For $(2n+1)$ observation ${ x }{ 1 },-{ x }{ 1 },{ x }{ 2 },-{ x }{ 2 },.....{ x }{ n },-{ x }{ n }$ and $0$, where ${ { x }_{ i } }^{ \displaystyle 8 }$ are distinct, non-zero numbers. Let S.D. and M.D. denote the standard deviation and median respectively. then, which one of the following is always true?

The median of this symmetric dataset is 0. The mean deviation about the median is the average of the absolute values of the observations, which is the sum of the absolute values of the x_i terms divided by (2n+1). The standard deviation is the square root of the average of the squares of the observations, calculated as the square root of (2 times the sum of x_i squared divided by (2n+1)). Applying the Cauchy-Schwarz inequality to these sums proves that the square root term is strictly greater than the linear average for distinct, non-zero terms, meaning the standard deviation is greater than the mean deviation.

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