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