Multiple choice

Let ${ x }{ 1 },{ x }{ 2 },.....{ x }_{ n }$ be $n$ observations and $\overline { X } $ be their arithmetic mean. The formula for the standard deviation is given by

  1. $\sum _{ i=1 }^{ n }{ { { (x }_{ i }-\overline { X } ) }^{ 2 } }$
  2. $\cfrac { 1 }{ n } \sum _{ i=1 }^{ n }{ { { (x }_{ i }-\overline { X } ) }^{ 2 } }$
  3. $\sqrt { \cfrac { 1 }{ n } \sum _{ i=1 }^{ n }{ { { (x }_{ i }-\overline { X } ) }^{ 2 } } }$
  4. $\sqrt { \cfrac { 1 }{ n } \sum _{ i=1 }^{ n }{ { { x }_{ i } }^{ 2 }+{ \overline { X } }^{ 2 } } } $
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C Correct answer
Explanation

The standard deviation is defined as the square root of the variance. The variance is the average of the squared differences from the mean, calculated as (1/n) * sum((xi - mean)^2).

AI explanation

The standard deviation measures the amount of variation or dispersion in a set of values. By definition, it is calculated by taking the square root of the average of the squared differences between each observation and the arithmetic mean. This directly corresponds to the formula of the square root of 1 over n times the sum from i equals 1 to n of (x_i minus the mean) squared.

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