energy of an object, the age of a person, the wealth of a person, the value of a percentage, and so on, can only take positive values. These characteristics, sometimes need to be grasped and used rather than neglected.

5. using the maximum information entropy (which I call the most complex), only with the algebraic mean to determine the invariant, can be pushed to export the probability density of the random variable can only obey the negative exponential distribution. With this knowledge tip, it is natural for you to count the algebraic averages. And you will find that the algebraic average of the other batch of samples is almost the same as the first, while the geometric average is different.

6. In addition, the use of information entropy maximum, only with the geometric mean (not algebraic average!) ) to determine the invariant, the probability density of the random variable can be pushed to the distribution of power rate only. With this knowledge tip, it is natural for you to count the geometric averages. It is also found that the geometric average of the other batch of samples is almost the same as the first 1 batches, while the algebraic mean is different. Yes, now the power distribution is very fashionable in fractal studies, why are some fractal phenomena satisfying the power distribution? Because the variable geometric mean of the system is conservative (unchanged), and satisfies the maximum entropy (most chaotic, most complex).

7. The preceding two paragraph shows that the randomness in the system embodies the maximum information entropy, and there is only one constraint, if the condition is (and only) the algebraic mean is not changed to the distribution of negative exponential distribution, if it is (and only) the geometric average is not changed, then the power rate is met.

8. If a variable does not reflect a negative exponential distribution, it is not a power rate, but a so-called Gamma distribution, which corresponds to the statistical characteristics of the description? The answer is: The system embodies the most information entropy at the same time (it can be exaggerated that this is the embodiment of the second law of thermodynamics, I used this mysterious law!) is constrained by two (and only two) conditions: the algebraic mean of the variable and (also) the geometric mean is also invariant (the two averages do not need to be the same, but the geometric average is necessarily less than the algebraic average). That is, the probability distribution is no longer negative exponent or power rate but gamma distribution. At this point, you will find that taking two batches of different large samples, the algebraic mean of both sides should be equal, and the geometric average of the two sides is equal. The algebraic mean and geometric average of the variables here are equally important. And these realizations provide a basis for you to explain from the theory why this distribution happens. The selection of the appropriate average value may be a reasonable springboard for theoretical analysis.

9. These understandings come from the most systematic analysis of the distribution and complexity in the composition theory. Here is the donuts. The above considerations are consistent with the Li Xiaowen Teacher's consideration of distribution issues.

What is the average of the hundred meter grades of the 3 grade students in junior high school? Here you need to ask the hundred meter results in seconds, to find the average, or to calculate the average speed. Be aware that speed is the inverse of the current number of seconds to calculate the score of hundred meters! Are you the average of the statistical speed or the inverse of the statistical velocity? This is linked to the use of harmonic averages in statistics. If you are more likely to get a theoretical explanation from the point of view of the harmonic mean (including the distribution), then the harmonic mean should be counted. It all depends on the subsequent analysis needs to see which averages are the right ladder for you to step on.

One . combined with the current is nearly tens of thousands of students college entrance examination, its students score if the normal distribution, statistical algebraic average value can be. If it conforms to the Gamma distribution (biased), I think it is necessary to analyze the algebraic mean, the geometric mean, and give the theoretical explanation (where the theoretical conclusion is prepared) with the constant of the algebraic mean and the invariant of the geometric mean.

The above description may not be suitable for further discussion.

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Question: On what occasions should geometric averages be used, rather than arithmetic averages?!