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