2.correlation and regression

Correlation and Regression

1. Definition of Correlation

2. Scatter Diagram,

3. Karl Pearson Coefficient of Correlation

4. Limits for Correlation Coefficient

5. Definition of Regression

6. Lines of Regression

7. Regression Curves

8. Regression coefficients

9. properties of Regression coefficients

10. Correlation Analysis vs. Regression Analysis.

Subject :- Statistics & Numerical Mathematics Created By Prof . Santosh Ashilwar

Definition of Correlation

Correlation is a statistical tool which studies the relationship

between the two variable under study

Correlation analysis involves various method and techniques

for studying and measuring the extent of the relationship

between the two variable

If there is any relation between two variables i.e. when one

variable changes the other also changes in the same or in the

opposite direction, we say that the two variables are correlated.

It means the study of existence, magnitude and direction of the

relation between two or more variables.

The famous astronomist Bravais, Prof. Sir Fanci’s Galton,

Karl Pearson Prof. Neiswang and so many others have

contributed to this great subject.


Negative Correlation–as x increases, y decreases

x = hours of training (horizontal axis)

y = number of accidents (vertical axis)


Scatter Plots and Types of Correlation

60

50

40

30

20

10

0

0 2 4 6 8 10 12 14 16 18 20

Hours of Training

Accid

en

ts

Positive Correlation–as x increases, y increases

x = SAT score

y = GPA

GP

A



4.003.753.50

3.002.752.502.252.00

1.501.75

3.25

300 350 400 450 500 550 600 650 700 750 800

Math SAT

No linear correlation

x = height y = IQ



160

150

140

130

120

110

100

90

80

60 64 68 72 76 80

Height

IQ

Types of Correlation

Positive and negative correlation

Linear and non-linear correlation


Positive and negative correlation

If two variables change in the same direction, then this

is called a positive correlation. For example:

Advertising and sales.

If two variables change in the opposite direction then

the correlation is called a negative correlation. For

example: T.V. registrations and cinema attendance.


Linear and non-linear correlation

The nature of the graph gives us the idea of the linear type

of correlation between two variables. If the graph is in a

straight line, the correlation is called a "linear correlation"

and if the graph is not in a straight line, the correlation is

non-linear or curvi-linear.


Degrees of Correlation

Through the coefficient of correlation, we can measure the degree or

extent of the correlation between two variables.

On the basis of the coefficient of correlation we can also determine

whether the correlation is positive or negative and also its degree or

extent.

Perfect correlation: If two variables changes in the same direction and

in the same proportion, the correlation between the two is perfect

positive

Absence of correlation: If two series of two variables exhibit no

relations between them or change in variable does not lead to a change

in the other variable

Limited degrees of correlation: If two variables are not perfectly

correlated or is there a perfect absence of correlation, then we term the

correlation as Limited correlation


Degrees of Correlation

High degree, moderate degree or low degree are the three categories of

this kind of correlation. The following table reveals the effect of

coefficient or correlation.


Degrees Positive Negative

Absence of correlation

Zero 0

Perfect correlation + 1 -1

High degree + 0.75 to + 1 - 0.75 to -1

Moderate degree + 0.25 to + 0.75 - 0.25 to - 0.75

Low degree 0 to 0.25 0 to - 0.25

Methods Of Determining Correlation

We shall consider the following most commonly used

methods.

(1) Scatter Plot

(2) Kar Pearson’s coefficient of correlation


Scatter Plot Method

In this method the values of the two variables are

plotted on a graph paper.

One is taken along the horizontal ( (x-axis) and the

other along the vertical (y-axis).

By plotting the data, we get points (dots) on the

graph which are generally scattered and hence the

name ‘Scatter Plot’.

The manner in which these points are scattered,

suggest the degree and the direction of correlation.

The degree of correlation is denoted by ‘ r ’ and its

direction is given by the signs positive and negative.


Scatter Plot Method i) If all points lie on a rising straight line the correlation is

perfectly positive and r = +1 (see fig.1 )

ii) If all points lie on a falling straight line the correlation is

perfectly negative and r = -1 (see fig.2)

iii) If the points lie in narrow strip, rising upwards, the correlation

is high degree of positive (see fig.3)

iv) If the points lie in a narrow strip, falling downwards, the

correlation is high degree of negative (see fig.4)

v) If the points are spread widely over a broad strip, rising

upwards, the correlation is low degree positive (see fig.5)

vi) If the points are spread widely over a broad strip, falling

downward, the correlation is low degree negative (see fig.6)

vii) If the points are spread (scattered) without any specific pattern,

the correlation is absent. i.e. r = 0. (see fig.7)


Scatter Plot Method

i) If all points lie on a rising straight line the correlation is perfectly

positive and r = +1 (see fig.1 )

ii) If all points lie on a falling straight line the correlation is perfectly

negative and r = -1 (see fig.2)


Scatter Plot Method

iii) If the points lie in narrow strip, rising upwards, the correlation is high

degree of positive (see fig.3)

iv) If the points lie in a narrow strip, falling downwards, the correlation is

high degree of negative (see fig.4)


Scatter Plot Method

v) If the points are spread widely over a broad strip, rising upwards, the

correlation is low degree positive (see fig.5)

vi) If the points are spread widely over a broad strip, falling downward,

the correlation is low degree negative (see fig.6)

vii) If the points are spread (scattered) without any specific pattern, the

correlation is absent. i.e. r = 0. (see fig.7)Subject :- Statistics & Numerical Mathematics Created By Prof . Santosh Ashilwar

Scatter Plot Method

Though this method is simple and is a rough idea

about the existence and the degree of correlation, it

is not reliable. As it is not a mathematical method,

it cannot measure the degree of correlation.


Karl Pearson’s coefficient of correlation

It gives the numerical expression for the measure of

correlation. it is noted by ‘ r ’. The value of ‘ r ’ gives the

magnitude of correlation and sign denotes its direction. It is

defined as





Example Calculate the coefficient of correlation between the

heights of father and his son for the following data.


Height of father (cm):

165 166 167 168 167 169 170 172

Height of son (cm):

167 168 165 172 168 172 169 171

Solution: n = 8 ( pairs of observations )

Height of

father

xi

Height of

son

yi

x = xi-x y = yi-y xy x2 y2

165 167 -3 -2 6 9 4

166 168 -2 -1 2 4 1

167 165 -1 -4 4 1 16

167 168 -1 -1 1 1 1

168 172 0 3 0 0 9

169 172 1 3 3 1 9

170 169 2 0 0 4 0

172 171 4 2 8 16 4

Sxi=1344 Syi=1352 0 0 Sxy=24 Sx2=36 Sy2=44




Example From the following data compute the coefficient of

correlation between x and y.

.


Limits for Correlation Coefficient

Pearsonian correlation coefficient can not exceed 1

numerically. In other words it lies between -1 and +1 .

Symbolically , -1 <=r<=+1


Definition of Regression

1. Regression can be defined as a method that estimates the value of

one variable when that of other variable is known, provided the

variables are correlated.

2. The dictionary meaning of regression is "to go backward." It was

used for the first time by Sir Francis Galton in his research paper

"Regression towards mediocrity in hereditary stature."

3. Regression helps us to estimate one variable or the dependent

variable from the other variable or the independent variable

4. According to Blair “Regression is the measure of the average

relationship between two or more variable in terms of the original

units of data”

5. According to Wallis and Robert “It is often more important to find

out what the relation actually is , in order to estimate or predict one

variable(the depedent variable) and statistical techniques

appropriate in such cases is Called Regression Analysis ”


Lines of Regression1. In scatter plot, we have seen that if the variables are

highly correlated then the points (dots) lie in a narrowstrip. if the strip is nearly straight, we can draw astraight line, such that all points are close to it fromboth sides.

2. This line is called the line of best fit if it minimizes thedistances of all data points from it.

3. This line is called the line of regression. Nowprediction is easy because now all we need to do is toextend the line and read the value

4. Thus to obtain a line of regression, we need to have aline of best fit. But statisticians don’t measure thedistances by dropping perpendiculars from points on tothe line.

5. They measure deviations ( or errors or residuals asthey are called) (i) vertically and (ii) horizontally. Thuswe get two lines of regressions as shown in the figure(1) and (2).


Lines of Regression

They are obtained by

(I)Graphically - by Scatter plot

(II)Mathematically - by the method of least squares.


Lines of Regression By - Graphically - by Scatter plot Method

Line of regression of y on x

Its form is y = a + b x

It is used to estimate y when x is givenWhere a is intercept of the line and b is the slope of line x on y.

(2) Line of regression of x on y

Its form is x = a + b y

It is used to estimate x when y is given.

Where a is intercept of the line and b is the slope of line y on x.


Lines of Regression By - by the method of least squares


Line of regression of y on x

Where

Line of regression of x on y

Where

Example on Regression By - by the method of least squares


A panel of two judges A and B graded dramatic performance by independently awarding marks as follows:

Solution:-



Example Continue ….



Example Continue ….

Date post:	08-Jul-2015
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2.correlation and regression

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