Mean

A mean is a quantity representing the "center" of a collection of numbers and is intermediate to the extreme values of the set of numbers. There are several kinds of means (or "measures of central tendency") in mathematics, especially in statistics. Each attempts to summarize or typify a given group of data, illustrating the magnitude and sign of the data set. Which of these measures is most illuminating depends on what is being measured, and on context and purpose.

The arithmetic mean, also known as "arithmetic average", is the sum of the values divided by the number of values. The arithmetic mean of a set of numbers x₁, x₂, ..., x_n is typically denoted using an overhead bar, ${\displaystyle {\bar {x}}}$. If the numbers are from observing a sample of a larger group, the arithmetic mean is termed the sample mean (${\displaystyle {\bar {x}}}$) to distinguish it from the group mean (or expected value) of the underlying distribution, denoted ${\displaystyle \mu }$ or ${\displaystyle \mu _{x}}$.

Outside probability and statistics, a wide range of other notions of mean are often used in geometry and mathematical analysis; examples are given below.

In mathematics, the three classical Pythagorean means are the arithmetic mean (AM), the geometric mean (GM), and the harmonic mean (HM). These means were studied with proportions by Pythagoreans and later generations of Greek mathematicians because of their importance in geometry and music.

The arithmetic mean (or simply mean or average) of a list of numbers, is the sum of all of the numbers divided by their count. Similarly, the mean of a sample ${\displaystyle x_{1},x_{2},\ldots ,x_{n}}$, usually denoted by ${\displaystyle {\bar {x}}}$, 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: