Formal Arithmetic Mean Function
The Formal Arithmetic Mean Function is an arithmetic mean function that is a formal mathematical function.
- Context:
- It can be represented as:
- [math]\displaystyle{ f(X) = (a_i + … + a_n) / n }[/math].
- [math]\displaystyle{ \bar{x} = \frac{w_1 x_1 + w_2 x_2 + \cdots + w_n x_n}{w_1 + w_2 + \cdots + w_n} }[/math].
- It can range from being a Weighted Formal Arithmetic Mean Function to being an Unweighted Formal Arithmetic Mean Function.
- …
- It can be represented as:
- Example(s):
- as referenced in (Gini, 1921).
- as referenced in (Dalton, 1920).
- [math]\displaystyle{ E(Z) = Np }[/math], for a binomial distribution.
- …
- Counter-Example(s):
- See: Statistic Function, Summation Operator.
References
2018
- (Wikipedia, 2018) ⇒ https://en.wikipedia.org/wiki/Arithmetic_mean#Definition Retrieved:2018-10-27.
- The arithmetic mean (or mean or average), [math]\displaystyle{ \bar{x} }[/math] (read [math]\displaystyle{ x }[/math] bar), is the mean of the [math]\displaystyle{ n }[/math] values [math]\displaystyle{ x_1,x_2,\ldots,x_n }[/math] .
The arithmetic mean is the most commonly used and readily understood measure of central tendency in a data set. In statistics, the term average refers to any of the measures of central tendency. The arithmetic mean of a set of observed data is defined as being equal to the sum of the numerical values of each and every observation divided by the total number of observations. Symbolically, if we have a data set consisting of the values [math]\displaystyle{ a_1, a_2, \ldots, a_n }[/math], then the arithmetic mean [math]\displaystyle{ A }[/math] is defined by the formula: : [math]\displaystyle{ A=\frac{1}{n}\sum_{i=1}^n a_i=\frac{a_1+a_2+\cdots+a_n}{n} }[/math] (See summation for an explanation of the summation operator).
For example, let us consider the monthly salary of 10 employees of a firm: 2500, 2700, 2400, 2300, 2550, 2650, 2750, 2450, 2600, 2400. The arithmetic mean is : [math]\displaystyle{ \frac{ 2500+ 2700+ 2400+ 2300+ 2550+ 2650+ 2750+ 2450+ 2600+ 2400}{10}=2530. }[/math] If the data set is a statistical population (i.e., consists of every possible observation and not just a subset of them), then the mean of that population is called the population mean. If the data set is a statistical sample (a subset of the population), we call the statistic resulting from this calculation a sample mean.
- The arithmetic mean (or mean or average), [math]\displaystyle{ \bar{x} }[/math] (read [math]\displaystyle{ x }[/math] bar), is the mean of the [math]\displaystyle{ n }[/math] values [math]\displaystyle{ x_1,x_2,\ldots,x_n }[/math] .
1921
- (Gini, 1921) ⇒ Corrado Gini. (1921). “Measurement of Inequality of Incomes.” The Economic Journal 31, no. 121 DOI:10.2307/2223319
- QUOTE: … In the first of these the relations are brought to light which exist between the mean deviation from the arithmetic mean, the mean deviation from the median, and the area of concentration. …
1920
- (Dalton, 1920) ⇒ Hugh Dalton. (1920). “The Measurement of the Inequality of Incomes.” The Economic Journal 30, no. 119
- QUOTE: … The arithmetic mean is, indeed, easily calculated from perfect statistics, and fairly easily approximated to from imperfect statistics, but the corresponding calculations for the geometric and harmonic means are very laborious, when the number of individual incomes is large …