The mean of 5 observations is 15. If the mean of first three observations is 14 and that of the last three is 17, find the third observation.

By formula,

Mean = $\dfrac{\text{Total sum of observations}}{\text{No. of observations}}$

Given,

The mean of 5 observations = 15

$\therefore 15 = \dfrac{\text{Total sum of 5 observations}}{5} \\[1em] \Rightarrow \text{Total sum of observations} = 15 \times 5 \\[1em] \Rightarrow \text{Total sum of observations} = 75.$

Given,

Mean of first 3 observations = 14

$\therefore 14 = \dfrac{\text{Sum of first 3 observations}}{3} \\[1em] \Rightarrow \text{Sum of first three observations} = 14 \times 3 \\[1em] \Rightarrow \text{Sum of first three observations} = 42.$

Given,

Mean of last 3 observations = 17

$\therefore 17 = \dfrac{\text{Sum of last 3 observations}}{3} \\[1em] \Rightarrow \text{Sum of last three observations} = 17 \times 3 \\[1em] \Rightarrow \text{Sum of last three observations} = 51.$

Third observation = Sum of first three observations + Sum of last three observations - Total sum of observation
= 51 + 42 - 75 = 18.

Hence, the third observation = 18.