Elementary Statistics and Computer Application

Standard Deviation

• It is defined as the positive square-root of the arithmetic mean of the Square of the deviations of the given observation from their arithmetic mean.
• The standard deviation is denoted by s in case of sample and Greek letter σ(sigma) in case of population.
The formula for calculating standard deviation is as follows
for raw data
And for grouped data the formulas are
for discrete data
for continuous data
Where d =
C = class interval

Example 3
Raw Data

• The weights of 5 ear-heads of sorghum are 100, 102,118,124,126 gms. Find the standard deviation.
Solution

x	x2
100	10000
102	10404
118	13924
124	15376
126	15876
570	65580

Standard deviation =

Example 4
Discrete distribution
The frequency distributions of seed yield of 50 seasamum plants are given below. Find the standard deviation.

Seed yield in gms (x)	3	4	5	6	7
Frequency (f)	4	6	15	15	10


Solution

Seed yield in gms (x)	f	fx	fx2
3	4	12	36
4	6	24	96
5	15	75	375
6	15	90	540
7	10	70	490
Total	50	271	1537


Here n = 50
Standard deviation




= 1.1677 gms
Example 5
Continuous distribution

• The Frequency distributions of seed yield of 50 seasamum plants are given below. Find the standard deviation.
Seed yield in gms (x)	2.5-35	3.5-4.5	4.5-5.5	5.5-6.5	6.5-7.5
No. of plants (f)	4	6	15	15	10

Solution


Seed yield in gms (x)	No. of Plants f	Mid x	d=	df	d2 f
2.5-3.5	4	3	-2	-8	16
3.5-4.5	6	4	-1	-6	6
4.5-5.5	15	5	0	0	0
5.5-6.5	15	6	1	15	15
6.5-7.5	10	7	2	20	40
Total	50	25	0	21	77


• A=Assumed mean = 5
• n=50, C=1
• =1.1677
Merits and Demerits of Standard Deviation
Merits

1. It is rigidly defined and its value is always definite and based on all the observations and the actual signs of deviations are used.
2. As it is based on arithmetic mean, it has all the merits of arithmetic mean.
3. It is the most important and widely used measure of dispersion.
4. It is possible for further algebraic treatment.
5. It is less affected by the fluctuations of sampling and hence stable.
6. It is the basis for measuring the coefficient of correlation and sampling.

Demerits

1. It is not easy to understand and it is difficult to calculate.
2. It gives more weight to extreme values because the values are squared up.
3. As it is an absolute measure of variability, it cannot be used for the purpose of comparison.