Mean Deviation
Even though Range and Quartile Deviation give an idea about the spread of individual items of a series, they do not try to calculate their dispersion from its average. If the variations of items were calculated from the average, such a measure of dispersion would through light on the formation of the series and the spread of items round the central value. Mean deviation (M.D) is such a measure of dispersion.Mean deviation of a series is the arithmetic average of the deviations of various items from a measure of central tendency. In aggregating the deviations, algebraic signs of the deviations are not taken into account. It is because, if the algebraic signs were taken into account, the sum of deviations from the mean should be zero and that from median is nearly zero. Theoretically the deviations can be taken from any of the three averages, namely, arithmetic mean, median or mode; but, mode is usually not considered as it is less stable. Between mean and median, the latter is supposed to be better because, the sum of the deviations from the median is less than the sum of the deviations from the méan.
While doing problems, if the type of the average is mentioned, we take that average: otherwise we consider mean or median as the case may be.
This measure of dispersion has found favour with economists and business men due to its simplicity in calculation. For forecasting of business cycles, this measure has been found more useful than others. it is also good for small sample studies where elaborate statistical analysis is not needed.
Where D represents deviations from mean or median, ignoring signs, and N the total number of items.
MD is an absolute measure of dispersion. The relative measure of MD is coefficient of MD, defined as:
$$ \mathbf {Coefficient\,of\,MD\,=\,{{{\frac{MD}{Mean}} }}} $$
 It is based on all items
 A change in even one value will affect it
 Value will be least, if we are calculating it from median
 Value will be higher, if calculated from the mean
 Since it ignores signs of deviations, it is not suitable for openend distribution
Mean Deviation from Arithmetic Mean
Individual Series

Find Mean using the equation \( {{{\frac{ΣX}{N}} }} \)

Take deviations of individual values from mean, d (modulus) = (x  X̄), ignoring signs

MD_{X̄} = \( {{{\frac{ΣD}{N}} }} \) (N = number of items)
Table 6.11  

Roll No.  Marks 
1  12 
2  18 
3  23 
4  18 
5  25 
6  15 
7  9 
8  14 
9  6 
10  23 
11  19 
12  10 
Table 6.12  

Roll No.  X (Marks)  D = X  X̄ = X  16 
1  12  4 
2  18  2 
3  23  7 
4  18  2 
5  25  9 
6  15  1 
7  9  7 
8  14  2 
9  6  10 
10  23  7 
11  19  3 
12  10  6 
N = 12  ΣD = 60 
Discrete Series
STEPS:

Find Mean using the equation \( {{{\frac{ΣfX}{Σf}} }} \)

Take deviations of individual values from mean, d (modulus) = (x  X̄), ignoring signs

Find fd and ΣfD(modulus) = (x  X̄), ignoring signs

MD_{X̄} = \( {{{\frac{ΣfD}{Σf}} }} \)
Table 6.13  

Value  Frequency 
8  2 
13  5 
15  9 
21  14 
24  7 
28  7 
29  4 
30  2 
Table 6.14  

Value  f  fx  D = X  X̄ = X  21  fD 
8  2  16  13  26 
13  5  65  8  40 
15  9  135  6  54 
21  14  294  0  0 
24  7  168  3  21 
28  7  196  7  49 
29  4  116  8  32 
30  2  60  9  18 
N = 50  ΣfX = 1050  ΣfD = 240 
Continuous Series
In order to calculate MD and its coefficient for continuous series, we use the same method described earlier. Here we the devition from midvalues of classes. That is, we take midpoint as X here.STEPS:

Find Mean using the equation \( {{{\frac{Σfm}{Σf}} }} \)

Take deviations of mid points from mean, d (modulus) = (m  X̄), ignoring signs

Find fd and ΣfD

MD = \( {{{\frac{ΣfD}{Σf}} }} \)
Table 6.15  

Marks  No. of Students 
0  10  2 
10  20  2 
20  30  5 
30  40  5 
40  50  3 
50  60  2 
60  70  1 
Table 6.16  

Class  f  X = m  fm  D = m  X̄ = m  32.5  fD 
010  2  5  10  27.5  55 
1020  2  15  30  17.5  35 
2030  5  25  125  7.5  37.5 
3040  5  35  175  2.5  12.5 
4050  3  45  135  12.5  37.5 
5060  2  55  110  22.5  45 
6070  1  65  65  32.5  32.5 
Σf = 20  Σfm = 650  ΣfD = 255 
Mean Deviation from Median
Individual Series
STEPS:

Arrange the data in ascending order

Compute the median
Median = Size of \( {{{\frac{N + 1}{2}} }}^{th} \) item

Take deviation of individual values from median. i.e., d = X  Median (ignoring signs)
Coefficient of MD = \( {{{\frac{MD}{Median}} }} \)
4000, 4200, 4400, 4600, 4800
$$ Median\, =\, {{{\frac{N + 1}{2}} }}^{th} item $$ $$ =\, {{{\frac{5 + 1}{2}} }}^{th} item $$ $$ =\, {{{\frac{6}{2}} }}^{th} item $$ $$ =\, 3^{rd} item $$ $$ =\, 4400 $$
Table 6.17  

Deviation from Median 4400  
Income  D 
4000  400 
4200  200 
4400  0 
4600  200 
4800  400 
N = 5  ΣD = 1200 
Discrete Series
STEPS:

Arrange the data in ascending order

Find out cumulative frequency

Find median; Median = \( \Biggl[{{\frac{N + 1 }{2}}}\Biggl]^{th} item \)

Take deviation of individual values from median. i.e., d = X  Median (ignoring signs)
Coefficient of MD = \( {{{\frac{MD_{Median}}{Median}} }} \)
Table 6.18  

x  f 
2  1 
4  4 
6  6 
8  4 
10  1 
$$ Median\, =\, {{{\frac{N + 1}{2}} }}^{th} item $$ $$ =\, {{{\frac{16 + 1}{2}} }}^{th} item $$ $$ =\, {{{\frac{17}{2}} }}^{th} item $$ $$ =\, 8.5^{th} item $$ $$ =\, 6 $$ $$ ∴\, Median\,=\, 6 $$
Table 6.19  

x  f  D  fD  cf 
2  1  4  4  1 
4  4  2  8  5 
6  6  0  0  11 
8  4  2  8  15 
10  1  4  4  16 
Continuous Series
STEPS:

Find Median

Median class = Size of \( {{{\frac{N}{2}} }}^{th} item \)

Median = \( { L + \frac{\frac{N}{2}  {cf}}{f} × h} \)

Find out d = x  Median
MD_{Median} = \( {{{\frac{ΣfD}{Σf}} }} \)
Coefficient of MD = \( {{{\frac{MD_{Median}}{Median}} }} \)
Table 6.20  

Age  No. of Person 
0  10  6 
10  20  9 
20  30  20 
30  40  5 
40  50  10 
Table 6.21  

Class  f  cf  m  D = m  median  fD 
010  6  6  5  20  120 
1020  9  15  15  10  90 
2030  20  35  25  0  0 
3040  5  40  35  10  50 
4050  10  50  45  20  200 
Σf = 50  ΣfD = 460 
$$ Median \,= \,{ L + \frac{\frac{N}{2}  {cf}}{f} × h} $$ $$ = \,{ 20 + \frac{{25}  {15}}{20} × 10} $$ $$ = \,{ 20 \,+ \,5 \,=\, 25} $$ $$ MD_{Median} \,=\, {{{\frac{ΣfD}{Σf}} }} $$ $$ =\, {{{\frac{460}{50}} }} $$ $$ =\, 9.2 $$ $$ Coefficient \,of \,MD \,= {{{\frac{MD_{Median}}{Median}} }} $$ $$ =\, {{{\frac{9.2}{25}} }} $$ $$ =\, 0.368 $$
MERITS OF MEAN DEVIATION
 It is rigidily defined.
 The calculation is very simple.
 It is based on all values.
 It is not affected by extreme items.
 It truly represents the average deviations of the items.
 It has practical utilities in the fields of Business and Commerce.
DEMERITS OF MEAN DEVIATION
 The algebraic signs are ignored while taking the deviation of items.
 It is not capable of further algebraic tratment.
 It is not often useful for statistical inference.
 It will not give accurate result when deviations are taken from mode.
 Very much affected by sampling fluctuations.