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Students taking an exam are divided into two groups, P and Q such that each group has the same number of students. The performance of each of the students in a test was evaluated out of $200$ marks. It was observed that the mean of group P was $105$, while that of group Q was $85$. The standard deviation of group P was $25$, while that of group Q was $5$. Assuming that the marks were distributed on a normal distribution, which of the following statements will have the highest probability of being TRUE?

  1. No student in group Q scored less marks than any student in group P.
  2. No student in group P scored less marks than any student in group Q.
  3. Most students of group Q scored marks in a narrower range than students in group P.
  4. The median of the marks of group P is $100$.
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As Both group $P$ and $Q$ have same number of students so  let suppose they both contain $2$ students

Given $P_{1}+P_{2}=210$   and  $Q_{1}+Q_{2}=170$

We can see that both above equations don't comment anything about their comparative value so option A and B are false.

Standard Deviation is a measure that is used to quantify the amount of variation or dispersion of a set of data values.

Group $Q$ has lesser standard deviation than group $P$ this clearly shows that $Q_{1}$ and $Q_{2}$ are more close to each other than $P1_{1}$ and $P_{2}$

Normal Distribution :- $Mean=Median=Mode$

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Group Q students have less standard deviation than group P , means most students in group Q got less marks than group P but Not all students in group Q got less marks than group P .

That makes statement A and B  incorrect .

Mean, Median and Mode of Normal Distribution is same , so option D is wrong .

Only option C is correct.
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6 votes

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source-https://www.geeksforgeeks.org/mathematics-probability-distributions-set-3-normal-distribution/

within one standard deviation probability of both is 0.68

range of P=80-130

range of Q=80-90

so Q has a narrower ranger than P

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Very well explained
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