# Examples of Standard Deviation

Examples of Standard Deviation

This tutorial covers some examples of standard deviation using all methods which are discussed in the previous 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 assumed 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 a 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 the 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