Question

Let \[x_1 , x_2 , . . . , x_n\]  be n observations and  \[X\]  be their arithmetic mean. The standard deviation is given by

 

Options

• \[\sum^n_{i = 1} \left( x_i - X \right)^2\]

•  \[\frac{1}{n}\sum^n_{i = 1}\left( x_i - X \right)^2\]

• \[\sqrt{\frac{1}{n} \sum^n_{i = 1} \left( x_i - X \right)^2}\]

•  \[\sqrt{\frac{1}{n} \sum^n_{i = 1} x_i^2 - X^2}\]

Answer 1

It is given that \[x_1 , x_2 , . . . , x_n\]  are n observations and  \[X\] is their arithmetic mean.
The standard deviation of given observations is \[\sqrt{\frac{1}{n} \sum^n_{i = 1} \left( x_i - X \right)^2}\]

Also,

\[\sqrt{\frac{1}{n} \sum^n_{i = 1} \left( x_i - X \right)^2}\] = \[\sqrt{\frac{1}{n} \sum^n_{i = 1} x_i^2 - X^2}\]