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Concept
Mean
Summary
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set.
For a data set, the arithmetic mean, also known as "arithmetic average", is a measure of central tendency of a finite set of numbers: specifically, the sum of the values divided by the number of values. The arithmetic mean of a set of numbers x1, x2, ..., xn is typically denoted using an overhead bar, . If the data set were based on a series of observations obtained by sampling from a statistical population, the arithmetic mean is the sample mean () to distinguish it from the mean, or expected value, of the underlying distribution, the population mean (denoted or ).
Outside probability and statistics, a wide range of other notions of mean are often used in geometry and mathematical analysis; examples are given below.
Pythagorean means
The arithmetic mean (or simply mean) of a list of numbers, is the sum of all of the numbers divided by the number of numbers. Similarly, the mean of a sample , usually denoted by , is the sum of the sampled values divided by the number of items in the sample.
For example, the arithmetic mean of five values: 4, 36, 45, 50, 75 is:
The geometric mean is an average that is useful for sets of positive numbers, that are interpreted according to their product (as is the case with rates of growth) and not their sum (as is the case with the arithmetic mean):
For example, the geometric mean of five values: 4, 36, 45, 50, 75 is:
The harmonic mean is an average which is useful for sets of numbers which are defined in relation to some unit, as in the case of speed (i.e., distance per unit of time):
For example, the harmonic mean of the five values: 4, 36, 45, 50, 75 is
Inequality of arithmetic and geometric means
AM, GM, and HM satisfy these inequalities:
Equality holds if all the elements of the given sample are equal.
Official source
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