Find the mean of 8, 12, 16, 22, 10 and 4. Find the resulting mean, if each of the observations, given above, be :

(i) multiplied by 3.

(ii) divided by 2.

(iii) multiplied by 3 and then divided by 2.

(iv) increased by 25 %.

(v) decreased by 40 %.

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

= $\dfrac{8 + 12 + 16 + 22 + 10 + 4}{6}$

= $\dfrac{72}{6}$

= 12

(i) According to property 4, if each observation is multiplied by a quantity a, then the mean is also multiplied by the same quantity a.

If each observation is multiplied by 3, then the mean is also multiplied by 3 ( 12 x 3 = 36).

Hence, the mean of the new observations is 36.

(ii) According to property 5, if each observation is divided by a quantity a, then the mean is also divided by the same quantity a.

If each observation of the data is divided by 2, then the mean is also divided by 2 $\Big(\dfrac{12}{2} = 6\Big)$.

Hence, the mean of the new observations is 6.

(iii) If each observation of data is multiplied by 3, then the mean is also multiplied by 3 ( 12 x 3 = 36).

Similarly, if each observation of the data is divided by 2, then the mean is also divided by 2 $\Big(\dfrac{36}{2} = 18\Big)$.

Hence, the mean of the new observations is 18.

(iv) According to property 2, if each observation is increased by a quantity a, then the mean is also increased by the same quantity a.

If each observation is increased by 25 %, then the mean is also increased by 25 %.

= 12 + $\dfrac{25}{100} \times 12$

= 12 + 3

= 15

Hence, the mean of the new observations is 15.

(v) According to property 3, if each observation is decreased by a quantity a, then the mean is also decreased by the same quantity a.

If each observation is decreased by 40 %, then the mean is also decreased by 40 %.

= 12 - $\dfrac{40}{100} \times 12$

= 12 - 4.8

= 7.2

Hence, the mean of the new observations is 7.2.