Functional Relationship

8/9/2019 Functional Relationship

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Functional Relationship

Two variables x and y are functionally related

when the value of the second variable can becalculated when the value of the first is known.

Example

If a stone is dropped, the formula that allows usto accurately calculate the height if it is a function

of elapsed time is:

h = g t!.

Statistical Relationship

Two variables x and y are related statisticallywhen the value of the second variable can be

approximately estimated when the first is known.

Examples

Income and expenses for a family.

"roduction and sales for a factory.

#dvertising expenses and profits for a corporate

enterprise.

If each pair of values is represented as the

coordinates of a point, the set of all points on a

graph is called a scatter plot or a scatter diagram.

# line can be drawn on a scatter plot which bestrepresents the trend of the points. This line is calleda regression line.

Examples


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The scores of $% students in their &athematics and "hysics classes are:

Mathematics 2 3 4 4 5 6 6 7 7 8 10 10

Physics 1 3 2 4 4 4 6 4 6 7 9 10

Coa!iance

The coa!iance is the a!i thmetic mean of the p!o"#cts o$

"eiations of each a!ia%le to their respective means.

Coa!iance is denoted by co&'()* .


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The covar iance ind icates the s ign o f the co!!elationbetween the

variables.

If co&'( )* + 0 the co!!elation is positie.

If co&'( )* , 0 the co!!elation is ne-atie.

#s a disadvantage, its value depends on the chosen scale. That is, the

covariance wil l vary if the height is expressed in meters or feet. It wil l also

vary if money is expressed in euros or dollars.

Examples

The scores of $% students in their mathematics and physics classes are:

Mathematics 2 3 4 4 5 6 6 7 7 8 10 10

Physics 1 3 2 4 4 4 6 4 6 7 9 10

'ind the covariance of the distribution.

x i y i x i . y i

2 1 2

3 3 9

4 2 8

4 4 16

http://www.vitutor.com/statistics/regression/correlation.htmlhttp://www.vitutor.com/statistics/regression/correlation.html


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5 4 20

6 4 24

6 6 36

7 4 28

7 6 42

8 7 56

10 9 90

10 10 100

72 60 431

#fter tabulating the data, the arithmetic means can be found:

The values of the two variables ( and ) are distributed according to the

following table:

)/' 0 2 4

1 2 1 3

2 1 4 2

3 2 5 0


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*ext, find the covariance of the distribution.

+onvert the double entry table into a s imple table and compute the

arithmetic means.

x i y i $ i x i . $ i y i . $ i x i . y i . $ i

0 1 2 0 2 0

0 2 1 0 2 0

0 3 2 0 6 0

2 1 1 2 1 2

2 2 4 8 8 16

2 3 5 10 15 30

4 1 3 12 3 12

4 2 2 8 4 16

20 40 41 76

Co!!elation

Co!!elation is a !elationship or dependency that exists %eteen to

a!ia%les.


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If a correlation exists, it is said that the variables are correlated or

there is a correlation between them.

ypes o$ Co!!elation

Positie Co!!elation

# positive correlation occurs when an increase in one variable increases

the other variable.

The line corresponding to the scatter plot is an increasing line.

e-atie Co!!elation

# negative correlation occurs when an increase in one variable decreases

the other.

The line corresponding to the scatter plot is a decreasing line.


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o Co!!elation

*o correlation occurs when there is no l inear dependency between the

variables.

Pe!$ect Co!!elation

# perfect correlat ion occurs when there is a funcional dependency

between the variables.

In this case all the points are in a straight line.


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St!on- Co!!elation

# correlation is stronger the closer the points are located to one another

on the line.

ea Co!!elation

# correlation is weaker the farther apart the points are located to one

another on the line.


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http://www.vitutor.com/statistics/regression/correlation.html

inea! Co!!elation Coe$$icient

The linea! co!!elation coe$$icient is the !atio between

the coa!iance and the p!o"#ct o$ stan"a!" "eiations of both variables.

The linea! co!!elation coe$$icient is denoted by the letter !.

"roperties of the +orrelation +oefficient

1 The correlation coefficient does not change the measurement scale.

That is, i f the height is expressed in meters or feet, the correlation

coefficient does not change.

2 The sign of the correlation coefficient is the same as the coa!iance .

3 The linear correlation coefficient is a real number between $ and $.

1 ! 1

http://www.vitutor.com/statistics/regression/correlation.htmlhttp://www.vitutor.com/statistics/regression/covariance.htmlhttp://www.vitutor.com/statistics/regression/correlation.htmlhttp://www.vitutor.com/statistics/regression/covariance.html


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4 I f the l inea! co!!elation coe$$icient takes values c loser to 1 ,

the co!!elation is st!on- an" ne-atie, and w il l become s tronger the

closer ! approaches $.

5 If the l inea! co!!e lation coe$$ ic ient takes values close

to 1 the co!!elation is st!on- an" positie, and wi ll become stronger the

closer r approaches $

6 If the l inea! co!!elation coe$$icient takes values c lose to 0,

the co!!elation is ea.

7 If ! 1 or ! 1, there is pe!$ect co!!elation and the line on the

scatter plot is increasing or decreasing respectively.

8 If ! 0, there is no linea! co!!elation .

Example


Mathematics 2 3 4 4 5 6 6 7 7 8 10 10

Physics 1 3 2 4 4 4 6 4 6 7 9 10

'ind the correlation coefficient distribution and interpret it.


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x i y ix i . y

i

x i2 y i

2

2 1 2 4 1

3 3 9 9 9

4 2 8 16 4

4 4 16 16 16

5 4 20 25 16

6 4 24 36 16

6 6 36 36 36

7 4 28 49 16

7 6 42 49 36

8 7 56 64 49

1

09 90

10

081

1

0

1

0100

10

0

10

0

7

2

6

0431

50

4

38

0


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1: 'ind the a!ithmetic means .

2: +alculate the coa!iance .

3: +alculate the stan"a!" "eiations .

4: #pply the formula for the linear correlation coefficient.

The correlation is positie .

#s the corre lation coeff ic ient i s very c lose to $, the correlat ion is

very st!on- .

The values of the two variables ( and ) are distributed according to the

following table:

)/' 0 2 4

1 2 1 3

2 1 4 2

3 2 5 0

+alculate the correlation coefficient.

http://www.vitutor.com/statistics/descriptive/arithmetic_mean.htmlhttp://www.vitutor.com/statistics/regression/covariance.htmlhttp://www.vitutor.com/statistics/descriptive/standard_deviation.htmlhttp://www.vitutor.com/statistics/regression/correlation.html#pchttp://www.vitutor.com/statistics/regression/correlation.html#schttp://www.vitutor.com/statistics/descriptive/arithmetic_mean.htmlhttp://www.vitutor.com/statistics/regression/covariance.htmlhttp://www.vitutor.com/statistics/descriptive/standard_deviation.htmlhttp://www.vitutor.com/statistics/regression/correlation.html#pchttp://www.vitutor.com/statistics/regression/correlation.html#sc


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Turn the double entry table into a single table.

x i y i $ ix i .

$ ix i

2 . $ i y i . $ i y i2

. $ ix i . y i .

$ i

0 1 2 0 0 2 2 0

0 2 1 0 0 2 4 0

0 3 2 0 0 6 18 0

2 1 1 2 4 1 1 2

2 2 4 8 16 8 16 16

2 3 5 10 20 15 45 30

4 1 3 12 48 3 3 12

4 2 2 8 32 4 8 16

20 40 120 41 97 76


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The correlation is ne-atie .

#s the corre lation coeff ic ient i s very c lose to -, the correlat ion is

very ea .

inea! Re-!ession

The regression line is the line that best fits or represents the data on the

scatter plot.

ine of /egression of ) on (

The regression line of y on x is used to estimate the values of y from x.

The slope of the line is the 0uotient between

thecoa!iance and a!iance of the variable (.

ine of /egression of ( on )

The regression line of x on y is used to estimate the values of x from the

values of y.

The slope of the line is the 0uotient between the covariance and variance

of the variable y.

http://www.vitutor.com/statistics/regression/correlation.html#nchttp://www.vitutor.com/statistics/regression/correlation.html#wchttp://www.vitutor.com/statistics/regression/covariance.htmlhttp://www.vitutor.com/statistics/descriptive/variance.htmlhttp://www.vitutor.com/statistics/regression/correlation.html#nchttp://www.vitutor.com/statistics/regression/correlation.html#wchttp://www.vitutor.com/statistics/regression/covariance.htmlhttp://www.vitutor.com/statistics/descriptive/variance.html


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If ! 0 the !e-!ession lines are pe!pen"ic#la! to each other, and

their e0uations are:

y =

x =

Example


Mathematics 2 3 4 4 5 6 6 7 7 8 10 10

Physics 1 3 2 4 4 4 6 4 6 7 9 10

'ind the regression lines and represent them.

x i y ix i . y

i

x i2 y i

2

2 1 2 4 1

3 3 9 9 9

4 2 8 16 4

4 4 16 16 16

5 4 20 25 16

6 4 24 36 16

6 6 36 36 36


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7 4 28 49 16

7 6 42 49 36

8 7 56 64 49

1

09 90

10

081

1

0

1

0100

10

0

10

0

7

2

6

0431

50

4

38

0

1 'ind the a!ithmetic means .

2 +alculate the coa!iance .

3 +alculate the a!iances .

4inear regression of y on x.

5inear regression of x on y.

http://www.vitutor.com/statistics/descriptive/arithmetic_mean.htmlhttp://www.vitutor.com/statistics/regression/covariance.htmlhttp://www.vitutor.com/statistics/descriptive/variance.htmlhttp://www.vitutor.com/statistics/descriptive/arithmetic_mean.htmlhttp://www.vitutor.com/statistics/regression/covariance.htmlhttp://www.vitutor.com/statistics/descriptive/variance.html


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Functional Relationship