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Question

3. 13, 17, 16,14, 11,13, 10, 16,11,18,12, 17

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Solution

The given data is: 13, 17, 16, 14, 11, 13, 10, 16, 11, 18, 12, 17.

Arrange the given data in ascending order,

10, 11, 11, 12, 13, 13, 14, 16, 16, 17, 17, 18

The total numbers of observations are 12. The formula to calculate the median for the even number is the mean of ( n 2 ) th and ( n 2 +1 ) th observation, where n is the number of observation.

M= ( n 2 ) th + ( n 2 +1 ) th 2 (1)

Substitute 12 for n in equation (1).

M= ( 12 2 ) th + ( 12 2 +1 ) th 2 = 6 th + 7 th 2

Since sixth term is 13 and seventh term is 14, thus the median becomes.

M= 13+14 2 = 27 2 =13.5

Therefore, the median of the given data is 13.5.

The formula to calculate the deviations of the respective observations of the data from the mean x ¯ is,

x i M (2)

Substitute 13.5 for M and 10, 11, 11, 12, 13, 13, 14, 16, 16, 17, 17, 18 respectively for x i in equation (2).

3.5,2.5,2.5,1.5,0.5,0.5,0.5,2.5,2.5,3.5,3.5,4.5

The formula to calculate the absolute deviation of the respective observations of the data is,

| x i M | (3)

Substitute 3.5,2.5,2.5,1.5,0.5,0.5,0.5,2.5,2.5,3.5,3.5,4.5 for x i M respectively in equation (3).

3.5, 2.5, 2.5, 1.5, 0.5, 0.5, 0.5, 2.5, 2.5, 3.5, 3.5, 4.5

The formula to calculate the mean deviation about the median is the ratio of the sum of absolute values of deviations to the number of deviations.

M.D.= i=1 n | x i M | n (4)

Substitute 12 for n and 3.5, 2.5, 2.5, 1.5, 0.5, 0.5, 0.5, 2.5, 2.5, 3.5, 3.5, 4.5for | x i M | respectively in equation (4).

M.D.= 3.5+2.5+2.5+1.5+0.5+0.5+0.5+2.5+2.5+3.5+3.5+4.5 12 = 28 12 =2.33

Thus, the mean deviation of the given data is 2.33.


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