Information about Average

In mathematics, an average, or central tendency of a data set refers to a measure of the "middle" or "expected" value of the data set. There are many different descriptive statistics that can be chosen as a measurement of the central tendency of the data items. The most common method is the arithmetic mean, but there are more than one type of average (median being another common example).

Colloquially, people often use the term average to refer to an intuitive central tendency without having a specific measurement of central tendency in mind, or use terms such as "the average person". However, the phrase "there's no such thing as an average citizen" emphasizes that the average is a number, not a person or some other object. The average is calculated by combining the measurements related to a group of people or objects, to compute a number as being the average of the group.

Please see the table of mathematical symbols for explanations of the symbols used. In statistics, the term central tendency is used in some fields of empirical research to refer to what statisticians sometimes call "location". A "measure of central tendency" is either a location parameter or a statistic used to estimate a location parameter.

Measures of central tendency

There are many kinds of averages. The fundamental concept they have in common is that they are all ways of preserving a property of a list, which is symetric with permutations of the list, when each element of the list is replaced by the constant value of the average. For instance, that arithmetic average preserves the sum of a list when each element of the list is replaced by the arithmetic average. Thus, the arithmetic average, A, of 2 and 8 is obtained by solving: 2 + 8 = A + A. The geometric average perserves the product of a list when each element of the list is replaced by the geometric average. Thus, the geometric average, G, of 2 and 8 is obtained by solving: 2*8 = G*G. Different kinds of averages are described below.

Name	Equation or description
Arithmetic mean
Median	The middle value that separates the higher half from the lower half of the data set
Geometric median	A rotation invariant extension of the median for points in Rn
Mode	The most frequent value in the data set
Geometric mean
Harmonic mean
Quadratic mean (or RMS)
Generalized mean
Heronian mean	where j≠i
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
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
Annualization	Annualization of a set of returns is a variation on the geometric average that provides the intensive property of a return per year corresponding to a list of returns. Define the return factor as one plus the return. The annualized return factor of a list of returns is the Tth root of the product of their return factors, where T is the sum of the durations of the periods of the returns. For instance, if the return for one year is 10% and the return for a subsequent half a year is 5% then the annualized return of both of these returns together is obtained by taking the product of 1.1 and 1.05, and then taking the result to the power of one over 1.5 and then subtracting one, which gives approximately 10.08%.

Other averages

Other more sophisticated averages are: trimean, trimedian, and normalized mean. These are usually more representative of the whole data set.

One can create one's own average metric using 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. Another example, expmean (exponential mean) is a mean using the function f(x) = e^x, and it is inherently biased towards the higher values. However, this method for generating means is not general enough to capture all averages. A more general method for defining an average, y, takes any function of a list g(x_1, x_2, ..., x_n), which is symmetric under permutation of the members of the list, and equates it to the same function with the value of the average replacing each member of the list: g(x_1, x_2, ..., x_n) = g(y, y, ..., y). 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.

Average applied to a data stream

The concept of an average can be applied to a stream of data as well as a bounded set, the goal being to find a value about which recent data is in some way clustered. The stream may be distributed in time, as in samples taken by some data acquisition system from which we want to remove noise, or in space, as in pixels in an image from which we want to extract some property. An easy-to-understand and widely used application of average to a stream is the simple moving average in which we compute the arithmetic mean of the most recent N data items in the stream. To advance one position in the stream, we add 1/N times the new data item and subtract 1/N times the data item N places back in the stream.

Derivation of the name

The original meaning of the word average is "damage sustained at sea": the same word is found in Arabic as awar, in Italian as avaria and in French as avarie. Hence an average adjuster is a person who assesses an insurable loss.

Marine damage is either particular average, which is borne only by the owner of the damaged property, or general average, where the owner can claim a proportional contribution from all the parties to the marine venture. The type of calculations used in adjusting general average gave rise to the use of "average" to mean "arithmetic mean".

External links

• Median as a weighted arithmetic mean of all Sample Observations
• Calculations and comparison between arithmetic and geometric mean of two values