Examples of Standard Deviation

 Examples of Standard Deviation:

This tutorial is about some examples of standard deviation using all methods which are discussed in the pervious tutorial.

Example:

Calculate the standard deviation for the following sample data using all methods: 2, 4, 8, 6, 10, and 12.
Solution:
            Method-I: Actual Mean Method

X

{\left( {X - \overline X    } \right)^2}

2

{(2 - 7)^2} = 25

4

{(4 - 7)^2} = 9

8

{(8 - 7)^2} = 1

6

{(6 - 7)^2} = 1

10

{(10 - 7)^2} = 9

12

{(12 - 7)^2} = 25

\sum X = 42

\sum {\left( {X - \overline X } \right)^2} = 70

 

\overline X = \frac{{\sum X}}{n} = \frac{{42}}{6} = 7
S = \sqrt {\frac{{\sum {{\left( {X - \overline X }  \right)}^2}}}{n}}
S = \sqrt {\frac{{70}}{6}} = \sqrt {\frac{{35}}{3}} = 3.42


            Method-II: Taking Assumed Mean as 6

X

D = \left( {X - 6}    \right)

{D^2}

2

 - 4

16

4

 - 2

4

8

2

4

6

0

0

10

4

16

12

6

36

Total

\sum D = 6

\sum {D^2} = 76

 

S = \sqrt {\frac{{\sum {D^2}}}{n} - {{\left(  {\frac{{\sum D}}{n}} \right)}^2}}
S = \sqrt {\frac{{76}}{6} - {{\left( {\frac{6}{6}}  \right)}^2}} = \sqrt {\frac{{70}}{6}}
S = \sqrt {\frac{{35}}{3}} = 3.42


            Method-III: Taking Assume Mean as Zero

X

{X^2}

2

4

4

16

8

64

6

36

10

100

12

144

\sum X = 42

\sum {X^2} = 364

S = \sqrt {\frac{{\sum {X^2}}}{n} - {{\left(  {\frac{{\sum X}}{n}} \right)}^2}}
S = \sqrt {\frac{{364}}{6} - {{\left(  {\frac{{42}}{6}} \right)}^2}}
S = \sqrt {\frac{{70}}{6}} = \sqrt {\frac{{35}}{3}} = 3.42


            Method-IV: Taking 2as common divisor or factor

X

U = \left( {X - 4}    \right)/2

{U^2}

2

 - 1

1

4

0

0

8

2

4

6

1

1

10

3

9

12

4

16

Total

\sum U = 9

\sum {U^2} = 31

 

S = \sqrt {\frac{{\sum {U^2}}}{n} - {{\left(  {\frac{{\sum U}}{n}} \right)}^2}} \times c
S = \sqrt {\frac{{31}}{6} - {{\left( {\frac{9}{6}}  \right)}^2}} \times 2
S = \sqrt {2.92} \times 2 = 3.42

Example:
            Calculate standard deviation from the following distribution of marks by using all the methods.

Marks

No. of Students

1 - 3

40

3 - 5

30

5 - 7

20

7 - 9

10

Solution:
            Method-I: Actual Mean Method

Marks

f

X

fX

{\left( {X - \overline X } \right)^2}

f{\left( {X - \overline X } \right)^2}

1 - 3

40

2

80

4

160

3 - 5

30

4

120

0

0

5 - 7

20

6

120

4

80

7 - 9

10

8

80

16

160

Total

100

 

400

 

400

            \overline X = \frac{{\sum fX}}{{\sum f}} =  \frac{{400}}{{100}} = 4
            S = \sqrt {\frac{{\sum  f{{\left( {X - \overline X } \right)}^2}}}{{\sum f}}} = \sqrt {\frac{{400}}{{100}}} = \sqrt 4  = 2 Marks

            Method-II: Taking assumed mean as 2

Marks

f

X

D = \left( {X - 2} \right)

fD

f{D^2}

1 - 3

40

2

0

0

0

3 - 5

30

4

2

60

120

5 - 7

20

6

4

80

320

7 - 9

10

8

6

60

160

Total

100

 

 

200

800

            S = \sqrt {\frac{{\sum  f{D^2}}}{{\sum f}} - {{\left( {\frac{{\sum fD}}{{\sum f}}} \right)}^2}} = \sqrt {\frac{{800}}{{100}} - {{\left(  {\frac{{200}}{{100}}} \right)}^2}}
            S = \sqrt {8 - 4} = \sqrt 4  = 2 Marks

            Method-III: Using assumed mean as Zero

Marks

f

X

fX

f{X^2}

1 - 3

40

2

80

160

3 - 5

30

4

120

480

5 - 7

20

6

120

720

7 - 9

10

8

80

640

Total

100

 

400

2000

            S = \sqrt {\frac{{\sum  f{X^2}}}{{\sum f}} - {{\left( {\frac{{\sum fX}}{{\sum f}}} \right)}^2}} = \sqrt {\frac{{2000}}{{100}} - {{\left(  {\frac{{400}}{{100}}} \right)}^2}}
            S = \sqrt {20 - 16} = \sqrt 4  = 2 Marks

            Method-IV: By taking 2 as the common divisor

Marks

f

X

U = \left( {X - 2} \right)/2

fU

f{U^2}

1 - 3

40

2

 - 2

 - 80

160

3 - 5

30

4

 - 1

 - 30

30

5 - 7

20

6

0

0

0

7 - 9

10

8

1

10

10

Total

100

 

 

 - 100

200

            S = \sqrt {\frac{{\sum  f{U^2}}}{{\sum f}} - {{\left( {\frac{{\sum fU}}{{\sum f}}} \right)}^2}} \times h = \sqrt {\frac{{200}}{{100}} -  {{\left( {\frac{{ - 100}}{{100}}} \right)}^2}} \times 2
            S = \sqrt {2 - 1} \times 2 = \sqrt 1 \times 2 = 1 \times 2 = 2 Marks

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