Mean

In mathematics, an average of a collection or group is a value that is most central, common, or typical in some sense, and represents its overall position. In mathematics, it most commonly refers to the arithmetic mean, but may also refer to other measures such as other types of mean, the median, or the mode.

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 most commonly used definition of the average is the arithmetic mean, also known as "arithmetic average" i.e. the sum divided by the count, so the "average" of the list of numbers [2, 3, 4, 7, 9] is generally considered to be (2+3+4+7+9)/5 = 25/5 = 5. The arithmetic mean of a set of numbers x₁, x₂, ..., x_n is typically denoted using an overhead bar, \(\bar{x}\). If the numbers are from observing a sample of a larger group, the arithmetic mean is termed the sample mean (\(\bar{x}\)) to distinguish it from the group mean (or expected value) of the underlying distribution, denoted \(\mu\) or \(\mu_x\).

However, other meanings are sometimes used depending on the context, which can lead to confusion; for instance, in teaching, "average" sometimes refers to "the three Ms": mean, median, and mode.

The median, defined as the value in the center after sorting the group, is usually used as the average in situations where the data is skewed or has outliers, in order to focus on the main part of the group rather than the long tail. For example, the average personal income is usually given as the median income, so that it represents the majority of the population rather than being overly influenced by the much higher incomes of the few rich people.

The harmonic mean, defined as the reciprocal of the mean of the reciprocals, is used in a variety of situations involving rates or ratios, such as computing the average speed from multiple measurements taken over the same distance. Indeed, unlike an arithmetic mean or median of speeds, a harmonic mean of speeds will give the value of the constant speed that would cause one to travel the same distance in the same amount of time.

The mode represents the most common value found in the group. It can be used when the data is categorical rather than numeric, when the frequency of each value is relevant (such as where a histogram, bar chart, or probability density function is being referenced), or to find a value that represents the majority of the group.

Other statistics that can be used as an average include the mid-range, the quadratic mean or the geometric mean, but they are rarely referred to as "the average".

In statistics, a central tendency (or measure of central tendency) is a central or typical value for a probability distribution. Informally, measures of central tendency are often called 'averages'. The term central tendency dates from the late 1920s.

The most common measures of central tendency are the arithmetic mean, the median, and the mode. A middle tendency can be calculated for either a finite set of values or for a theoretical distribution, such as the normal distribution. Occasionally authors use central tendency to denote "the tendency of quantitative data to cluster around some central value."

The central tendency of a distribution is typically contrasted with its dispersion or variability; dispersion and central tendency are the often characterized properties of distributions. Analysis may judge whether data has a strong or a weak central tendency based on its dispersion.

All averages of a collection are somewhere within its bounding box (and so for real numbers, between its maximum and minimum). Therefore, if a collection consists entirely of the same value, any average of it is that value.

Most averages are monotonic, i.e. moving a member of it in one direction causes the average to move in the same direction, or equivalently, if two collections of numbers A and B have the same number of elements, and they can be arranged such that each entry in A ≥ the corresponding entry in B, then the average of A ≥ the average of B.

All commonly-used averages are linearly homogeneous, i.e. multiplying every value by the same scale factor multiplies the average by that same scale factor.

Most averages remain identical when the list of items is permuted, i.e. the ordering does not matter.

In descriptive statistics, the mean may be confused with the median, mode or mid-range, as any of these may colloquially be called an "average" (more formally, a measure of central tendency). The mean of a set of observations is the arithmetic average of the values; however, for skewed 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 the exponential and Poisson distributions.

For unimodal distributions the following bounds are known and are sharp:

\(\frac{| \theta - \mu |}{ \sigma } \le \sqrt{ 3 } ,\)

\(\frac{| \nu - \mu |}{ \sigma } \le \sqrt{ 0.6 } ,\)

\(\frac{| \theta - \nu |}{ \sigma } \le \sqrt{ 3 } ,\)

where μ is the mean, ν is the median, θ is the mode, and σ is the standard deviation.

For every distribution,

\(\frac{| \nu - \mu |}{ \sigma } \le 1.\)

Even though perhaps not an average, the \(\tau\)th quantile (another summary statistic that generalizes the median) can similarly be expressed as a solution to the optimization problem

\(\underset{x \in \mathbb{R}}{\operatorname{argmin}}\, \sum_{i=1}^n \max\big((1-\tau)(x_i - x),\, \tau(x - x_i)\big) = \underset{x \in \mathbb{R}}{\operatorname{argmin}}\, \sum_{i=1}^n \big(|x - x_i| + (1 - 2\tau)\,x\big)\),

which aims to minimize the total tilted absolute value loss (or quantile loss or pinball loss).

Other more sophisticated averages are: trimmore sophistiean, trimedian, and normalized mean, with their generalizations.

In a more general fashion, one can create their own average metric using the generalized f-mean:

\(y = f^{-1}\left(\frac{1}{n}\left[f(x_1) + f(x_2) + \cdots + f(x_n)\right]\right)\)

where f is any invertible function. The harmonic mean is an example of this using f(x) = 1/x, and the geometric mean is another, using f(x) = log x.

However, this method for generating means is not general enough to capture all averages. A more general method for defining an average takes any function g(x₁, x₂, ..., x_n) of a list of arguments that is continuous, strictly increasing in each argument, and symmetric (invariant under permutation of the arguments). The average y is then the value that, when replacing each member of the list, results in the same function value: g(y, y, ..., y) = g(x₁, x₂, ..., x_n). This most general definition still captures the important property of all averages that the average of a list of identical elements is that element itself. The function g(x₁, x₂, ..., x_n) = x₁+x₂+ ··· + x_n provides the arithmetic mean. The function g(x₁, x₂, ..., x_n) = x₁x₂···x_n (where the list elements are positive numbers) provides the geometric mean. The function g(x₁, x₂, ..., x_n) = (x₁+x₂+ ··· + x_n)) (where the list elements are positive numbers) provides the harmonic mean.

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 generalized mean, also known as the power mean or Hölder mean, abstracts several other means. It is defined for positive numbers \(x_1, \dots, x_n\) by

\(M_p(x_1, \dots, x_n) = \left( \frac{1}{n} \sum_{i=1}^n x_i^p \right)^{1/p}.\)

This, as a function of \(p\), is well defined on \(\mathbb{R}\setminus \{0\}\), but can be extended continuously to \(\mathbb{R} \cup \{-\infty, +\infty\}\). By choosing different values for \(p\), other well known means are retrieved.

A similar approach to the power mean is the \(f\)-mean, also known as the quasi-arithmetic mean. For an injective function \(f \colon I \rightarrow \mathbb{R}\) on an interval \(I \subset \mathbb{R}\) and real numbers \(x_1, \dots, x_n \in I\) we define their \(f\)-mean as

\(M_f(x_1, \dots, x_n) = f^{-1}\left({\frac{1}{n} \sum_{i=1}^n{f\left(x_i\right)}}\right).\)

By choosing different functions \(f\), other well known means are retrieved.

The weighted arithmetic mean (or weighted average) is used if one wants to combine average values from different sized samples of the same population, and is define by

\(\bar{x} = \frac{\sum_{i=1}^n {w_i x_i}}{\sum_{i=1}^n w_i},\)

where \(x_i\) and \(w_i\) are the mean and size of sample \(i\) 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. Often, outliers are erroneous data caused by artifacts. In this case, one can use a truncated 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. A specific example of a truncated mean is the interquartile mean.

In some circumstances, mathematicians may calculate a mean of an infinite (or even an uncountable) set of values. This can happen when calculating the mean value \(y_\text{avg}\) of a function \(f(x)\). 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 by integration. The integration formula is written as:

\(y_\text{avg}(a, b) = \frac{1}{b - a} \int_a^b f(x)\,dx.\)

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 require modular arithmetic to add and otherwise combine numbers. These quantities can be averaged using the circular mean. In all these situations, it is possible that no mean exists, for example if all points being averaged are equidistant. Consider a color wheel, there is no mean to the set of all colors. Additionally, there may not be a unique 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.