Question 19:

Given Σ x = 99, n = 9 and Σ (x - 10)² = 79. Find Σ x² and Σ (x - x̄)².

Solution:

x̄ = (Σ x/n)

= (99/9)

= 11

Σ (x - 10)² = 79

Σ (x²  + 10² - 2 x (10)) = 79

Σ (x²  + 100 - 20 x) = 79

Σ x²  + 100 Σ - 20 Σ x = 79

Σ x²  + 100 (9) - 20 (99) = 79

Σ x²  + 900 - 1980 = 79

Σ x² - 1080 = 79

Σ x² = 79 + 1080

Σ x² = 1159

Σ (x - x̄)² = Σ (x - 11)²

= Σ (x²  + 11² - 2 x (11))

= Σ (x²  + 121 - 22 x)

= Σx² + 121Σ - 22 Σx

= 1159 + 121(9) - 22 (99)

= 1159 + 1089 - 2178

=  2248 - 2178

= 70

Question 20:

Two marks scored by two students A,B in a class are given below.

A          58          51          60          65          66

B          56          87          88          46          43

Solution:

Student A

x̄ = (Σ x/n)

x̄ = (58 + 51 + 60 + 65 + 66)/5

= 300/5

= 60

 x d = x - 60 d² 51 51 - 60 = -9 81 58 58 - 60 = -2 4 60 60 - 60 = 0 0 65 65 - 60 = 5 25 66 66 - 60 = 6 36 Σ x = 300 Σd²=146

Student B

x̄ = (Σ x/n)

x̄ = (56 + 87 + 88 + 46 + 43)/5

= 320/5

= 64

 x d = x - 64 d² 43 43 - 64 = -21 441 46 46 - 64 = - 18 324 56 56 - 64 = -8 64 87 87 - 64 = 23 529 88 88 - 64 = 24 576 Σ x = 320 Σd²=1934
 Student Aσ = √Σd²/n   = √146/5   = √29.2   = 5.4C.V = (σ/x̄) x 100       = (5.4/60) x 100      = 9 Student Bσ = √Σd²/n   = √1934/5   = √386.8   = 19.67C.V = (σ/x̄) x 100       = (19.67/64) x 100      = 30.73

Coefficient of variation of student A is less than student B. From this we can decide that student A is more consistent than student B.