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A professor has recorded exam grades for 30 students in his class, one of the 30 grades is unavailable. 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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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

2 Answers 2

$$ 82=\text{mean score}=\frac{\text{sum of all scores}}{30}=\frac{\text{missing score}+\text{sum of all others}}{30}. $$ 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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