### Linear Regression

This model says how dependent one variable is on another.

For example cost (X) and volume (Y).

These variables are called the **dependent** and **independent variables**.

The Dependent variable’ value depends on the value of the other variable.

E.g. Sales may be dependent on marketing spend

You would then need to determine the strength of the relationship between these two variables in order to forecast sales.

For example, if the marketing budget increases by 1%, how much will your sales increase?

#### Regression Equation

This helps us predict the variable we require.

The formula for a simple linear regression is as follows:

**Y = a + bx**

where:

**Y** is the value we are trying to forecast (dependent)

**“b”** is the slope of the regression,

**“x”** is the value of our independent value, and

**“a” **represents the y-intercept. (the value we are trying to

forecast when the independent value is 0)

A simpler way to picture this might be thinking of variable costs and fixed costs.

We are trying to forecast **TOTAL COSTS**

So

Y = Total costs

b = Variable cost per unit

a = Fixed Costs

x = Amount of units produced

In this graph, the dots represent the actual date.

Linear regression attempts to estimate a line that best fits the data, and the equation of that line results in the regression equation

Once a linear relationship is identified and quantified using linear regression analysis, values for (a) and (b) are obtained and these can be used to make a forecast for the budget such as a sales budget or cost estimate for the budgeted level of activity

#### Covariance

#### shows the direction of the relationship between 2 variables as well as its relative strength.

If one variable increases and the other variable tends to also increase, then we experience positive covariance.

If one variable increases and the other tends to decrease, then the covariance would be negative.

#### Correlation

This is concerned with establishing how strong the relationship is:

**Perfect Correlation**refers to a correlation where all pairs of values lie on a straight line and there is an exact linear relationship between the two variables.

**Partial Correlation**refers to a correlation where there is not an exact relationship, but low values of (x) tend to be associated with low values of (y), and high values of (x) tend to be associated with high values of (y).

They may also have low values of (x) associated with high values of (y) and vice versa (negative correlation)

**No Correlation**refers to a situation where the two variables seem to be completely unconnected

#### Correlation Coefficient

The correlation calculation simply takes the covariance and divides it by the product of the standard deviation of the two variables.

#### The degree of correlation can be measured (r).

This gives a value of -1 and +1.

A correlation of +1 can be interpreted to suggest that both variables move perfectly positively with each other, and a -1 implies they are perfectly negatively correlated.

#### Coefficient of Determination (r2)

This measures how good the estimated regression equation is and is designated as r2 and has the range of values between 0 and 1.

The higher the r2, the more confidence in the equation.

#### Limitations of Simple Linear Regression Analysis

Assumes a linear relationship between variables;

Measures only the relationship between two variables where in reality many variables exist;

Assumes that the historical behaviour of the data continues into the foreseeable future;

Interpolated predictions are only reliable if there is a significant correlation between the data.