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A professor has recorded exam grades for $30$ students in his class, $1$ of the $30$ grades is unreadable. The mean score on the exam was $82$, and the mean score of the $29$ available scores is $84$,

What is the value of the unreadable score?

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closed as off-topic by Bookend, Arctic Char, RecklessReckoner, USER91500, Claude Leibovici Jun 19 '15 at 4:55

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possible duplicate of How the sample mean changes when you add a new observation – joriki Oct 21 '12 at 22:17
Note that while one of the questions is expressed in terms of adding a new value and the other in terms of a value going missing, in both cases the mean of $n+1$ values is to be calculated based on the $(n+1)$-th value and the mean of $n$ values. – joriki Oct 21 '12 at 22:19

$$ 82=\text{mean score}=\frac{\text{sum of all scores}}{30}=\frac{\text{missing score}+\text{sum of all others}}{30}. $$ Therefore $$ 30\cdot82 = \text{missing score}+\text{sum of all others}. $$ $$ 30\cdot82=\text{missing score}+\left(29\cdot\text{mean of all others}\right)= \text{missing score}+(29\cdot84). $$ So $$ 30\cdot82=\text{missing score}+(29\cdot84). $$ Can you find the missing score given that?

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If our missing person's grade was $84$, the average would be $84$. But the average is $82$. To bring down the average to $82$, the missing person must have received a grade of $(2)(30)$ below $84$.

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