Arithmetic mean Arithmetic mean In mathematics and statistics the arithmetic mean of a set of numbers is the sum of all the members of the set divided by the number of items in the set cardinality The word set is used perhaps somewhat loosely for example the number could occur more than once in such a set The arithmetic mean is what pupils are taught very early to call the average If the set is a statistical population then we speak of the population mean If the set is a statistical sample we call the resulting statistic a sample mean.
The mean may be conceived of as an estimate of the median When the mean is not an accurate estimate of the median the set of numbers or frequency distribution is said to be skewed.
We denote the set of data by X x x x n The symbol Greek mu is used to denote the arithmetic mean of a population We use the name of the variable X with a horizontal bar over it as the symbol X bar for a sample mean Both are computed in the same way.
 bar x rm arithmetic mean cdots x_n n.
The arithmetic mean is greatly influenced by outliers For instance reporting the average net worth in Redmond Washington as the arithmetic mean of all annual net worths would yield a surprisingly high number because of Bill Gates These distortions occur when the mean is different from the median and the median is a superior alternative when that happens.
In certain situations the arithmetic mean is the wrong concept of average altogether For example if a stock rose in the first year in the second year and fell in the third year then it would be incorrect to report its average increase per year over this three year period as the arithmetic mean the correct average in this case is the geometric mean which yields an average increase per year of only.
If X is a random variable then the expected value of X can be seen as the long term arithmetic mean that occurs on repeated measurements of X This is the content of the law of large numbers As a result the sample mean is used to estimate unknown expected values.
Note that several other means have been defined including the generalized mean the generalized f mean the harmonic mean the arithmetic geometric mean and the weighted mean.
Alternate notations
The arithmetic mean may also be expressed using the sum notation.
 bar x sum_ i n x_i.
Pronunciation
When used as a noun the word arithmetic is pronounced with the emphasis on the second syllable aRITHmetic but when used as an adjective as in the title of this article the emphasis is on the third syllable arithMETic.
See also
mean average summary statistics variance central tendency standard deviation inequality of arithmetic and geometric means Muirhead's inequality
External links
 Calculations and comparisons between arithmetic and geometric mean between two numbers
 Arithmetic and geometric means