Regression Analysis

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Linear Regression Analysis

Linear regression analysis is based on working out an equation for the line of best fit.

The line of best fit will be of the form 
y = a + bx
where 
y is the value of the dependent variable (on vertical axis)
a is the intercept - fixed costs
b is the slope of the line - variable costs
x is the value of the independent variable (on horizontal axis)

i. How do we calculate 'a' and 'b'?

ACCA MA C2c Linear Regression Analysis

How do we calculate ‘a’ and ‘b’?

‘a’ is the fixed cost per period
‘b’ is the variable cost per unit
‘x’ is the activity level (independent variable)’
‘y’ is the total cost = fixed cost + variable cost
‘n’ is the sample size – number of pairs of data

These formulae are given in the exam. Remember always start working ‘b’, then move to ‘a’.

Illustration

Let's work out A and B with the formulas above.

  • Remember that b must always be calculated before a

Sales people (X - Independent Variable)  Sales (Y - Dependent variable) XY X² 
1 100 100 1
2 180 360 4
3 350 1,050 9
4 420 1,680 16
5 500 2,500 25
6 550 3,300 36
= 21 = 2,100 = 8,990 = 91

Solution

n = 6 (as we have 6 data sets)

B = (6 x 8,990) - (21 x 2,100) / (6 x 91) - (21²) = 93.7

A = 350 - (93.7 x 3.5) = 22

Average of y = 2,100 / 6 = 350

Average of x = 21/ 6 = 3.5

Illustration

Now that we have a and b, let us forecast the total revenue if we have 10 salespeople

  • Solution

    Y = a + bx

    a = 22
    b = 93.7
    x = 10

    Therefore, Y = 22 + (93.7 x 10) = 959

    The total revenue expected with 10 sales people will be $959

Correlation

Correlation measures the strength of the relationship between two variables.

One way of measuring ‘how correlated’ two variables are, is by drawing the ‘line of best fit’ on a scatter graph.  When correlation is strong, the estimated line of best fit should be more reliable.

Another way of measuring ‘how correlated’ two variables are, is to calculate a correlation coefficient, r.

Different degrees of correlation

ACCA MA C2c Different degrees of correlation graph ACCA MA C2c Moderate Positive Correlation ACCA MA C2c No Correlation

The correlation coefficient (r)

The correlation coefficient measures the strength of a linear relationship between two variables.  It can only take on values between -1 and +1.

r = +1 indicates perfect positive correlation

r = 0 indicates no correlation

r = -1 indicates perfect negative correlation

The correlation coefficient is calculated as follows

This formula is also given in the exam

ACCA MA C2C The correlation coefficient graph

For the following illustration, we can see that there is positive correlation as every time a sales person is increased, the sales also increase by $100.

So let us calculate R (the level of correlation) for our illustration above.

Sales people (X - Independent Variable)  Sales (Y - Dependent variable) XY X²  Y² 
1 100 100 1 10,000
2 200 400 4 40,000
3 300 900 9 90,000
4 400 1,600 16 160,000
5 500 2,500 25 250,000
6 600 3,600 36 360,000
= 21 = 2,100 = 9,100 = 91 = 910,000

Solution

R = (6 x 9,100) - (21 x 2,100) / √ (6 x 91) - (21)² (6 x 910,000 - 2,100² ) =  +1

This is a perfect positive correlation as we expected, as every time one sales person increased, sales also increased by $100.

Coefficient of determination (r2)

The coefficient of determination is the square of the correlation coefficient. 

It measures how much of the variation in the dependent variable is ‘explained’ by the variation of the independent variable. 

The value of r2 can be between 0 and 1.

For example, if r = 0.97, r2 = 0.94 or 94%.

94% of the variation in the dependent variable (y) is due to variations in the independent variable (x). 6% of the variation is due to random fluctuations. 

Therefore, there is high correlation between the two variables.

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