The mean of 25 observations is 36. The mean of the first 13 of them is 32 and the mean of the last 13 is 39. Find the 13th observation.

All 25 observations add to $25\times 36=900$. The first thirteen add to $13\times 32=416$ and the last thirteen to $13\times 39=507$. Their sum 416 + 507 = 923 covers the 13th observation TWICE, since it is both the last of the first thirteen and the first of the last thirteen, so the 13th observation is 923 − 900 = 23. Averaging the two given means gives 35.5 and adding them gives 71, and neither uses the overall mean at all; 13 is the position of the observation, not its value.