Standard Deviation

Standard Deviation


Standard deviation is the positive square root of the arithmetic mean of the squares of the deviations of the given observation from their arithmetic mean.

Among all the methods of finding out dispersion, standard deviation is regarded as the best. It is free from the defects which the earlier methods range, quartile deviation and mean deviation suffer. Its value is based upon each and every item of the series and it also take into account algebraic signs. Standard deviation is also known as ‘Root Mean-Square Deviation’ because it is the square root of the arithmetic mean of the square of the deviations. It is denoted by the σ (sigma).

 Individual Series:

The standard deviation of the set of n observation is given by
standard deviation for individual series

Discrete and Continuous Series:

For a discrete and continuous frequency distribution, the standard deviation is given by        
Standard Deviation for Discrete and Continuous Series

Where N is the total number of observations of the given data. In case of continuous distribution x is taken as the mid-value of the corresponding class.

If the values of the observations and their corresponding frequencies are large, then the calculation of standard deviation is tedious and time consuming. In such cases, we can calculate the standard deviation by taking deviation as follows:

Individual Series:

When the values of the observations are very large in the individual series, the standard deviation can be calculated by using the following formula,   
S.D. for individual series by deviation method

Where d=x–A  and A is assumed mean.

Discrete or Continuous Series:

When the values of the observations and the frequencies are large in case of discrete or continuous series, the calculation of standard deviation can be made easy by using the following formula.            
S.D. for discrete and continuous series by deviation method

Where d=x–A  and A is assumed mean.

When the value of variables or mid-values have some common factor, the calculation of the standard deviation can be made still easier by taking the deviation as follows:
S.D. by taking common factor of variables


Coefficient of Standard Deviation

The relative measure or the coefficient of standard deviation is given by the formula,
Coefficient of Standard Deviation


Variance and Coefficient of Variance


Workout Examples

Example 1: The weekly expenditure of 5 families in rupees are given below:
Family
A
B
C
D
E
Expenditure (Rs.)
300
325
350
375
425

Calculate the standard deviation and its coefficient by:
a.    Direct method
b.    Deviation method

Solution: Here,
a.    Direct method
Families
Expenditure
x2
A
B
C
D
E
300
325
350
375
425
90000
105625
122500
140625
180625

∑x = 1775
∑x2 = 639375
            
Example 1 a. Standard Deviation and Coefficient of S.D. by direct method


b.    Deviation method
Family
Exp. (x)
d = x – A(350)
d2
A
B
C
D
E
300
325
350
375
425
-50
-25
0
25
75
2500
625
0
625
5625


25
9375

Example 1 b. Standard Deviation and Coefficient of S.D. by deviation method



Example 2: Find the standard deviation and its coefficient from the following frequency distribution.

Marks
20
24
30
35
38
40
No. of students
8
7
10
12
6
3

Solution: Here,

Marks
No. of students
fx
fx2
20
24
30
35
38
40
8
7
10
12
6
3
160
168
300
420
228
120
3200
4032
9000
14700
8664
4800

N = 46
∑fx = 1396
∑fx2 = 44396

Example 2. Standard Deviation and Coefficient of S.D.



Example 3: Following are the marks obtained by students in a test exam.

Marks
20-30
30-40
40-50
50-60
60-70
70-80
80-90
No. of students
4
6
10
17
11
9
3

Calculate the standard deviation and its coefficient by
a.    Direct method
b.    Deviation method

Solution: Here,
a.    Direct method

Marks
No. of std(f)
Mid value(x)
fx
fx2
20-30
30-40
40-50
50-60
60-70
70-80
80-90
4
6
10
17
11
9
3
25
35
45
55
65
75
85
100
210
450
935
715
675
255
2500
7350
20250
51425
46475
50625
21675

N = 60

∑fx = 3340
∑fx2 = 200300
            
Example 3 a. Standard Deviation and Coefficient of S.D. by direct method


b.    Deviation method

Marks
f
Mid value(x)
d=(x-55)/10
fd
fd2
20-30
30-40
40-50
50-60
60-70
70-80
80-90
4
6
10
17
11
9
3
25
35
45
55
65
75
85
-3
-2
-1
0
1
2
3
-12
-12
-10
0
11
18
9
36
24
10
0
11
36
27

N = 60


∑fd = 4
∑fd2 = 144

Example 3 b. Standard Deviation and Coefficient of S.D. by deviation method



Example 4: Find the variance and its coefficient from the following data.

Age in yrs
10
15
20
25
30
35
No. of people
5
7
15
25
10
8

Solution: Here,

Age (x)
No. of people (f)
fx
fx2
10
15
20
25
30
35
5
7
15
25
10
8
50
105
300
625
300
280
500
1575
6000
15625
9000
9800

N = 70
∑fx = 1660
∑fx2 = 42500

Example 4: Variance and Coefficient of Variance


You can comment your questions or problems regarding standard deviation here.

No comments:

Powered by Blogger.