MEASURES OF DISPERSION | MEAN DEVIATION

MEAN DEVIATION

Mean deviation is a measure of dispersion based on all items in distribution. Clark and Schkade defined mean deviation as ‘the average amount of scatter of the item in a distribution from either the mean or the median; ignoring the signs of the deviation. The average that is taken of the scatter is an arithmetic mean, which accounts for the fact that this measure is often called the mean deviation’. While calculating mean deviation we must ignore the minus sign of deviation.

Mean deviation (MD) = Sum (x-A)

                                              N

Where,

A = any one of the average mean x, Median (M), and mode (z)

D = deviation

i.e., Mean deviation from

 

mean            MD = Sum (xi-x)                   = Sum (d)

                                        n                                   n

 

median         MD = Sum (xi – M)              = Sum (dM)

                                          n                                  n

 

mode            MD = Sum (x1-z)               = Sum (dz)

                                        n                                  n

 

where,

MD = Mean deviation

X1 = i^th value of x

N = number of items

X = mean

M = median

Z = mode

 

Mean deviation calculated from any measure of central tendency is an absolute measure. But a relative mean deviation is required to compare variation among different series which are expressed in the same or different units. The relative mean deviation is known as coefficient of mean deviation.

The coefficient of mean deviation can be obtained by dividing mean deviation by the average used to find out the mean deviation itself.

Coefficient of mean deviation =    Mean deviation

                                                         Mean or median or mode

 

Example:

 

Calculate the mean deviation from mean and median for the following data

Price: 20, 22, 25, 38, 40, 50, 65, 70, 75.

Solution:

Price (x)
20
22
25
38
40
50
65
70
75
Sum x = 409 N = 9

 

Mean (x1) = Sum x / n

                   = 405/9

                  = 45

 

Median (M) = (N + 1) / 2

                      = (9+1) / 2

                      = 5^th item = 40

M = 40

 

Price (x)	(dx1) = (x-x1)	(dM) = (x-M)
20	25	20
22	23	18
25	20	15
38	7	2
40	5	0
50	5	10
65	20	25
70	25	30
75	30	35
Sum x = 409 N = 9	Sum (dx1) = 160	Sum (dM) = 155

 

MD from Mean

MDx1 = Sum (dx1) / n

           = 160/9

          = 17.78

 

MD from Median

MDm = Sum (dm) / n

= 155/9

= 17.22

Find out the mean deviation of the following data

Height (inches)	58	59	60	61	62	63	64	65	66
No. of Children	8	4	20	17	20	30	28	18	15

 

Solution

Height x	No of children f	Fx
58	8	464
59
60
61
62
63
64
65
66
	N =160	Sum fx = 10,019

 

X1 = sum fx / n

= 10019/160

= 62.61

 

Mean deviation

Height x1	No of children f	Fx	Dx = x – x1	F (dx)
58	8	464	4.61	36.88
59
60
61
62
63
64
65
66
	N =160	Sum fx = 10,019	Sum dx1 = 20.61	Sum f(dx) = 287.58

 

MDx1 = sum f (dx1) / n

= 287.58/160

= 1.80