1. Linear regression and correlation
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Linear regression and correlation
Learning outcomes
This work will help you
To draw a scatter diagrams.
Draw linear regression lines y on x and x on y and work out their equations.
Identify the types of correlation and calculate product moment correlation coefficient.
Use technology to find all of the above.

2. Linear Regression
When looking for a linear relationship between two sets of data we can plot what is known as a scatter diagram. y x
Looking at the graph we can see that there is some positive correlation.

3. First lets consider y on x regression line.
It is possible to draw a line called a regression line. There are two types y on x and x on y.
First lets consider y on x regression line. x y
y on x
The y on x line, draws the regression line by keeping the sum of the squares of the vertical distance to a minimum.
Note: The equation of the line is called “The Equation of the Least Squares regressions Lines”

4. Now consider the x on y regression line.
The x on y line, draws the regression line by keeping the sum of the squares of the horizontal distance to a minimum.

5. Drawing both graphs on the same graph we have
x on y x y
y on x
We should note that both graphs will pass through the means of both sets of data,

6. It is possible to calculate the equations of the y on x and x on y regression lines.
Important formulae
y on x regression line is of the form and can be calculated by using the formula.
Where is called the covariance and links the x and y data.
is the variance of the x data

7. x on y regression line is of the form and can be calculated by using the formula.
Where is called the covariance and links the x and y data.
is the variance of the y data

8. Example
In the table below are the results of ten students in both their Mathematics and Physics examinations. The teacher thinks there might be a relationship between the two. His hypothesis is “a student who has Mathematical ability also has ability in Physics.”
Mathematics Mark /100 (x)
Physics Mark /100 (y) 61 56 34 45 24 15 89 92 47 67 57 82 75 6 8 53 76

9. Drawing a scatter graphx y 61 56 34 45 24 15 89 92 47 67 57 82 75 6 8 53 76

10. Maths/100

11. Finding y on x using technology
Product-Moment Correlation Coefficient

12. Finding x on y using technology
Remember you have to interchange
the x and y when writing down the x
on y regression line.
Product-Moment Correlation Coefficient

13. Example
In the table below are the results of ten students in both their Mathematics and Physics examinations. The teacher thinks there might be a relationship between the two. His hypothesis is “a student who has Mathematical ability also has ability in Physics.”
Mathematics Mark /100 (x)
Physics Mark /100 (y) 61 56 34 45 24 15 89 92 47 67 57 82 75 6 8 53 76

14. Now calculating the regression lines76 89 47 8 75 57 61 92 15 45 56 y 53 6 82 67 24 34 x
5.8
2.8
33.64
7.84
16.24
-21.2
-8.2
449.44
67.24
173.84
-31.2
-38.2
973.44
33.8
38.8
-8.2
7.8
67.24
60.84
-63.96
11.8
3.8
139.24
14.44
44.84
26.8
21.8
718.24
475.24
584.24
-49.2
-45.2
-2.2
-6.2
4.84
38.44
13.64
33.8
22.8
519.84
770.64
552
532

15. Using alternate formulae and the TI-nspire Calculator
Variance of x
Variance of y

16. Covariance
Having done the 2-variable stats
calculation the actual value of variance
(which is the standard deviation
squared) can be found using the “Var”
menu on the calculator.

17. For regression line y on x which has form

18. For regression line x on y which has form

19. Plotting both lines on the scatter diagram
y on x,
and for
x on y,
x on y
y on x
Note: For x on y line, remember to rearrange it into the following form before trying to plot

20. (standard deviation of x)
Correlation
We need a way to determine if there is linear correlation or not. So we calculate what is known as the Product-Moment Correlation Coefficient (r).
(covariance),
(standard deviation of x)
(standard deviation of y).
We can see that the quantity r from the following five sets of data above tells us something about the degree of scatter of the two sets of data, if we are looking for a linear relationship.

21. Table 1 x 5 10 15 20 25 30 35 y 38 28 26 19 17 8 1
y on x
x on y
The product moment correlation coefficient
In table 1 we notice that the two regressions lines (y on x and x on y) nearly coincide and that as the x-data increases the y-data decreases. The value of r is , which is close to –1. Here we have what is called strong negative linear correlation.

22. Table 2 x 5 10 15 20 25 30 35 y 23 32 2
y on x
x on y
The product moment correlation coefficient
In table 2, the two regression lines are further apart although there is weak negative linear correlation. The value of r is and it is getting closer to 0.

23. Table 3 x 5 10 15 20 25 30 35 y 31 19 23 32 6
y on x
x on y
The product moment correlation coefficient
In table 3, the two regression lines are virtually perpendicular and there is no linear correlation. The value of r is and it is very close to 0.

24. Table 4 x 5 10 15 20 25 30 35 y 12 17 23 9 38 18 40
y on x
x on y
The product moment correlation coefficient
In table 4, the two regression lines are further apart but we notice that as the x-data increases the y-data increases. We say there is weak positive linear correlation. The value of r is and it is moving away from 0 and getting closer to 1.

25. Table 5 x 5 10 15 20 25 30 35 y 2 4 12 16 18 26 27 32
y on x
x on y
The product moment correlation coefficient
In table 5, we notice that the two regressions lines (y on x and x on y) nearly coincide and that as the x-data increases the y-data increases. The value of r is 0.990, which is very close to 1. Here we have what is called strong positive linear correlation.

26. r is called Product-Moment Correlation Coefficient.
The value of r determines the degree of linear scatter of the two sets of data and
- indicates that the data have perfect negative linear correlation,
- indicates that the data has no linear correlation,
- indicates that the data have perfect positive linear correlation.
r is called Product-Moment Correlation Coefficient.

27. Returning to our example
y on x
x on y
So we can conclude that as r is close to 1, that the results show that his hypothesis that “a student who has Mathematical ability also has ability in Physics’” might be true.

28. Maths/100