Everything about Average totally explained
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 many other types of averages.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.
Calculation
Arithmetic mean
An average is a single value that's meant to typify a list of values. If all the numbers in the list are the same, then this number should be used. If the numbers are not all the same, an easy way to get a representative value from a list is to randomly pick any number from the list. However, the word 'average' is usually reserved for more sophisticated methods that are generally found to be more useful.
The most common type of average is the
arithmetic mean, often simply called the mean. The arithmetic mean of two numbers, such as 2 and 8, is obtained by finding a value A such that 2 + 8 = A + A. It is then simple to find that A = (2 + 8)/2 = 5. Switching the order of 2 and 8 to read 8 and 2 doesn't change the resulting value obtained for A. The mean 5 isn't less than the minimum 2 nor greater than the maximum 8. If we increase the number of terms in the list for which we want an average, we get, for example, that the arithmetic mean of 2, 8, and 11 is found by solving for the value of A in the equation 2 + 8 + 11 = A + A + A. It is simple to find that A = (2 + 8 + 11)/3 = 7.
Again, changing the order of the three members of the list doesn't change the result: A = (8 + 11 + 2)/3 = 7, and that 7 is between 2 and 11. This summation method is easily generalized for lists with any number of elements. However, the mean of a list of integers isn't necessarily an integer. "The average family has 1.7 children" is a jarring way of making a statement that's more appropriately expressed by "the average number of children in the collection of families examined is 1.7".
Geometric mean
With
geometric mean, instead of adding numbers, the numbers are multiplied. Thus, the geometric mean of 2 and 8 is obtained by solving for G in the following equation:
. Thus, the geometric mean of 2 and 8 is
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) =
ex, and it's inherently biased towards the higher values. However, this method for generating means isn't 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.
The function g(x
1, x
2, ..., x
n) =x
1+x
2+ ...+ x
n provides the arithmetic mean.
The function g(x
1, x
2, ..., x
n) =x
1·x
2· ...· x
n provides the geometric mean.
The function g(x
1, x
2, ..., x
n) =x
1−1+x
2−1+ ...+ x
n−1 provides the harmonic mean. (See John Bibby (1974) “Axiomatisations of the average and a further generalisation of monotonic sequences,” Glasgow Mathematical Journal, vol. 15, pp. 63–65.)
In data streams
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.
Etymology
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".
Footnotes
Further Information
Get more info on 'Average'.
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