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I have once practice statics question, which I'm not able to solve. The question is as follows :

Average score of ten students was $5.4$ for a particular test. Seven of these students scored $8, 7, 5, 5, 2, 1, 3$. What are the possible values of the mode of the marks of all ten students? Mark one or more correct options

(A) 4    (B) 5     (C) 6    (D) 7     (E) 8

The correct answer is B, D, E, i.e. 5, 7, 8. I don't know how to solve this.

My approach :

Since average = $5.4$, sum of all scores = $54$.
Total score of $7$ students = $31$, so, the remaining three students have in total $23$ marks. Now any information about maximum score is not given in the question. So, I assumed all scores less than 23 are possible in this case as long as the condition "Total score of remaining three students = $23$" is satisfied. If I assume that, option A can be answer if remaining three students have scores $4, 4, 15$ right? Then why option A is not in the answer? I'm sure I'm doing something wrong here, but I don't know what. Any help will be highly appreciated.

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    $\begingroup$ Presumably there is a maximum score on the test. I'd guess $10$? $\endgroup$
    – lulu
    Commented May 2, 2021 at 14:30
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    $\begingroup$ Another ambiguity arises when there are several modes. If the remaining scores were, say, $\{3,10,10\}$ then you have two $5's$ and two $10's$, so I would say that both $5,10$ were modes. Perhaps the question was intended to read "which of the following are possible values for the mode, given that there is a unique mode" though of course that should have been spelled out. $\endgroup$
    – lulu
    Commented May 2, 2021 at 14:35

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All ten scores add to 54, the seven scores shown add to 31, leaving a total of 23 for the remaining three.

Mode could be 5: three additional might be 11, 12, 0.

Mode could be 7: three additional might be 7,7,9.

Mode could be 8: three additional might be 8,8,7.

However, you correctly show a mode could be 4. Clearly, there must be an unstated restriction that there is only one unique mode. Or an unstated restriction on the possible test scores. @lulu has mentioned both possibilities.

So check your definition of mode. Statistical literature is full of references to multi-modal samples and distributions. But maybe that doesn't fit the definition of your course. When it comes to discussions of 'mode', many elementary texts have incomplete or misleading definitions. Try to make sense of what your text says.

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  • $\begingroup$ Thanks, I think that unstated constraint is about max score limit. As, @lulu mentioned, if we assume max score to be 10, then we can prove that answer is of course B, D and E. $\endgroup$
    – Ruchit Vithani
    Commented May 3, 2021 at 9:07

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