The mean of 25 observations is 36. If the mean of the first 13 observations is 32 and that of the last 13 observations is 39, find the 13th observation.

Let the 13^th observation be x

Given that mean of 25 observations = 36

∴ sum of 25 observations = 25 × 36 = 900

Mean of the first 13 observations = 32

Sum of first 13 observations = 13 × 32 = 416

Mean of the last 13 observations = 39

Sum of the last 13 observations = 13 × 39 = 507

Therefore,

Sum of first 13 observations + sum of last 13 observations – 13^th observation = sum of 25 observations

416 + 507 - x = 900

⇒ 923 – x = 900

⇒ x = 23