1. CORRELATON & REGRESSION
Correlation and regression are concerned with the investigation of relationships between two or more variables.

2. We consider just two associated variables. We might want to know:
If a relationship exists between those variables
If so, how strong that relationship is
What form that relationship takes
Can we make use of that relationship for predictive purposes i.e. forecasting?

3. General method for investigating the relationship between 2 variables:
Correlation is used to find the strength of the relationship
Regression describes the relationship itself in the form of an equation which best fits the data
General method for investigating the relationship between 2 variables:

4. For an initial insight into the relationship between two variables:
plot a scatter diagram
If there appears to be a linear relationship, quantify it:
calculate the correlation coefficient
This is a measure of the strength of this linear
relationship.
Its symbol is 'r' and its value lies between
-1 and +1

5. If the relationship is found to be significantly strong:
find the equation of the ‘line of best fit’ through the data, using linear regression
The 'goodness of fit' statistic can be calculated to see how useful the regression equation is likely to be
Once defined by an equation, the relationship can be used for predictive purposes.

6. The data represents a sample of advertising
Example
The data represents a sample of advertising
expenditures and sales for ten randomly
selected months. See slide 12 for complete
data.
Month Advertising Sales
expenditure (£0.000’s) y
(£0,000’s) x
etc.
Plot a scatter diagram of the data

7. The graph suggests a linear relationship between
Note scales are not started at zero
The graph suggests a linear relationship between
sales and advertising expenditure.
The larger the amount spent on advertising the higher the sales in general.

8. If there is a relationship,
we need to be able to measure the strength of that relationship.
i.e. calculate the value of the correlation coefficient

9. Pearson's Product Moment Correlation
Coefficient (r)
is a measure of how close a linear relationship there is between x and y.
can be produced directly from a calculator in LR (linear regression) mode
For the sales and advertising data the correlation coefficient: r =
The value of r is always between + 1 and -1

10. r = -1 perfect negative correlation
r = 0 no correlation
r = +0.8
r = +1 perfect positive correlation

11. Formula for correlation coefficient, r
r = Sxy
Sxx Syy
where
Sxx = Sx2 - Sx Sx n
Syy = Sy2 - Sy Sy
Sxy = Sx2 - Sx Sy

12. Longhand calculations for correlation coefficient r.
Step 1

13. Step 2 Sxx = Sx2 - Sx Sx = 9.28 - 9.4 x 9.4 = 0.444 n 10
Therefore:
Sxx = Sx2 - Sx Sx = x = n
Syy = Sy2 - Sy Sy = x = n
Sxy = Sxy - Sx Sy = x = n
Step 3
Therefore: r = Sxy = =
Sxx Syy x

14. Hypothesis test for the value of r
We shall not go into the details here!
Null hypothesis (H0): A linear relationship does not
exist between sales and advertising
Alternative hypothesis(H1): A linear relationship does
exist between sales and advertising.
If we calculate a test statistic and critical value we discover that test statistic > critical value
so we reject H0
Conclude that a linear relationship exists between sales and amount spent on advertising.

15. The Goodness of Fit Statistic (R2)
This also measures of the closeness of the relationship between x and y
R2 = 100r2
R2 tells us what percentage of the total variation in y (here sales) is explained by the variation in x (here advertising expenditure)

16. Interpretation:
If r = +1 or –1, then R2 =100%
So 100% of the variation in y is explained by the variation in x.
If r = 0, then R2 = 0%
So none of the variation in y is explained by the variation in x
For the data above the goodness of fit statistic R2 = 100 r2 = x
= 76.6%

17. 76.6% of the variation in sales is explained by the variation in the amount spent on advertising.
The remaining 23.4% of the variation is explained by other factors:
e.g. price
competitor’s prices etc.

18. Regression equation
Since we know, for the sample data, that
there is a significant relationship between
the two variables,
the next obvious step is to find its equation.
We can then add the regression line to the
scatter diagram and use it to predict future
sales, given advertising expenditure for a
particular month.
The regression equation can be produced
directly from a calculator in LR mode.

19. The regression line has the equation:
y = a + bx
x is the independent variable
y is the dependent variable
a is the intercept on the y-axis
b is the gradient or slope of the line.

20. For the sales and advertising data, the
values of a and b are 46.5 and 52.6.
So regression equation is:
y = x
Sales = advertising
(a and b can be found using LR mode on your calculator or by calculation)

21. Formula for a and b
This is found by calculating the square of the differences between actual and expected values.
We chose a and b so that the total difference is minimizied:
b = Sxy a = y - b x
Sxx ( x , y )
is called the
centroid
Where x , y are the means of the x and y data
and the S’s are defined as previously.

22. Calculations for the regression equation.
In the regression equation y = a + bx
b = Sxy = =
Sxx
a = y - b x = x = 46.5
(As y = Sy = and x = Sx = = 0.94)
n n
Therefore the regression equation is
y = x

23. Plotting the regression equation on the
scatter diagram.
The line y = a + bx can be plotted on the scatter
diagram by plotting three points.
The centroid ( x , y ) and any other two points,
which satisfy the regression equation.
From the data (x, y) = (0.94, 95.9)
When x = 0.6, y = ( x 0.6) =
When x = 1.2, y = (52.6 x 1.2) =
Plot (0.94,95.9)
Plot (0.6, 78.6)
Plot (1.3, 109.6)

24. x x x x

25. Note
regression equation y = a + bx can only be used to calculate an estimate for y given the value of x
The linear relationship y = a + bx can only be assumed to exist between y and x for the range of values within the sample

26. Interpreting the coefficients in the
regression equation -
first the a value
The intercept (a) is the estimate of
y when x = 0, but care is needed if using this – why?
y = x
Sales = advertising
When x = 0, y = 46.5
i.e. When nothing is spent on advertising,
sales would be expected on average to be 46.5 units = x £10,0000
=£ 465,000

27. If x = 0 y = 46.5, but care is needed here!
the b value
y = x
If x = 0 y = 46.5, but care is needed here!
If x = y = (52.6)(0.6) =
If x = y = (52.6)(0.8) =
If x = 1 y = =
If x = 1.2 y = (52.6)(1. 2) =
If x = 2 y = x 2 but care is needed
here also!
etc.
So if advertising expenditure is increased
by 1 unit, sales will be increased by 52.6
units on average.

28. For each additional £10,000 spent on
advertising, sales will increase by
£52.6 x £10,000 = £526,000 on average.
But we cannot estimate sales outside the range:
E.g. we should not try to estimate sales
for x = 5 using this method.