Question

In a series of 2n observation, half of them equal 'a' and remaining half equal '-a'.If the standard deviation of the observations is 2,then find the value of |a|.

Answer 1

Given, 

N = 2a,

Mean,

\(\bar{x}=\frac{a+a+...+a+(-a)+(-a)+...(-a)}{2n}\)

\(=\frac{0}{2n}\)

 = 0

Standard deviation, 
\(σ=2\)

⇒ \(σ^2=2^2\)

\(⇒\frac{\displaystyle\sum {x_i}^2}{N}-(\frac{\displaystyle\sum {x_i}}{N})^2=4\)

\(⇒\frac{\displaystyle\sum {x_i}^2}{N}-x^{-2}=4\)

\(⇒\frac{\displaystyle\sum {x_i}^2}{N}-0=4\) 

\(⇒\displaystyle\sum {x_i}^2=8n\)

⇒ a² + a² + ⋯ + a² (to 2n terms) 2a² = 8n

⇒ a² = 4

⇒ a = ± 2

∴ |a| = |± 2|

= 2