Amean is a quantity representing the "center" of a collection of numbers and is intermediate to the extreme values of the set of numbers.[1] There are several kinds ofmeans (or "measures ofcentral tendency") inmathematics, especially instatistics. Each attempts to summarize or typify a given group ofdata, illustrating themagnitude andsign of thedata set. Which of these measures is most illuminating depends on what is being measured, and on context and purpose.[2]
Thearithmetic 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 numbersx1,x2, ..., xn is typically denoted using anoverhead bar,.[note 1] If the numbers are from observing asample of alarger group, the arithmetic mean is termed thesample mean () to distinguish it from thegroup mean (orexpected value) of the underlying distribution, denoted or.[note 2][3]
Outside probability and statistics, a wide range of other notions of mean are often used ingeometry andmathematical analysis; examples are given below.
In mathematics, the three classicalPythagorean means are thearithmetic mean (AM), thegeometric mean (GM), and theharmonic mean (HM). These means were studied with proportions byPythagoreans and later generations of Greek mathematicians[4] because of their importance in geometry and music.
Thearithmetic mean (or simplymean oraverage) of a list of numbers, is the sum of all of the numbers divided by their count. 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:
Thegeometric 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:
Theharmonic mean is an average which is useful for sets of numbers which are defined in relation to someunit, as in the case ofspeed (i.e., distance per unit of time):
For example, the harmonic mean of the five values: 4, 36, 45, 50, 75 is
If we have five pumps that can empty a tank of a certain size in respectively 4, 36, 45, 50, and 75 minutes, then the harmonic mean oftells us that these five different pumps working together will pump at the same rate as five pumps that can each empty the tank in minutes.
AM, GM, and HM ofnonnegativereal numbers satisfy these inequalities:[5]
Equality holds if all the elements of the given sample are equal.
Indescriptive statistics, the mean may be confused with themedian,mode ormid-range, as any of these may incorrectly be called an "average" (more formally, a measure ofcentral tendency). The mean of a set of observations is the arithmetic average of the values; however, forskewed distributions, the mean is not necessarily the same as the middle value (median), or the most likely value (mode). For example, mean income is typically skewed upwards by a small number of people with very large incomes, so that the majority have an income lower than the mean. By contrast, the median income is the level at which half the population is below and half is above. The mode income is the most likely income and favors the larger number of people with lower incomes. While the median and mode are often more intuitive measures for such skewed data, many skewed distributions are in fact best described by their mean, including theexponential andPoisson distributions.
The mean of aprobability distribution is the long-run arithmetic average value of arandom variable having that distribution. If the random variable is denoted by, then the mean is also known as theexpected value of (denoted). For adiscrete probability distribution, the mean is given by, where the sum is taken over all possible values of the random variable and is theprobability mass function. For acontinuous distribution, the mean is, where is theprobability density function.[7] In all cases, including those in which the distribution is neither discrete nor continuous, the mean is theLebesgue integral of the random variable with respect to itsprobability measure. The mean need not exist or be finite; for some probability distributions the mean is infinite (+∞ or−∞), while for others the mean isundefined.
Thegeneralized mean, also known as the power mean or Hölder mean, is an abstraction of thequadratic, arithmetic, geometric, and harmonic means. It is defined for a set ofn positive numbersxi by
By choosing different values for the parameterm, the following types of means are obtained:
This can be generalized further as thegeneralizedf-mean
and again a suitable choice of an invertiblef will give
Theweighted arithmetic mean (or weighted average) is used if one wants to combine average values from different sized samples of the same population:
Where and are the mean and size of sample respectively. In other applications, they represent a measure for the reliability of the influence upon the mean by the respective values.
Sometimes, a set of numbers might contain outliers (i.e., data values which are much lower or much higher than the others). Often, outliers are erroneous data caused byartifacts. In this case, one can use atruncated mean. It involves discarding given parts of the data at the top or the bottom end, typically an equal amount at each end and then taking the arithmetic mean of the remaining data. The number of values removed is indicated as a percentage of the total number of values.
Theinterquartile mean is a specific example of a truncated mean. It is simply the arithmetic mean after removing the lowest and the highest quarter of values.
assuming the values have been ordered, so is simply a specific example of a weighted mean for a specific set of weights.
In some circumstances, mathematicians may calculate a mean of an infinite (or even anuncountable) set of values. This can happen when calculating the mean value of a function. Intuitively, a mean of a function can be thought of as calculating the area under a section of a curve, and then dividing by the length of that section. This can be done crudely by counting squares on graph paper, or more precisely byintegration. The integration formula is written as:
In this case, care must be taken to make sure that the integral converges. But the mean may be finite even if the function itself tends to infinity at some points.
Angles, times of day, and other cyclical quantities requiremodular arithmetic to add and otherwise combine numbers. These quantities can be averaged using thecircular mean. In all these situations, it is possible that no mean exists, for example if all points being averaged are equidistant. Consider acolor wheel—there is no mean to the set of all colors. Additionally, there may not be aunique mean for a set of values: for example, when averaging points on a clock, the mean of the locations of 11:00 and 13:00 is 12:00, but this location is equivalent to that of 00:00.
TheFréchet mean gives a manner for determining the "center" of a mass distribution on asurface or, more generally,Riemannian manifold. Unlike many other means, the Fréchet mean is defined on a space whose elements cannot necessarily be added together or multiplied by scalars.It is sometimes also known as theKarcher mean (named after Hermann Karcher).
In geometry, there are thousands of differentdefinitions forthe center of a triangle that can all be interpreted as the mean of a triangular set of points in the plane.[8]
This is an approximation to the mean for a moderately skewed distribution.[9] It is used inhydrocarbon exploration and is defined as:
where, and are the 10th, 50th and 90th percentiles of the distribution, respectively.
under the guidance of Clark Kimberling, an electronic encyclopedia of triangle centers (ETC) has been developed, it contains more than 7000 centers and many properties of these points
