1. Statistics Correlation

2. Inputs and Outputs
In formulas, some variables are “input” variables The thing you are calculating with the formula is the “output”

3. Inputs and Outputs
In a factory, the inputs are the components

4. Inputs and Outputs
The output is the final product

5. Inputs and Outputs
The input variable is ALWAYS put on the x-axis (horizontal axis) No reason – it’s a TRADITION!

6. Inputs and Outputs
It can sometimes be hard to figure out which is the input variable and which the output variable

7. Inputs and Outputs
Input variables sometimes CAUSE the result

8. Inputs and Outputs
Sometimes input variables occur before the out variables

9. Inputs and Outputs
Frequently the input variables cannot be controlled by us (time, for example) For this reason, input variables are frequently called “independent” variables

10. Inputs and Outputs
And, the output variables (which depend on the values of the independent input variables) are called “dependent” variables

11. Questions?

12. Correlation
A relationship can be seen by graphing the independent and dependent variables in a scatter graph

13. Correlation
A linear relationship is very common

14. Correlation
When we calculated a correlation coefficient, we said it was a measure of the closeness to a linear relationship between the two variables

15. Line of Best Fit
That means, we could find the formula for a line that would be the best fit for the two variables

16. Line of Best Fit
We “fit” a line to the data

17. Line of Best Fit
Real-world data rarely lands exactly on a straight line

18. Line of Best Fit
But we fit the “best” line to the data

19. Line of Best Fit
When you graph two variables on an x-y plot, you can fit a line through the data called a “trend line”

20. Line of Best Fit
This trend line is a “line of best fit” to the data

21. Regression
The “line of best fit” is created by minimizing the total distance of all the points to the line (deviations)

22. Regression
The line of best fit is called the “regression” line

23. Regression
Because it is a line, it has an equation: y = b + mx m = slope b = y-intercept

24. Regression
The slope “m” and the correlation coefficient “r” will both have the same sign

25. Regression
R2 tells how closely the regression line “fits” the data – “goodness of fit”

26. Regression
As you can imagine, the calculations for correlation and the regression line are scary

27. Regression
Hooray for Excel!

28. Regression
Francis Galton

30. Questions?

31. Regression in Excel
Does coming to class affect my grade?

32. What Does It Say?

33. Regression in Excel
Edwin Hubble gathered and analyzed data from astronomical objects He used regression to show that the universe is expanding

34. Regression in Excel
Let’s take a look!

36. Regression in Excel
What do we do first?

37. Regression in Excel
What do we do first? GRAPH THE DATA!

38. Regression in Excel

39. Regression in Excel
Does it look like a straight line would fit the data well?

40. Regression in Excel
Now we’re going to go to: Data Data Analysis Regression

41. Regression in Excel
They want “y” first (I HATE this…)

42. Regression in Excel
Let’s use “distance” for “x” and “velocity” for “y”

43. Regression in Excel
Eeek! What’s all this????

44. Regression in Excel
Here’s the RSQ. What is the %?

45. Regression in Excel
For the trend line, you need:

46. Regression in Excel
This (believe it or not) is the equation of the line of best fit!

47. Regression in Excel
Line of best fit: y = mx + b

48. Regression in Excel
Our equation is: Vel = x Dist

49. Regression in Excel
Highlight and copy:

50. Regression in Excel
Paste on the “Hubble” page

51. Regression in Excel
Add a new column heading: Trend

52. We’re going to calculate our line:

53. Copy it down…

54. Oops! That doesn’t look right!

55. The reference cells are changing for each row We need to make those constant

56. Go back to the first entry Add a $ before the row numbers you want to keep constant

57. Now, copy it down!

58. Much better!

59. Regression in Excel
Create a new graph:

60. Regression in Excel
Make it purty! To make the trend line a line: change Marker Options to “none” change line color to “solid line”

61. TAH-DAH!

62. What Does It Say?

63. Regression
Note: some people use the symbol “ŷ” for the trend data corresponding to the “y” observation data

64. Regression in Excel
Summary of Regression Analysis: Data/Data Analysis/Regression Enter “y” first Copy the first two coefficients in the bottom table Trend line is: =Coeff*Data+Intercept Make a graph Change the trend dots into a line

65. Questions?