JEE Mains · Maths · STD 11 - 13. statistics

Let \( \bar x , M\) and \(\sigma^2\) be respectively the mean, mode and variance of \(n\) observations \(x_1 , x_2,...,x_n\) and \(d_i\, = - x_i - a, i\, = 1, 2, .... , n\), where \(a\) is any number.
Statement \(I\): Variance of \(d_1, d_2,.....d_n\) is \(\sigma^2\).
Statement \(II\) : Mean and mode of \(d_1 , d_2, .... d_n\) are \(-\bar x -a\) and \(- M - a\), respectively

A Statement \(I\) and Statement \(II\) are both false
B Statement \(I\) and Statement \(II\) are both true
C Statement \(I\) is true and Statement \(II\) is false
D Statement \(I\) is false and Statement \(II\) is true

Verified Solution

Answer & Solution

Correct Answer

(B) Statement \(I\) and Statement \(II\) are both true

Step-by-step Solution

Detailed explanation

\(\left( b \right)\,\,\bar x = \frac{{{x_1} + {x_2} + {x_3} + ... + {x_n}}}{n}\) \({\sigma ^2} = \frac{1}{n}\sum\limits_{i = 1}^n {{{\left( {{x_i} - \bar x} \right)}^2}} \) Mean of \({d_1},{d_2},{d_3},......,{d_n}\) \( = \frac{{{d_1} + {d_2} + {d_3} + ...... + {d_n}}}{n}\)…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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