The mean of 25 observations was worked out as 48. It was then found that one observation had been entered as 65 by mistake. When the entry was corrected the mean rose to 49.2. What is the correct value of that observation?

Work through the totals, since only one entry moved. The wrong total was $25\times 48=1200$ and the right total is $25\times 49.2=1230$, so the total gained 30. That whole gain belongs to the single corrected entry, which must therefore be 65 + 30 = 95. The value 35 = 65 − 30 moves the entry the wrong way, and can be ruled out at once: the mean ROSE, so the corrected value has to be larger than the one it replaced. The value 66.2 = 65 + 1.2 adds the change in the MEAN to the observation, forgetting that a change of 1.2 in the mean of 25 readings requires a change of $25\times 1.2=30$ in one reading. The value 30 is that change in the total, quoted as though it were the observation itself.