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Sunday, 28 April 2013

Average


Average


In colloquial language average usually means the sum of a list of numbers divided by the size of the list, in other words the arithmetic mean. However it can alternatively mean the median, themode, or some other central or typical value. In statistics, these are all known as measures of central tendency.

Pythagorean means

The three most common averages are the Pythagorean means – the arithmetic mean, the geometric mean, and the harmonic mean.


Geometric mean can be thought of as the antilog of the arithmetic mean of the logs of the numbers.


Harmonic mean
reciprocal of the arithmetic mean of the reciprocals of the ai's:
HM = \frac{1}{\frac{1}{n}\sum_{i=1}^n \frac{1}{a_i}}=\frac{n}{\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}}.

AM \ge GM \ge HM. \,



The mode has the advantage that it can be used with non-numerical data (e.g., red cars are most frequent), while other averages cannot.

 File:Comparison mean median mode.svg
Comparison of arithmetic mean, median and mode of twolog-normal distributions with different skewness.
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