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.
Definitions
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, ..., xn is typically denoted using an overhead bar, . If the numbers are from observing a sample of a larger group, the arithmetic mean is termed the sample mean () to distinguish it from the group mean (or expected value) of the underlying distribution, denoted or .Underhill, L.G.; Bradfield d. (1998) Introstat, Juta and Company Ltd. p. 181
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".
Central tendency
In statistics, a central tendency (or measure of central tendency) is a central or typical value for a probability distribution.Weisberg H.F (1992) Central Tendency and Variability, Sage University Paper Series on Quantitative Applications in the Social Sciences, p.2 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."Upton, G.; Cook, I. (2008) Oxford Dictionary of Statistics, OUP (entry for "central tendency")Dodge, Y. (2003) The Oxford Dictionary of Statistical Terms, OUP for International Statistical Institute. (entry for "central tendency")
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.
General properties 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.
Statistical location
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.
Mean of a probability distribution
The mean of a probability distribution is the long-run arithmetic average value of a random variable having that distribution. If the random variable is denoted by , then the mean is also known as the expected value of (denoted ). For a discrete probability distribution, the mean is given by , where the sum is taken over all possible values of the random variable and is the probability mass function. For a continuous distribution, the mean is , where is the probability density function. In all cases, including those in which the distribution is neither discrete nor continuous, the mean is the Lebesgue integral of the random variable with respect to its probability measure. The mean need not exist or be finite; for some probability distributions the mean is infinite ( or ), while for others the mean is undefined.
Relationships between the mean, median and mode
For unimodal distributions the following bounds are known and are sharp:Johnson NL, Rogers CA (1951) "The moment problem for unimodal distributions". Annals of Mathematical Statistics, 22 (3) 433–439
where μ is the mean, ν is the median, θ is the mode, and σ is the standard deviation.
For every distribution,Hotelling H, Solomons LM (1932) The limits of a measure of skewness. Annals Math Stat 3, 141–114Garver (1932) Concerning the limits of a mesuare of skewness. Ann Math Stats 3(4) 141–142
Possible averages
Name Equation or description As solution to optimization problem Arithmetic mean (AM) For example, the arithmetic mean of five values: 4, 36, 45, 50, 75 is: || Median A middle value that separates the higher half from the lower half of the data set; may not be unique if the data set contains an even number of points Geometric median A rotation invariant extension of the median for points in Tukey median Another rotation invariant extension of the median for points in —a point that maximizes the Tukey depth Mode The most frequent value in the data set Geometric mean (GM) an average that is useful for sets of positive numbers, interpreted according to their product and not their sum For example, the geometric mean of five values: 4, 36, 45, 50, 75 is: || Harmonic mean (HM) For example, the harmonic mean of the five values: 4, 36, 45, 50, 75 is || Contraharmonic mean Lehmer mean Quadratic mean(or RMS) Cubic mean Generalized mean Quasi-arithmetic mean is monotonic Weighted mean Truncated mean The arithmetic mean of data values after a certain number or proportion of the highest and lowest data values have been discarded Interquartile mean A special case of the truncated mean, using the interquartile range. A special case of the inter-quantile truncated mean, which operates on quantiles (often deciles or percentiles) that are equidistant but on opposite sides of the median. Midrange Winsorized mean Similar to the truncated mean, but, rather than deleting the extreme values, they are set equal to the largest and smallest values that remain Medoid A representative object of a set of objects with minimal sum of dissimilarities to all the objects in the set, according to some dissimilarity function .
Even though perhaps not an average, the th quantile (another summary statistic that generalizes the median) can similarly be expressed as a solution to the optimization problem
,
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:
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, ..., xn) 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: . 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 provides the arithmetic mean. The function (where the list elements are positive numbers) provides the geometric mean. The function (where the list elements are positive numbers) provides the harmonic mean.
Pythagorean means
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.
Relationship between AM, GM, and HM
AM, GM, and HM of nonnegative real numbers satisfy these inequalities:
Equality holds if all the elements of the given sample are equal.
Power mean
The generalized mean, also known as the power mean or Hölder mean, abstracts several other means. It is defined for positive numbers by
This, as a function of , is well defined on , but can be extended continuously to .P. S. Bullen: Handbook of Means and Their Inequalities. Dordrecht, Netherlands: Kluwer, 2003, pp. 176. By choosing different values for , other well known means are retrieved.
Name Exponent Value Minimum Harmonic mean Geometric mean Arithmetic mean Root mean square Cubic mean Maximum
Quasi-arithmetic mean
A similar approach to the power mean is the -mean, also known as the quasi-arithmetic mean. For an injective function on an interval and real numbers we define their -mean as