एक विद्यार्थी पाँच परीक्षाओं में निम्न अंक प्राप्त करता है : \(45,54,41,57,43\). उसके द्वारा छटी परीक्षा में प्राप्त अंक ज्ञात नहीं है। यदि छः परीक्षाओं में प्राप्त अंकों का माध्य \(48\) है तो छः परीक्षाओं में प्राप्त अंको का मानक विचलन है

A \(\frac{10}{3}\)
B \(\frac{100}{3}\)
C \(\frac{{100}}{{\sqrt 3 }}\)
D \(\frac{{10}}{{\sqrt 3 }}\)

Verified Solution

Answer & Solution

Correct Answer

(D) \(\frac{{10}}{{\sqrt 3 }}\)

Step-by-step Solution

Detailed explanation

\(AM = \frac{{41 + 45 + 54 + 57 + 43 + x}}{6} = 48\) \( \Rightarrow x = 48\) \({\sigma ^2} + {48^2} = \frac{1}{6}\left( {{{41}^2} + {{45}^2} + {{54}^2} + {{57}^2} + {{43}^2} + {{48}^2}} \right)\) \({\sigma ^2} = \frac{{14024}}{6} - 2304\) \( = \frac{{100}}{3}\)

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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