Mean, median and mode indicate central point of distribution or data set. Let \({P}_{X}\) denotes distribution of a random variable \(X\). Any reasonable rule \(\mathcal{O} = \mathcal{O}({P}_{X})\) indicating a point \(\mathcal{O}\) to be the center of \({P}_{X}\) should satisfy the following postulates:
A1 If \(P(a \leq X \leq b) = 1\) then \(a \leq \mathcal{O}({P}_{X}) \leq b\)
A2 \(\mathcal{O}({P}_{X+c}) = \mathcal{O}({P}_{X}) + c\) for any constant \(c\) [transitivity]
A3 \(\mathcal{O}({P}_{cX}) = c\mathcal{O}({P}_{X})\) for any constant \(c\) [homogeneity]
The mean is a synonym of the first moment, i.e. the expected value \(EX\). For a continuous random variable \(X\) it may be expressed in terms of density function \(f(x)\), as the integral \(EX ={ \int \nolimits \nolimits }_{-\infty }^{+\infty }xf(x)dx\). In discrete case it is defined as the sum of type \(EX ={ \sum \nolimits }_{i}{x}_{i}{p}_{i}\), where \({x}_{i}\) is a possible value of \(X\), \(i \in I\), while \({p}_{i} =...
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References and Further Reading
Cramér H (1946) Mathematical methods of statistics. Princeton University Press, Princeton
Joag-Dev K (1989) MAD property of median. A simple proof. Am Stat 43:26–27
Prokhorov AW (1982a) Expected value. In: Vinogradov IM (ed) Mathematical encyclopedia, vol 3. Soviet Encyclopedia, Moscow, pp 600–601 (in Russian)
Prokhorov AW (1982b) Mode. In: Vinogradov IM (ed) Mathematical encyclopedia, vol 3. Soviet Encyclopedia, Moscow p 763 (in Russian)
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Stepniak, C. (2011). Mean, Median and Mode. In: Lovric, M. (eds) International Encyclopedia of Statistical Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04898-2_354
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