The marks obtained by 15 students in a class test are 12, 14, 07, 09, 23, 11, 08, 13, 11, 19, 16, 24, 17, 03, 20. Find :

(i) the mean of their marks.

(ii) the mean of their marks when the marks of each student are increased by 4.

(iii) the mean of their marks when 2 marks are deducted from the marks of each student.

(iv) the mean of their marks when the marks of each student are doubled.

(i) The sum of marks of all students = 12 + 14 + 07 + 09 + 23 + 11 + 08 + 13 + 11 + 19 + 16 + 24 + 17 + 03 + 20 = 207.

$\therefore \text{The mean of marks} = \dfrac{\text{Sum of marks of all students}}{\text{No. of students}} = \dfrac{∑x_i}{n} \\[1em] = \dfrac{207}{15} \\[1em] = 13.8$

Hence, the mean of marks of 15 students is 13.8

(ii) When the marks of each student are increased by 4, then the sum of their marks increases by 15 × 4 i.e. by 60.

∴ The new sum of marks of all students = 207 + 60 = 267.

$\therefore \text{The new mean of marks} = \dfrac{\text{ New sum of marks}}{\text{No. of students}} \\[1em] = \dfrac{267}{15} \\[1em] = 17.8$

Hence, the mean of marks of 15 students, when the marks of each student are increased by 4 is 17.8

(iii) When the marks of each student is decreased by 2, then the sum of their marks decreases by 15 × 2 i.e. by 30.

∴ The new sum of marks of all students = 207 - 30 = 177.

$\therefore \text{The new mean of marks} = \dfrac{\text{ New sum of marks}}{\text{No. of students}} \\[1em] = \dfrac{177}{15} \\[1em] = 11.8$

Hence, the mean of marks of 15 students, when the marks of each student is decreased by 2 is 11.8

(iv) When the marks of each student are doubled, then the sum of their marks will also be doubled.

∴ The new sum of marks of all students = 207 × 2 = 414.

$\therefore \text{The new mean of marks} = \dfrac{\text{ New sum of marks}}{\text{No.of students}} \\[1em] = \dfrac{414}{15} \\[1em] = 27.6$

Hence, the mean of marks of 15 students, when the marks of each student is doubled is 27.6