Determine the mean and standard deviation for the following distribution: Marks 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Frequency 1 6 6 8 8 2 2 3 0 2 1 0 0 0 1

x f i f i x i d i = x i - x ¯ f i d i f i d i 2 2 1 2 - 4 - 4 16 3 6 18 - 3 - 18 54 4 6 24 - 2 - 12 24 5 8 40 - 1 - 8 8 6 8 48 0 0 0 7 2 14 1 2 2 8 2 16 2 4 8 9 3 27 3 9 27 10 0 0 4 0 0 11 2 22 5 10 50 12 1 12 6 6 36 13 0 0 7 0 0 14 0 0 8 0 0 15 0 0 9 0 0 16 1 16 10 10 100 N = 40 ∑ f i x i = 239 ∑ f i d i = - 1 ∑ f i d i 2 = 325 Mean x ¯ = 239 40 = 5 . 975 ≈ 6 ∴ S D = σ = ∑ f i d i 2 N - ∑ f i d i N 2 = 325 40 - - 1 40 2 = 8 . 125 - 0 . 000625 = 8 . 124375 = 2 . 85 Hence, the required mean = 6 and MD = 2 . 85 .

The weights of coffee in 70 jars is shown in the following table: Weight (in g ) Frequency 200 - 201 13 201 - 202 27 202 - 203 18 203 - 204 10 204 - 205 1 205 - 206 1 Determine variance and standard deviation of the above distribution.

Class-interval f i x i d i = x i - A f i d i f i d i 2 200 - 201 13 200 . 5 - 2 - 26 52 201 - 202 27 201 . 5 - 1 - 27 27 202 - 203 18 202 . 5 = A 0 0 0 203 - 204 10 203 . 5 1 10 10 204 - 205 1 204 . 5 2 2 4 205 - 206 1 205 . 5 3 3 9 N = 70 ∑ f i d j = - 38 ∑ f i d i 2 = 102 ∴ Variance = σ 2 = ∑ f i d i 2 N - ∑ f i d i N 2 = 102 70 - - 38 70 2 = 1 . 457 - 0 . 292 = 1 . 165 ∴ S . D . σ = 1 . 165 = 1 . 08 g

Following are the marks obtained, out of 100 , by two students Ravi and Hashina in 10 tests. Ravi 25 50 45 30 70 42 36 48 35 60 Hashina 10 70 50 20 95 55 42 60 48 80 Who is more intelligent and who is more consistent?

Case I: For Ravi x i d i = x i - 45 d i 2 25 - 20 400 50 5 25 45 = A 0 0 30 - 15 225 70 25 625 42 - 3 9 36 - 9 81 48 3 9 35 - 10 100 60 15 225 Total ∑ d i = - 9 ∑ d i 2 = 1699 ∴ σ = ∑ d i 2 N - ∑ d i N 2 = 1699 10 - - 9 10 2 = 169 . 9 - 0 . 81 = 13 . 003 and x ¯ = A + ∑ d i N = 45 - 9 10 = 44 . 1 Now for Hashina x i ' d i ' = x i ' - 55 d i ' 2 10 - 45 2025 70 15 625 50 - 5 25 20 - 35 1225 95 40 1600 55 = A ' 0 0 42 - 13 169 60 5 25 48 - 7 49 80 25 625 Total ∑ d i ' = - 20 ∑ d i ' 2 = 6368 Assumed mean A ' = 55 ∴ σ ' = ∑ d i ' 2 N = 6368 10 = 25 . 2 and x ' ¯ = A ' + ∑ d i ' N = 55 - 20 10 = 53 For Ravi C V = σ x ¯ × 100 = 13 . 003 44 . 1 × 100 = 29 . 48 For Hashina, CV = σ ' x ' ¯ × 100 = 25 . 2 53 × 100 = 47 . 55 Hence, Hashina is more consistent and intelligent.