QR II Lect 15 (Bivariate analysis and scatter plot, correlation).pptx
- 1. 11/13/2024 1
- 2. 11/13/2024 2 Quantitative Reasoning-II By Nazia Aslam Designation: Lecturer Niazi Medical & Dental College Sargodha
- 3. 11/13/2024 3 Content • Bivariate Analysis and Scatter Plot • Variables • Types of relationship • Correlation Coefficient • Scatter plot
- 4. Bivariate Analysis and Scatter Plot • Bivariate analysis is a statistical method that involves the analysis of two variables to determine the empirical relationship between them. It helps in understanding the association, correlation and causation between the variables. A scatter plot is a graphical representation used in bivariate analysis to visualize the relationship between two continuous variables,.
- 5. Variables • There are two types of variables: • Dependent variable (X) • Independent variable (Y) • Dependent variable (X): The variables that is presumed to cause or influence the dependent variable. • Independent variable (Y): The variable that is presumed to be affected by changes in the independence variable.
- 6. Types of relationship There are three types of relationship: Positive Relationship Negative Relationship No Relationship Positive Relationship: as the value of X increase, the value of Y also increase Negative Relationship: as the value of X increase, the value of Y decrease No Relationship: there is no discernible pattern between X and Y
- 7. Correlation Coefficient • Correlation coefficient (r): a statistical measure that describe the strength and direction of a linear relationship between two variables. It ranges from -1 to 1. • r = 1 : perfect positive correlation • r = -1 : perfect negative correlation • r = 0 : no correlation
- 8. Scatter plot • A scatter plot is a type of graph that display individual data points based on two variables each point represent an observations in the data set with its positions determine by the value of the two variables. • A scatter plot is a graph of the ordered pairs (x, y) of numbers consisting of the independent variable x and the dependent variable y. • The scatter plot is a visual way to describe the nature of the relationship between the independent and dependent variables. The scales of the variables can be different, and the coordinates of the axes are determined by the smallest and largest data values of the variables.
- 9. Example: Car Rental Companies • Construct a scatter plot for the data shown for car rental companies in the United States for a recent year.
- 10. Solution: • Step 1: Draw and label the x and y axes. • Step 2: Plot each point on the graph, as shown in Figure.
- 11. Interpretation • This graph suggests a positive relationship, since as the number of cars rented increases, revenue tends to increase also suggest a linear relationship, since the points seem to fit a straight line.
- 13. Example: Absences and Final Grades • Construct a scatter plot for the data obtained in a study on the number of absences and the final grades of seven randomly selected students from a statistics class. The data are shown here.
- 14. Solution • Step 1: Draw and label the x and y axes. • Step 2: Plot each point on the graph, as shown in Figure.
- 15. Interpretation • The plot of the data shown in above Figure suggests a negative relationship, since as the number of absences increases, the final grade decreases also suggest a linear relationship, since the points seem to fit a straight line.
- 16. Example: Age and Wealth • A researcher wishes to see if there is a relationship between the ages and net worth of the wealthiest people in America. The data for a specific year are shown.
- 17. Solution • Step 1: Draw and label the x and y axes. • Step 2: Plot each point on the graph, as shown in Figure.
- 18. Interpretation • The plot of the data shown in above Figure shows no specific type of relationship, since no pattern is discernible.
- 19. Correlation • Correlation Coefficient: As stated in the Introduction, statisticians use a measure called the correlation coefficient to determine the strength of the linear relationship between two variables. There are several types of correlation coefficients. The one explained in this section is called the Pearson product moment correlation coefficient (PPMC), named after statistician Karl Pearson, who pioneered the research in this area. • The correlation coefficient computed from the sample data measures the strength and direction of a linear relationship between two quantitative variables. The symbol for the sample correlation coefficient is r. The symbol for the population correlation coefficient is r (Greek letter rho).
- 20. • The range of the correlation coefficient is from 1 to 1. If there is a strong positive linear relationship between the variables, the value of r will be close to +1. If there is a strong negative linear relationship between the variables, the value of r will be close to -1. When there is no linear relationship between the variables or only a weak relation ship, the value of r will be close to 0.
- 23. Example: Car Rental Companies Compute the correlation coefficient for the data in following data:
- 24. Solution • Step 1 Make a table as shown here.
- 25. • Step 2 Find the values of xy,, and and place these values in the corresponding columns of the table. The completed table is shown.
- 26. • Step 3 Substitute in the formula and solve for r.
- 27. 11/13/2024 27 Reference • “Introductory Statistics Modeling” by Prem S. Mann, 7th Eddition, Willey, 2010. • “Applied statistical Modeling” by Salvatore Baboones, 1st Edition, SAGE Publications Ltd,2013. • “Applied mathematics for business, economics and social sciences” by frank S Budinck, 4th Edition, McGraw Hill. • “Elementary Statistics: A Step-by-Step approach” by Allan Bliman, 10th edition, McGraw Hill, 2017.
- 28. 11/13/2024 28