CORRELATION AND RELATION

Group Presentation

Correlation and Regression

Submitted to - DR. ABHISHEK KUMAR

GROUP MATES

BASUDEV PAL 20BAI10364

MIMANSA BHAGAVA 20BHI10041

ANAND K. MISHRA 20BAC10044

MAHI 20BAI10342

ASHFIYA KHAN 20BHI10057

JAIVARDHAN SINGH 20BAI10360

VIVEK AGARWAL 20BAI10303

JAYA PANDEY 20BAI10261

INDEX

Methods to find correlation coefficiant

Introduction

Types of corelation

Errors of correlation Coeficiants

Real Life applications of correlations

Properties of correlation coefficient

Types of Regression

Introduction to Regression

Rank correlation

Properties of Regression Coeffiecients

11

Real life Example

10

Thanks

12

Introduction

What is Correlation ?

Correlation

A correlation is a relationship between two variables. The data can be represented by the ordered pairs (x, y) where x is the independent (or explanatory) variable, and y is the dependent (or response) variable. A scatter plot can be used to determine whether a linear (straight line) correlation exists between two variables y

Types Of Correlation

1. Positive Correlation 2.Negative Correlation 3. Zero or no Correlation

METHODS TO FIND CORRELATION

There are two methods to compute the correlation coefficient Scatter diagram Karl Pearson coefficient of correlation

Scatter Diagram

If the values of the variables X and Y are plotted along the x-axis and y-axis respectively in the xy plane, the diagram of dots obtained is known as the scatter diagram.

Karl Pearson Coefficient of Correlation

Karl Pearson’s coefficient of correlation is an extensively used mathematical method in which the numerical representation is applied to measure the level of relation between linearly related variables. The coefficient of correlation is expressed by “r”.

Properties of Correlation Coeficients

• It is independent of change of order and change of origin i.e. if u=x-h/a , v=x-k/b then r(x,y)=r(u,v)
• Limit of r(x,y) is -1 to 1
• r(x,y) only provides a measure of the linear relationship between these two variables; it is not suitable for non-linear relation.
• Two independent variables are uncorrelated as for independent variable cov(x,y)=0.
• Two uncorrelated variables (zero correlation and zero covariance) are not independent.
• Zero correlation implies linear independence.

ERRORS OF CORRELATION COEFICIENTS

The Probable Error of Correlation Coefficient helps in determining the accuracy and reliability of the value of the coefficient that in so far depends on the random sampling. In other words, the probable error (P.E.) is the value which is added or subtracted from the coefficient of correlation (r) to get the upper limit and the lower limit respectively, within which the value of the correlation expectedly lies.

RANK CORRELATION

Let us take a group of n individuals arranged in order of merit or proficiency in possession of two characteristics A and B. These ranks in the two characteristics will, in general, be different Pearsonian coefficient of correlation between the ranks Xi's and Yi's is called the rank correlation coefficient between A and B for that group of individuals.

Spearman’s rank correlation is nothing but the Karl Pearson coefficient of correlation between the ranks so it can be interpreted in the same as Karl Pearson coefficient of correlation. Spearman’s rank correlation is easy to calculate compared to the Karl Pearson coefficient of correlation. While dealing with qualitative data, Spearman's correlation coefficient should be used.

REAL LIFE EXAMPLES OF CORRELATION

Time Spent Running vs. Body Fat

Height vs. Weight

Example 1

Example 2

REAL LIFE EXAMPLES OF CORRELATION

Coffee Consumption vs. Intelligence

Shoe Size vs. Movies Watched

Example 4

Example 3

INTRODUCTION TO REGRESSION

Regression analysis is a measure of the average measure of the relationship between two variables. It means “Stepping back towards the Average”. In regression analysis, there are two types of variables Which is influenced or is to be predicted by another variable called dependent variable (regressed, explained). Which influences another variable called the independent variable (regressor, predictor and explanatory).

Let us take a set of bivariate data, let’s say x and y. So if x and y are related then we will find that the points in the scatter diagram will cluster round some curve called the "curve of regression". If the curve is a straight line we call it “line of regression”. The line of regression is the line that gives the best estimate to the value of one variable for any specific value of the other variable. Thus the line of regression is the line of "best /it" and is obtained by the principles of least squares.

APPLICATIONOF REGRESSION ANALYSIS

REAL LIFE EXAMPLES OF REGRESSION

Businesses often use linear regression to understand the relationship between advertising spending and revenue. For example, they might fit a simple linear regression model using advertising spending as the predictor variable and revenue as the response variable

Depending on the value of β1, a company may decide to either decrease or increase their ad spending.

REAL LIFE EXAMPLES OF REGRESSION

Medical researchers often use linear regression to understand the relationship between drug dosage and blood pressure of patients. For example, researchers might administer various dosages of a certain drug to patients and observe how their blood pressure responds. They might fit a simple linear regression model using dosage as the predictor variable and blood pressure as the response variable.

Depending on the value of β1, researchers may decide to change the dosage given to a patient.

PROPERTIES OF REGRESSION

" The correlation coefficient is the geometric mean between the regression coefficients, the sign to be taken before the square root is that of the regression coefficients."

" In a particular case of perfect correlation, positive or negative, i.e., r ± I, the equation of the line of regression of Y on X becomes the equation of the line of regression of X on Y "

2nd PROPERTY

1st PROPERTY

"If one of the regression coefficients is greater than unity, the other must be less than unity."

"The arithmetic mean of the regression coefficient is greater than the correlation coefficient r, provided r>0"

3rd PROPERTY

4th PROPERTY

THANK YOU