(a) Find the mean of the following data :

18, 33, 30, 21 and 13.

Also, find the sum of deviations of this data from the mean.

(b) If 150 is the mean of 200 observations and 100 is the mean of some 300 other observations, find the mean of the combination.

(a)

$\text{Mean} = \dfrac{\text{Sum of all observations}}{\text{Total number of observations}}\\[1em] = \dfrac{18 + 33 + 30 + 21 + 13}{5}\\[1em] = \dfrac{115}{5}\\[1em] = 23$

Sum of deviations = (18 - 23) + (33 - 23) + (30 - 23) + (21 - 23) + (13 - 23)

= (-5) + 10 + 7 + (-2) + (-10)

= 0

Hence, mean = 23 and sum of deviation = 0.

(b) Mean of 200 observation = 150

Mean = $\dfrac{\text{Sum of all observations}}{\text{Total number of observations}}$

⇒ 150 = $\dfrac{\text{Sum of all observations}}{200}$

⇒ Sum of all observations = 150 x 200

⇒ Sum of all observations = 30,000

Mean of 300 observations = 100

⇒ 100 = $\dfrac{\text{Sum of all observations}}{300}$

⇒ Sum of all observations = 100 x 300

⇒ Sum of all observations = 30,000

Mean of combined observations = $\dfrac{30,000 + 30,000}{200 + 300}$

= $\dfrac{60,000}{500}$

= 120

Hence, the mean of the combination = 120.