Find the mean deviation about the mean for the following data: x i 2 4 7 9 12 15 f i 10 2 4 9 8 7

We have, x i 2 4 7 9 12 15 f i 10 2 4 9 8 7 Mean x ¯ = 20 + 8 + 28 + 81 + 96 + 105 40 = 8 . 45 Mean deviation is = 10 2 - 8 . 45 + 2 4 - 8 . 45 + 4 7 - 8 . 45 + 9 9 - 8 . 45 + 8 12 - 8 . 45 + 7 15 - 8 . 45 40 = 64 . 5 + 8 . 9 + 5 . 8 + 13 . 05 + 28 . 4 + 45 . 85 40 = 166 . 5 40 = 4 . 1625

Find the mean deviation about the median for the following data: x i 9 10 11 12 13 16 18 f i 3 4 9 7 4 6 7

We have, x i 9 10 11 12 13 16 18 f i 3 4 9 7 4 6 7 N = ∑ f i = 3 + 4 + 9 + 7 + 4 + 6 + 7 ⇒ N = 40 even Hence, median is the average of 40 2 th and 40 2 + 1 th observations i.e., 20 th & 21 th observations.Hence, median is M = 12 + 12 2 = 12 Mean deviation = ∑ f i x i - M N = 3 9 - 12 + 4 10 - 12 + 9 11 - 12 + 7 12 - 12 + 4 13 - 12 + 6 16 - 12 + 7 18 - 12 40 = 9 + 8 + 9 + 4 + 24 + 42 40 = 96 40 = 2 . 4

Find the mean deviation about the median for the following data: Class 0 - 6 6 - 12 12 - 18 18 - 24 Frequency 3 1 2 4

We make the following table from the given data: Class Frequency f i cumulative frequency cf x i x i f i 0 - 6 3 3 3 9 6 - 12 1 4 9 9 12 - 18 2 6 15 30 18 - 24 4 10 21 84 N = ∑ f i = 10 Since, number of observations is even, therefore median is the average of 10 2 th and 10 2 + 1 th observation i.e., 5 th & 6 th observation which is in the interval 12 - 18 . Therefore, median is Median = l + N 2 - cf f × h ⇒ M = 12 + 10 2 - 4 2 × 6 ⇒ M = 12 + 3 = 15 Mean deviation about median M . D = ∑ f i x i - M ∑ f i = 3 3 - 15 + 1 9 - 15 + 2 15 - 15 + 4 21 - 15 10 = 36 + 6 + 24 10 = 66 10 = 6 . 6 Therefore, mean deviation about median 6 . 6 .

Find the mean deviation about the mean for the following data: Class 2 - 4 4 - 6 6 - 8 8 - 10 Frequency 4 3 1 2

We have, Class f i x i x i f i 2 - 4 4 3 12 4 - 6 3 5 15 6 - 8 1 7 7 8 - 10 2 9 18 N = ∑ f i = 10 ∑ x i f i = 12 + 15 + 7 + 18 = 52 Mean M = ∑ x i f i ∑ f i = 52 10 = 5 . 2 Mean deviation = ∑ f i x i - M N = 4 × 2 . 2 + 3 × 0 . 2 + 1 × 1 . 8 + 2 × 3 . 8 10 = 8 . 8 + 0 . 6 + 1 . 8 + 7 . 6 10 = 18 . 8 10 = 1 . 88

Find the mean and variance of the following data: x i 4 8 11 17 20 24 32 f i 3 5 9 5 4 3 1

We have, x i 4 8 11 17 20 24 32 f i 3 5 9 5 4 3 1 Mean x = ∑ f i x i ∑ f i = 4 × 3 + 8 × 5 + 11 × 9 + 17 × 5 + 20 × 4 + 24 × 3 + 32 × 1 3 + 5 + 9 + 5 + 4 + 3 + 1 = 12 + 40 + 99 + 85 + 80 + 72 + 32 30 = 420 30 = 14 Variance = ∑ f i x i - x 2 ∑ f i = 3 4 - 14 2 + 5 8 - 14 2 + 9 11 - 14 2 + 5 17 - 14 2 + 4 20 - 14 2 + 3 24 - 14 2 + 1 32 - 14 2 30 = 300 + 180 + 81 + 45 + 144 + 300 + 324 30 = 1374 30 = 45 . 8

The annual rainfall in centimetres at a place of over the period 2004 - 2009 was as follows: Year 2004 2005 2006 2007 2008 2009 Rain fall(in cm) 111 73 79 87 50 50 Calculate the standard deviation of rainfall.

We have, x i f i d = x i - a f i d f i d 2 2004 111 - 2 - 222 444 2005 73 - 1 - 73 73 2006 = a 79 0 0 0 2007 87 1 87 87 2008 50 2 100 200 2009 50 3 150 450 N = 450 ∑ f i d = 42 ∑ f i d 2 = 1254 Now, S . D . = 1 N ∑ f i d 2 - ∑ f i d N 2 ⇒ S . D . = 1254 450 - 42 450 2 ⇒ S . D . = 1 450 1254 × 450 - 42 2 ⇒ S . D . = 562536 450 = 750 . 024 450 = 1 . 666