# 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 $2$as 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