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# E-LESSON-14 , 15

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BBA/BCOM 2 All right are reserved with CU-IDOL Business Mathematics and Statistics Course Code: BBA102/BCM102 Semester: First e-Lesson: 7 SLM Unit: 14,15 www.cuidol.in Unit-14,15(BBA102/BCM-102)

Business Mathematics and Statistics 33 OBJECTIVES INTRODUCTION To make students learn about how to In this unit we are going to learn about calculate and interpret the correlation the concept of correlation between two variables. To make the understand about the different Under this you will be able to methods of calculating correlation. understand the methods of correlation To make the aware about the practicality of Students will be able understand the correlation. practicality of regression . www.cuidol.in Unit-14,15(BBA102/BCM-102) INSTITUTE OF DISTANACEll ArNigDhtOaNrLeINreEsLeErAvRedNIwNiGth CU-IDOL

TOPICS TO BE COVERED 4 Correlation Types of correlation Regression Analysis Meaning Relationship between correlation and regression www.cuidol.in Unit-14,15(BBA102/BCM-102) All right are reserved with CU-IDOL

CORRELATION ANALYSIS 5  Correlation a LINEAR association between two random variables  Correlation analysis show us how to determine both the nature and strength of relationship between two variables  When variables are dependent on time correlation is applied  Correlation lies between +1 to -1 www.cuidol.in Unit-14,15(BBA102/BCM-102) All right are reserved with CU-IDOL

6  A zero correlation indicates that there is no relationship between the variables  A correlation of –1 indicates a perfect negative correlation  A correlation of +1 indicates a perfect positive correlation www.cuidol.in Unit-14,15(BBA102/BCM-102) All right are reserved with CU-IDOL

Types of Correlation 7 There are three types of correlation Types Type 1 Type 2 Type 3 www.cuidol.in Unit-14,15(BBA102/BCM-102) All right are reserved with CU-IDOL

Type1 8 Positive Negative No Perfect  If two related variables are such that when one increases (decreases), the other also increases (decreases).  If two variables are such that when one increases (decreases), the other decreases (increases)  If both the variables are independent www.cuidol.in Unit-14,15(BBA102/BCM-102) All right are reserved with CU-IDOL

9 Type 2 Linear Non – linear  When plotted on a graph it tends to be a perfect line  When plotted on a graph it is not a straight line www.cuidol.in Unit-14,15(BBA102/BCM-102) All right are reserved with CU-IDOL

10 www.cuidol.in Unit-14,15(BBA102/BCM-102) All right are reserved with CU-IDOL

11 Type 3 Simple Multiple Partial  Two independent and one dependent variable  One dependent and more than one independent variables  One dependent variable and more than one independent variable but only one independent variable is considered and other independent variables are considered constant www.cuidol.in Unit-14,15(BBA102/BCM-102) All right are reserved with CU-IDOL

Methods of Studying Correlation 12  Scatter Diagram Method  Karl Pearson Coefficient Correlation of Method  Spearman’s Rank Correlation Method www.cuidol.in Unit-14,15(BBA102/BCM-102) All right are reserved with CU-IDOL

Correlation: Linear 13 Relationships Strong relationship = good linear fit 180Symptom Index 160 160 140 140 Symptom Index 120 120 100 100 50 100 150 200 250 50 100 150 200 250 80 80 60 60 40 40 20 20 0 0 0 0 Drug A (dose in mg) Drug B (dose in mg) Very good fit Moderate fit Points clustered closely around a line show a strong correlation. The line is a good predictor (good fit) with the data. The more spread out the points, the weaker the correlation, and the less good the fit. The line is a REGRESSSION line (Y = bX + a) www.cuidol.in Unit-14,15(BBA102/BCM-102) All right are reserved with CU-IDOL

Coefficient of Correlation 14  A measure of the strength of the linear relationship between two variables that is defined in terms of the (sample) covariance of the variables divided by their (sample) standard deviations  Represented by “r”  r lies between +1 to -1  Magnitude and Direction www.cuidol.in Unit-14,15(BBA102/BCM-102) All right are reserved with CU-IDOL

15  -1 < r < +1  The + and – signs are used for positive linear correlations and negative linear correlations, respectively www.cuidol.in Unit-14,15(BBA102/BCM-102) All right are reserved with CU-IDOL

Coefficient of Determination 16  Coefficient of determination lies between 0 to 1  Represented by r2  The coefficient of determination is a measure of how well the regression line represents the data  If the regression line passes exactly through every point on the scatter plot, it would be able to explain all of the variation  The further the line is away from the points, the less it is able to explain www.cuidol.in Unit-14,15(BBA102/BCM-102) All right are reserved with CU-IDOL

Regression Analysis 17  Regression analysis refers to assessing the relationship between the outcome variable and one or more variables. The outcome variable is known as the dependent or response variable and the risk elements, and cofounders are known as predictors or independent variables. The dependent variable is shown by “y” and independent variables are shown by “x” in regression analysis.  The sample of a correlation coefficient is estimated in the correlation analysis. It ranges between -1 and +1, denoted by r and quantifies the strength and direction of the linear association among two variables. The correlation among two variables can either be positive, i.e, a higher level of one variable is related to a higher level of another or negative, i.e, a higher level of one variable is related to a lower level of the other.  The sign of the coefficient of correlation shows the direction of the association. The magnitude of the coefficient shows the strength of the association.  For example, a correlation of r = 0.8 indicates a positive and strong association among two variables, while a correlation of r = -0.3 shows a negative and weak association. A correlation near to zero shows the non-existence of linear association among two continuous variables. www.cuidol.in Unit-14,15(BBA102/BCM-102) All right are reserved with CU-IDOL

Linear Regression 18  Linear regression is a linear approach to modelling the relationship between the scalar components and one or more independent variables. If the regression has one independent variable, then it is known as a simple linear regression. If it has more than one independent variables, then it is known as multiple linear regression. Linear regression only focuses on the conditional probability distribution of the given values rather than the joint probability distribution. In general, all the real world regressions models .involve multiple predictors. So, the term linear regression often describes multivariate linear regression www.cuidol.in Unit-14,15(BBA102/BCM-102) All right are reserved with CU-IDOL

Correlation and Regression 19 Differences There are some differences between Correlation and regression.  Correlation shows the quantity of the degree to which two variables are associated. It does not fix a line through the data points. You compute a correlation that shows how much one variable changes when the other remains constant. When r is 0.0, the relationship does not exist. When r is positive, one variable goes high as the other goes up. When r is negative, one variable goes high as the other goes down.  Linear regression finds the best line that predicts y from x, but Correlation does not fit a line.  Correlation is used when you measure both variables, while linear regression is mostly applied when x is a variable that is manipulated. www.cuidol.in Unit-14,15(BBA102/BCM-102) All right are reserved with CU-IDOL

Comparison Between Correlation 20 and Regression Basis Correlation Regression Meaning A statistical measure that defines co- Describes how an independent relationship or association of two variable is associated with the Dependent and Independent variables. dependent variable. variables Usage No difference Both variables are different. Objective To describe a linear relationship To fit the best line and estimate one between two variables. variable based on another variable. To find a value expressing the To estimate values of a random relationship between variables. variable based on the values of a fixed variable. www.cuidol.in Unit-14,15(BBA102/BCM-102) All right are reserved with CU-IDOL

Correlation and Regression 21 Statistics  The degree of association is measured by “r” after its originator and a measure of linear association. Other complicated measures are used if a curved line is needed to represent the relationship.  The graph represents the correlation.  The coefficient of correlation is measured on a scale that varies from +1 to -1 through 0. The complete correlation among two variables is represented by either +1 or -1. The correlation is positive when one variable increases and so does the other; while it is negative when one decreases as the other increases. The absence of correlation is described by 0. www.cuidol.in Unit-14,15(BBA102/BCM-102) All right are reserved with CU-IDOL

Multiple Choice Questions 22 1.The correlation coefficient is used to determine: a. A specific value of the y-variable given a specific value of the x-variable b. A specific value of the x-variable given a specific value of the y-variable c. The strength of the relationship between the x and y variables d. None of these 2. In regression analysis, the variable that is being predicted is the a. response, or dependent,variable b. independent variable c. intervening variable d. is usually x Answers: 1. c. 2. a. www.cuidol.in Unit-14,15(BBA102/ BCM102) All right are reserved with CU-IDOL

Multiple Choice Questions 23 3. The coefficient of correlation a. is the square of the coefficient of determination b.is the square root of the coefficient of determination c. is the same as r-square d.can never be negative Answers: 3.b. www.cuidol.in Unit-14,15(BBA102/ BCM102) All right are reserved with CU-IDOL

Summary 24  Correlation compares the relative position of cases along two variables. If an increase in level of one variable is associated with an increase in the other, the relationship is positive. If an increase in one is associated with a decrease in the other, the relationship is negative (an inverse correlation).  A correlation coefficient provides a more precise indication of the degree of the relationship between two variables. The value of a correlation coefficient can range from +1 (a perfect positive correlation) to -1 (a perfect negative correlation). The null hypothesis is that there is no predictable relationship between the two variables (correlation coefficient = 0).  While a significant correlation between variables allows us to make predictions from one to the other, it does NOT establish a causal relationship. www.cuidol.in Unit-14,15(BBA102/ BCM102) All right are reserved with CU-IDOL

Frequently Asked Questions 25 Q1. Can correlation and regression be used together? Ans: Regression describes how an independent variable is numerically related to the dependent variable. ... Correlation coefficient indicates the extent to which two variables move together. Regression indicates the impact of a unit change in the known variable (x) on the estimated variable (y). Q2. If two variables are correlated are they causally related? It is a common error to confuse correlation and causation. All that correlation shows is that the two variables are associated. There may be a third variable, a confounding variable that is related to both of them. For example, monthly deaths by drowning and monthly sales of ice-cream are positively correlated, but no-one would say the relationship was causal . Q3.Distinguish Between Correlation and Regression? Ans :A statistical measure which determines the co-relationship or association of two quantities is known as Correlation. Regression describes how an independent variable is numerically related to the dependent variable. Correlation is used to represent the linear relationship between two variables. On the contrary, regression is used to fit the best line and estimate one variable on the basis of another variable. www.cuidol.in Unit-14,15(BBA102/ BCM102) All right are reserved with CU-IDOL

REFERENCES 26 1. Veerarajan, T. (2016). Discrete Mathematics. New Delhi: Tata Macgraw Hill. 2. Singaravelu (2013). Allied Mathematics. Chennai: Meenakshi Agency. 3. Vittal, P.R. (2017). Allied Mathematics. Chennai:Margham Publications. 4. Venkatachalapathy, S.G. (2007). Allied Mathematic. Chennai: Margham Publications. www.cuidol.in Unit-14,15(BBA102/ BCM102) All right are reserved with CU-IDOL

27 THANK YOU For queries Email: [email protected] www.cuidol.in Unit-14,15(BBA102/ BCM102) All right are reserved with CU-IDOL