Formal Weighted Arithmetic Mean Function

A Formal Weighted Arithmetic Mean Function is an weighted arithmetic mean function that is a formal arithmetic mean function.

• Context:
  • …
• Counter-Example(s):
  • an Informal Weighted Arithmetic Mean Function.
  • a Formal Unweighted Arithmetic Mean Function.
• See: Weighted Summation, Simpson's Paradox.

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References

2018

• (Wikipedia, 2018) ⇒ https://en.wikipedia.org/wiki/Weighted_arithmetic_mean#Mathematical_definition Retrieved:2018-10-27.
  • Formally, the weighted mean of a non-empty set of data : [math]\displaystyle{ \{x_1, x_2, \dots , x_n\}, }[/math] (where x represents a set of mean values)

    with non-negative weights : [math]\displaystyle{ \bar{x} = \frac{ \sum\limits_{i=1}^n w_i x_i}{\sum\limits_{i=1}^n w_i}, }[/math] which means: : [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] Therefore, data elements with a high weight contribute more to the weighted mean than do elements with a low weight. The weights cannot be negative. Some may be zero, but not all of them (since division by zero is not allowed).

    The formulas are simplified when the weights are normalized such that they sum up to [math]\displaystyle{ 1 }[/math] , i.e.: : [math]\displaystyle{ \sum_{i=1}^n {w_i'} = 1 }[/math] .

    For such normalized weights the weighted mean is then: : [math]\displaystyle{ \bar {x} = \sum_{i=1}^n {w_i' x_i} }[/math] .

    Note that one can always normalize the weights by making the following transformation on the original weights: : [math]\displaystyle{ w_i' = \frac{w_i}{\sum_{j=1}^n{w_j}} }[/math] .

    Using the normalized weight yields the same results as when using the original weights: : [math]\displaystyle{ \begin{align} \bar{x} &= \sum_{i=1}^n w'_i x_i= \sum_{i=1}^n \frac{w_i}{\sum_{j=1}^n w_j} x_i = \frac{ \sum_{i=1}^n w_i x_i}{\sum_{j=1}^n w_j} \\ & = \frac{ \sum_{i=1}^n w_i x_i}{\sum_{i=1}^n w_i}. \end{align} }[/math] The ordinary mean [math]\displaystyle{ \frac {1}{n}\sum_{i=1}^n {x_i} }[/math] is a special case of the weighted mean where all data have equal weights, [math]\displaystyle{ w_i=1 }[/math] .

    The standard error of the weighted mean (unit input variances), [math]\displaystyle{ \sigma_{\bar{x}} }[/math] can be shown via uncertainty propagation to be: : [math]\displaystyle{ \sigma_{\bar{x}} = \left(\sqrt{\sum_{i=1}^n {w_i^2}} \right)^{-1} }[/math]

1950

• (Beckenbach, 1950) ⇒ E. F . Beckenbach. (1950). “A Class of Mean Value Functions.” The American Mathematical Monthly 57, no. 1
  • QUOTE: … first and second derivatives. The solutions are x= L:7-1a;/n, which is the arithmetic mean of (a), and (1) .. !/ y = L: a; L: a,., i-1 k=l which is a weighted arithmetic mean of (a) with each a; as its own weight. …

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