Standard Deviation
The technique of the calculation of mean deviation is mathematically illogical as in its calculation the algebraic signs are ignored. This drawback is removed in the calculation of standard deviation. One of the easiest ways of doing away with algebraic signs is to square the figures and this process is adopted in the calculation of standard deviation. In the calculation of SD, first the AM is calculated and the deviations of. various items from the AM are squared. The squared deviations are summed up and the sum is divided by the number of items, The positive square root of the number will give SD. That is, SD is the positive square root of the mean of squared deviations from mean.The concept of standard deviation was first used by Karl Pearson in the year 1893. It is the most commonly used measure of dispersion. It satisfies most of the properties laid down for an ideal measure of dispersion. Note that SD is calculated from AM only. Just as mean is the best measure of central tendency, standard deviation is the best measure of dispersion. Standard deviation is calculated on the basis of mean only.
" Standard deviation is defined as the square root of the arithmetic average of the squares of deviations taken from the arithmetic average of a series. "
It is also known as the rootmeansquare deviation for the reason that it is the square root of the mean of the squared deviations from AM.Standard deviation is denoted by the Greek letter σ (small letter ‘sigma’).
The term variance is used to describe the square of the standard deviation. The term was first used by R. A. Fisher in 1913.
Standard deviation is an absolute measure of dispersion. The corresponding relative measure is called coefficient of SD. Coefficient of variation is also a relative measure. A series with more coefficient of variation is regarded as less consistent or less stable than a series with less coefficient of variation.
Symbolically,
Individual Series
Different methods are used to calculate standard deviation of individual series. All these methods result in the same value of standard deviation. These are given below:

Actual Mean Method
\( \mathbf{σ\, = \,\sqrt{\frac {Σd^{2}}{N}}} \), where d = X  x̄

Assumed Mean Method
\( \mathbf{σ\, = \,\sqrt{\frac{Σd^{2}}{N}\,\,{\Bigl(\frac{Σd}{N}\Bigr)}^{2}} }\)

Direct Method
\( \mathbf{σ\, = \,\sqrt{\frac{Σx^{2}}{N}\,\,{{\overline{X}}}^{2}} }\) or \( \mathbf{σ\, = \,\sqrt{\frac{Σx^{2}}{N}\,\,{\Bigl(\frac{Σx}{N}\Bigr)}^{2}} }\)

Step Deviation Method
\( \mathbf{σ\, = \,\sqrt{\frac{Σd'^{2}}{N}\,\,{\Bigl(\frac{Σd'}{N}\Bigr)}^{2}} \,×\,c }\)
Actual Mean Method
Height: 160, 160, 161, 162, 163, 163, 163, 164, 164, 170.
We need to find d and d^{2}, it is shown in the below given table.
$$ {\overline{X}}\,= \, {{{\frac{ΣX}{N}} }} $$ $$ = \, {{{\frac{1630}{10}} }} $$ $$ = \, 163 $$
Table 6.22  

x  d = (X  x̅)=(X  163)  d^{2} 
160  3  9 
160  3  9 
161  2  4 
162  1  1 
163  0  0 
163  0  0 
163  0  0 
164  1  1 
164  1  1 
170  7  49 
ΣX = 1630, N = 10  Σd^{2} = 74 
Assumed Mean Method
Height: 160, 160, 161, 162, 163, 163, 163, 164, 164, 170.
We need to find d and d^{2}, it is shown in the below given table.
$$ Assumed \,Mean \,=\, 162 $$
Table 6.23  

x  d = (X  162)  d^{2} 
160  2  4 
160  2  4 
161  1  1 
162  0  0 
163  1  1 
163  1  1 
163  1  1 
164  2  2 
164  2  2 
170  8  64 
Σd = 10  Σd^{2} = 84 
Direct Method
Height: 160, 160, 161, 162, 163, 163, 163, 164, 164, 170.
We need to find x^{2}, it is shown in the below given table.
Table 6.24  

x  x^{2} 
160  25600 
160  25600 
161  25921 
162  26244 
163  26569 
163  26569 
163  26569 
164  26896 
164  26896 
170  28900 
Σx = 1630  Σx^{2} = 265764 
Step Deviation Method
5, 10, 25, 30, 50.
We need to find d, d', and d'^{2}. Deviations taken from 25 and common factor 5, it is shown in the below given table.
Table 6.25  

x  d = (X  25)  \( \mathbf{d'\, = \, {{{\frac{(x25)}{5}} }}} \)  d'^{2} 
5  20  4  16 
10  15  3  9 
25  0  0  0 
30  5  1  1 
50  25  5  25 
Σd' = 1  Σd'^{2} = 51 
Discrete Series
Standard deviation can be calculated in four ways:

Actual Mean Method
\( \mathbf{σ\, = \,\sqrt{\frac {Σfx^{2}}{Σf}}} \) or \( \mathbf{σ\, = \,\sqrt{\frac {Σfd^{2}}{Σf}}} \),
where d = X 
X 
Assumed Mean Method
\( \mathbf{σ\, = \,\sqrt{\frac{Σfd^{2}}{Σf}\,\,{\Bigl(\frac{Σfd}{Σf}\Bigr)}^{2}} } \)
where d = X  A

Direct Method
\( \mathbf{σ\, = \,\sqrt{\frac{Σfx^{2}}{Σf}\,\,{\Bigl(\frac{Σfx}{Σf}\Bigr)}^{2}} } \)

Step Deviation Method
\( \mathbf{σ\, = \,\sqrt{\frac{Σfd'^{2}}{Σf}\,\,{\Bigl(\frac{Σfd'}{Σf}\Bigr)}^{2}} \,×\,c } \)
where d = X  A
\( d'\, = \, {{{\frac{(xA)}{C}} }} \)
Actual Mean Method
Table 6.26  

x  f 
6  3 
7  6 
8  9 
9  13 
10  8 
11  5 
12  4 
$$ {\overline{X}}\,= \, {{{\frac{ΣfX}{Σf}} }} $$ $$ = \, {{{\frac{432}{48}} }} $$ $$ = \, 9 $$
Table 6.27  

x  f  fx  (X  
x^{2}  fx^{2} 
6  3  18  3  9  27 
7  6  42  2  4  24 
8  9  72  1  1  9 
9  13  117  0  0  0 
10  8  80  1  1  8 
11  5  55  2  4  20 
12  4  48  3  9  36 
Σf = 48  Σfx = 432  Σfx^{2} = 124 
Assumed Mean Method
Table 6.28  

x  f 
6  3 
7  6 
8  9 
9  13 
10  8 
11  5 
12  4 
Table 6.29  

x  f  (d = X  A)(A = 10)  d^{2}  fd  fd^{2} 
6  3  4  16  12  48 
7  6  3  9  18  54 
8  9  2  4  18  36 
9  13  1  1  13  13 
10  8  0  0  0  0 
11  5  1  1  5  5 
12  4  2  4  8  16 
Σf = 48  Σfd = 48  Σfd^{2} = 172 
Direct Method
Table 6.30  

x  f 
6  3 
7  6 
8  9 
9  13 
10  8 
11  5 
12  4 
Table 6.31  

x  f  fx  x^{2}  fx^{2} 
6  3  18  36  108 
7  6  42  49  294 
8  9  72  64  576 
9  13  117  81  1053 
10  8  80  100  800 
11  5  55  121  605 
12  4  48  144  576 
Σf = 48  Σfx = 432  Σfx^{2} = 4012 
Step Deviation Method
Table 6.32  

x  f 
10  2 
15  8 
20  10 
25  15 
30  3 
35  2 
Table 6.33  

x  f  d = X  A (A = 25)  \(\mathbf{d'\, = \, {{{\frac{(XA)}{C}} }}} \) C = 5  fd'  d'^{2}  fd'^{2} 
10  2  15  3  6  9  18 
15  8  10  2  16  4  32 
20  10  10  1  10  1  10 
25  15  0  0  0  0  0 
30  3  5  1  3  1  3 
35  2  10  2  4  4  8 
Σf = 40  Σfd' = 25  Σf'd^{2} = 71 
Continuous Series
In continuous series we have class intervals for the variable. So we have to find out the midpoint for the various classes. Then the problem becomes similar to those of discrete series.Standard deviation can be calculated in four ways:

Actual Mean Method
\( \mathbf{σ\, = \,\sqrt{\frac {Σfx^{2}}{Σf}}} \)

Assumed Mean Method
\( \mathbf{σ\, = \,\sqrt{\frac{Σfd^{2}}{Σf}\,\,{\Bigl(\frac{Σfd}{Σf}\Bigr)}^{2}} } \)
where d = X  A

Direct Method
\( \mathbf{σ\, = \,\sqrt{\frac{Σfm^{2}}{Σf}\,\,{\Bigl(\frac{Σfm}{Σf}\Bigr)}^{2}} } \)

Step Deviation Method
Deviation d can be converted into d' by multiplying it with the class interval, C.
\( \mathbf{σ\, = \,\sqrt{\frac{Σfd'^{2}}{Σf}\,\,{\Bigl(\frac{Σfd'}{Σf}\Bigr)}^{2}} \,×\,c } \)
where d = X  A
\( d'\, = \, {{{\frac{d}{C}} }} \)
Actual Mean Method
Table 6.34  

x  f 
40  50  2 
50  60  5 
60  70  12 
70  80  18 
80  90  8 
90  100  5 
$$ {\overline{X}}\,= \, {{{\frac{Σfm}{Σf}} }} $$ $$ = \, {{{\frac{3650}{50}} }} $$ $$ = \, 73 $$
Table 6.35  

x  f  m  fm  x (m  
fx  x^{2}  fx^{2} 
40  50  2  45  90  28  56  784  1568 
50  60  5  55  275  18  90  324  1620 
60  70  12  65  780  8  96  64  768 
70  80  18  75  1350  2  36  4  72 
80  90  8  85  680  12  96  144  1152 
90  100  5  95  475  22  110  484  2420 
Σf = 50  Σfm = 3650  0  Σfx^{2} = 7600 
Assumed Mean Method
Table 6.36  

x  f 
40  50  2 
50  60  5 
60  70  12 
70  80  18 
80  90  8 
90  100  5 
Table 6.37  

x  f  m  d (x  75)  d^{2}  fd  fd^{2} 
40  50  2  45  30  900  60  1800 
50  60  5  55  20  400  100  2000 
60  70  12  65  10  100  120  1200 
70  80  18  75  0  0  0  0 
80  90  8  85  10  100  80  800 
90  100  5  95  20  400  100  200 
Σf = 50  Σfd = 100  Σfd^{2} = 7800 
Direct Method
Table 6.38  

x  f 
40  50  2 
50  60  5 
60  70  12 
70  80  18 
80  90  8 
90  100  5 
Table 6.39  

x  f  m  fm  fm^{2} 
40  50  2  45  90  4050 
50  60  5  55  275  15125 
60  70  12  65  780  50700 
70  80  18  75  1350  101250 
80  90  8  85  680  57800 
90  100  5  95  475  45125 
Σf = 50  Σfm = 3650  Σfm^{2} = 274050 
Step Deviation Method
Table 6.40  

x  f 
40  50  2 
50  60  5 
60  70  12 
70  80  18 
80  90  8 
90  100  5 
Table 6.41  

x  f  m  \(\mathbf{d'\, = \, {{{\frac{(m75)}{10}} }}} \)  fd'  d'^{2}  fd'^{2} 
40  50  2  45  3  6  9  18 
50  60  5  55  2  10  4  20 
60  70  12  65  1  12  1  12 
70  80  18  75  0  0  0  0 
80  90  8  85  1  8  1  8 
90  100  5  95  2  10  4  20 
Σf = 50  Σfd' = 10  Σfd'^{2} = 78 
Properties of SD

SD is calculated from AM because; the sum of the squares of the deviations taken from the AM is least.

SD is independent of the change of origin. That is, if a constant A is added or subtracted from each of the items of series, then SD remains unchanged.

SD is affected by change of scale. That is, if each item of series is multiplied or divided by a constant, say, c, then the SD is also affected by the same constant c.
MERITS OF STANDARD DEVIATION
 Rigidly defined.
 Its value is always definite.
 Based on all items.
 It is capable of further algebraic treatment.
 It possesses many mathematical properties.
 It is less affected by sampling fluctuations.
 The difficulty about algebrfaic signs is not found here.
DEMERITS OF STANDARD DEVIATION
 Calculation is not easy.
 It is not understood by a layman.
 Much affected by extreme values.
 Gives much importance to extreme values than values near the mean (this happens because of taking square of the deviations).
Absolute and Relative Measures of Dispersion
Absolute measures of dispersion are expressed in the same statistical unit in which the original data are given such as rupees, tonnes, centimeters, etc. In case two sets of data are expressed in different units, absolute measures of dispersion are not comparable. In such cases, measures of relative dispersion should be used.A measure of relative dispersion is the ratio of measure of absolute dispersion to an appropriate average. It is sometimes called a coefficient of dispersion because coefficient means a pure number that is independent of the unit of measurement. Greater the value of coefficient of dispersion more is the variability in a distribution (less consistency).
Table 6.42  

Absolue Measure  Relative Measure 
\(\mathbf{Range\, = \, L\,\,S} \)  \(\mathbf{ Coefficient \,of \,Range \,= \,{{\frac{L  S }{L + S}}}} \) 
\(\mathbf{ Quartile \,Deviation\, =\, {{{\frac{Q_3  Q_1}{2}} }}} \)  \(\mathbf{ Coefficient\, of\, Quartile\, Deviation\, =\, {{{\frac{Q_3  Q_1}{Q_3 + Q_1}} }}} \) 
\(\mathbf{Mean\, Deviation\, =\, {{{\frac{ΣD}{N}} }} }\)  \(\mathbf{Coefficient\, of\, MD\, =\, {{{\frac{MD}{{Mean\,/\,Median\,/\,Mode}}} }}} \) 
\( \mathbf{Standard \, Deviation \, = \,\sqrt{\frac {Σx^{2}}{Σf}}} \); \( \mathbf{\sqrt{\frac{Σd^{2}}{Σf}\,\,{\Bigl(\frac{Σd}{Σf}\Bigr)}^{2}} }\); \( \mathbf{\sqrt{\frac{Σfd^{2}}{Σf}\,\,{\Bigl(\frac{Σfd}{Σf}\Bigr)}^{2}} }\)  \(\mathbf{Coefficient\, of\, SD\, =\, {{{\frac{σ}{\overline{X}}} }}} × 100 \) 