1. Regression
All about going back (regressing) from the points to the function from which they came.
We are looking at linear regression.

2. Regression Finding, using mathematics, the line of best fit.
Overview of GCSE
Fit by eye
Equal numbers of points above and below
Going through the mean

3. A mathematical approach
Residuals
Aim: to minimise Σei2

4. An anchor point

5. An anchor point

6. An anchor point

7. An anchor point

8. An anchor point

9. An anchor point

10. Least Squares Minimising Σ ei does not tend to help.
We seek instead to minimise Σ ei2 i.e. the sum of the squares of the ei’s.
This gives what is called the least squares regression line of y on x.

11. Least Squares Regression Line
It can be shown that minimising Σ ei2 happens when

12. b is known as the regression coefficient of y on x

31. Regression line of y on x

32. Regression line of y on x

33. Height and Weight

34. Summary Statistics

36. Line of regression of y on x

37. Height and Weight This is the point formed by the means50 60 70 80 90
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Height (cm)
Weight (kg)
y = 0.760x
This is the point formed by the means
Notice that this is not the y-intercept of – since the axis does not begin at 0

38. Use of the y on x line
To predict y values from known x values by substituting for x values.
Safe to use if interpolating.
When interpolating, results are usually reliable.
Dangerous to use if extrapolating.
When extrapolating, results are usually NOT reliable.

39. There is another line

40. There is another line
This time the aim is to minimise the sum of the squares of the horizontal distances from the points to the line.
The is called the line of regression of x on y.

41. The regression coefficient of x on y
Regression line of x on y

43. Line of regression of x on y

44. Line of regression of x on y

45. Height and Weight x on y y on x 50 60 70 80 90 100 110 150 160 170 180
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Height (cm)
Weight (kg)
y = 0.760x
x = 1.227y
x on y
y on x

46. Use of the x on y line
To predict x values from known y values by substituting for y values.
Safe to use if interpolating.
When interpolating, results are usually reliable.
Dangerous to use if extrapolating.
When extrapolating, results are usually NOT reliable.

47. Observation

48. Observation

49. Observation

50. Observation

51. Dependent and Independent Variables
If one variable changes as another one changes the former is called the dependent variable and the latter is called the independent variable.
In a given situation which is the dependent variable and which is the independent variable will depend on which variable is being used to predict which variable.
If you are using a regression line of y on x then x is the independent variable and y the dependent one.
If you are using a regression line of x on y then y is the independent variable and x the dependent one!

52. Controlled Variables
Be aware that sometimes one of the variables is controlled
e.g, suppose measurements of a variable y are taken every time the x variable has increased by 5. The x variable is a controlled variable.
In this case a regression line of x on y would make no sense since you could not minimise the horizontal distances of points from the line since these are fixed.
In this case whether you were predicting y from x, or predicting x from y, you should use the regression line of y on x in both cases.

53. Link between b, b' and r

54. Link between b, b' and r

55. Link between b, b' and r

56. Link between b, b' and r

57. Link between b, b' and r

58. Link between b, b' and r

59. When are the regression lines the same?
y on x line

60. When are the regression lines the same?
x on y line

61. When are the regression lines the same?
x on y line

62. When are the regression lines the same?
y on x line
x on y line

68. Final thoughts
Be ready to talk in context about the meaning of a and b in the regression equation of y on x, namely y = a + bx.
a is the value of y when x = 0.
b is the amount by which y increases every time x increases by 1.
Remember IN CONTEXT.