Definition

“The standard deviation is defined as the positive square root of the mean of the squared deviations of the values from their mean.”

Characteristics of Standard Deviation:

i. The standard deviation is affected by the values of every observations.

ii. The process of squaring the deviations before adding, avoids the algebraic fallacy of disregarding signs.

iii. In general it is less affected by fluctuations of sampling than the other measures of dispersion.

iv. It has a specific mathematical meaning and could be easily adapted according to nature of algebraic treatment.

v. It has great practical utility in sampling and statistical inference.

Standard Deviation for Ungrouped Data

The standard deviation of a set of n values, denoted by S. The standard deviation for ungrouped data is mathematically represented as follows;

For Example

Find the standard deviation for the values 2, 3, 6, 8 and 11.

Standard Deviation for Grouped Data

In case of a frequency distribution with as class marks and as the corresponding class frequencies, the standard deviation shall be calculated by using the following formula;

Where n =

For Example

Find the frequency distribution for the frequency distribution of marks obtained by students provided in following Table 25

Table 25

Marks	Frequency (f)	Class Marks (X)	fX
20 – 24	1	22	22
25 – 29	4	27	108
30 – 34	8	32	256
35 – 39	11	37	407
40 – 44	15	42	630
45 – 49	9	47	423
50 – 54	2	52	104

Marks	f	X
20 – 24	1	22	-17	-17	289	289
25 – 29	4	27	-12	-48	144	576
30 – 34	8	32	-7	-56	49	392
35 – 39	11	37	-2	-22	4	44
40 – 44	15	42	3	45	9	135
45 – 49	9	47	8	72	64	576
50 – 54	2	52	13	26	169	338
	n= ∑f=50

For Example

Find the standard deviation of the weights distribution of 120 students in a University.

Weight (lb)	Class Marks (X)	Frequency (f)
110 – 119	114.5	1	-41.7	1738.89	1738.89
120 – 129	124.5	4	-31.7	1004.89	4019.56
130 – 139	134.5	17	-21.7	470.89	8005.13
140 – 149	144.5	28	-11.7	136.89	3832.92
150 – 159	154.5	25	-1.7	2.89	72.25
160 – 169	164.5	18	8.3	68.89	1240.02
170 – 179	174.5	13	18.3	334.89	4353.57
180 – 189	184.5	6	28.3	800.89	4805.34
190 – 199	194.5	5	38.3	1466.89	7334.45
200 – 209	204.5	2	48.3	2332.89	4665.78
210 -219	214.5	1	58.3	3398.89	3398.89
	n = ∑f = 120

Short Method for Computing Standard Deviation:

From the above methods of calculating standard deviation, both grouped and ungrouped, we have noticed that computing standard deviation requires number of calculations which makes the task laborious. To make the difficult task easy, there is also a short method for computing standard deviation. Such short formulae are explained as follows;