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The second moment of a random variable attains the minimum value when taken around the first moment (., mean) of the random variable, . a r g m i n m E ( ( X − m ) 2 ) = E ( X ) {\displaystyle \mathrm {argmin} _{m}\,\mathrm {E} ((X-m)^{2})=\mathrm {E} (X)} . Conversely, if a continuous function φ {\displaystyle \varphi } satisfies a r g m i n m E ( φ ( X − m ) ) = E ( X ) {\displaystyle \mathrm {argmin} _{m}\,\mathrm {E} (\varphi (X-m))=\mathrm {E} (X)} for all random variables X , then it is necessarily of the form φ ( x ) = a x 2 + b {\displaystyle \varphi (x)=ax^{2}+b} , where a > 0 . This also holds in the multidimensional case. [7]

For example, each of the three populations {0, 0, 14, 14}, {0, 6, 8, 14} and {6, 6, 8, 8} has a mean of 7. Their standard deviations are 7, 5, and 1, respectively. The third population has a much smaller standard deviation than the other two because its values are all close to 7. It will have the same units as the data points themselves. If, for instance, the data set {0, 6, 8, 14} represents the ages of a population of four siblings in years, the standard deviation is 5 years. As another example, the population {1000, 1006, 1008, 1014} may represent the distances traveled by four athletes, measured in meters. It has a mean of 1007 meters, and a standard deviation of 5 meters.

Sample of an article critique

sample of an article critique

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