Bivariate Data – Scatter Plots and Correlation Coefficient……

1. Bivariate Data – Scatter Plots and Correlation Coefficient…… Section 3.1 and 3.2

2. 2 Quantitative Variables…… • We represent 2 variables that are quantitative by using a scatter plot. • Scatter Plot – a plot of ordered pairs (x,y) of bivariate data on a coordinate axis system. It is a visual or pictoral way to describe the nature of the relationship between 2 variables.

3. X: a. Input Variable b. Independent Var c. Controlled Var Y: a. Output Variable b. Dependent Var c. Results from the Controlled variable Input and Output Variables……

4. When dealing with height and weight, which variable would you use as the input variable and why? Answer: Height would be used as the input variable because weight is often predicted based on a person’s height. Example……

5. Constructing a scatter plot…… • Do a scatter plot of the following data:

6. What do we look for?...... • A. Is it a positive correlation, negative correlation, or no correlation? • B. Is it a strong or weak correlation? • C. What is the shape of the graph?

7. Answer……With TI

8. Notice the following: A. Strong Positive – as x increases, y also increases. B. Linear - it is a graph of a line. Notice……

9. Example 2……By Hand

10. Example 2……With TI

11. Notice the following: Strong Negative – As x increases, y decreases Linear – it’s the graph of a line. Notice……

12. Example 3……By Hand

13. Example 3……With TI

14. Notice: There seems to be no correlation between the hours or exercise a person performs and the amount of milk they drink. Notice……

15. Put x’s in L1 and y’s in L2 Click on “2nd y=“ Set scatter plot to look like the screen to the right. Press zoom 9 or set your own window and then press graph. Steps to see on Calculator……

16. Linear Correlation Section 3.2

17. Definition – a statistical method used to determine whether a relationship exists between variables. 3 Types of Correlation: A. Positive B. Negative C. No Correlation Correlation……

18. Positive Correlation: as x increases, y increases or as x decreases, y decreases. • Negative Correlation: as x increases, y decreases. • No Correlation: there is no relationship between the variables.

19. Linear Correlation Analysis …… • Primary Purpose: to measure the strength of the relationship between the variables. • *This is a test question!!!!

20. The numerical measure of the strength and the direction between 2 variables. This number is called the correlation coefficient. The symbol used to represent the correlation coefficient is “r.” Coefficient of Linear Correlation

21. The range of “r” values…… • The range of the correlation coefficient is -1 to +1. • The closer to 0 you get, the weaker the correlation.

22. Range…… • Strong Negative No Linear Relationship Strong Positive ____________________________________ -1 0 +1

23. Computational Formula using z-scores of x and y……

24. Example 1…… • Find the correlation coefficient (r) of the following example. • Use the lists in the calculator.

25. Since you will be using a formula that uses z-scores, you will need to know the mean and standard deviation of the x and y values. Put x’s in L1 Put y’s in L2 Run stat calc one var stats L1 – Write down mean & st. dev. Run stat calc one var stats L2 – Write down mean & st. dev. Find mean and st. dev first……

26. X values: Y values:

27. X Values: Mean = 2.8 St. Dev = 1.643167673 Y Values: Mean = 76 St. Dev = 11.40175425 Write down on your paper……You’ll use them later.

28. Calculator Lists……

29. From the lists….. n = 5 Calculate “r”……

30. Since r = 0.61, the correlation is a moderate correlation. Do we want to make predictions from this? It depends on how precise the answer needs to be. What does that mean?

31. Example 2…… • Find the correlation coefficient (r) for the following data. • Do you remember what we found from the scatter plot?

32. Let’s do this one together…… • Remember to use your lists in the calculator. • Don’t round numbers until your final answer. • Find the mean and st. dev. for x and y. • Explain what you found.

33. X Values: Y Values:

34. List values you should have……

35. Compute “r”……

36. Describe it…… • Since r = 0.897 Strong Positive Correlation

37. Example 3…… • Find the correlation coefficient for the following data. • Do you remember what we found from the scatter plot?

38. X Values: Y Values:

39. List Values you should have……

40. Compute “r”……

41. Describe it…… • Since r = -0.944 Strong Negative Correlation

42. Example 4…… • Find the correlation coefficient of the following data. • Do you remember what we found from the scatter plot?

43. X Values: Y Values:

44. List Values you should have……

45. Compute “r”……

46. Describe It…… • Since r = .067 No Correlation…..No correlation exists

47. What is • It is the coefficient of determination. • It is the percentage of the total variation in y which can be explained by the relationship between x and y. • A way to think of it: The value tells you how much your ability to predict is improved by using the regression line compared with NOT using the regression line.

48. For Example…… • If it means that 89% of the variation in y can be explained by the relationship between x and y. • It is a good fit.

49. Assignment…… • Worksheet