Least-Squares Regression

Regression is a technique used to predict future values based on known values. For instance, linear regression allows us to predict what an unknown Y value will be, given a series of known X and Y’s, and a given X value.

Given the following, it’s easy to see the pattern. But assuming no obvious pattern exists, regression can help us determine what the Y value will be given our known X values.

The X value is known as the independent variable, the “predictor variable”, while the Y value is the value you’re being predicted.

The linear regression (or “least squares regression”) equation is Y’ = a + bX

• Y’ (Y-prime) is the predicted Y value for the X value
• a is the estimated value of Y when X is 0
• b is the slope (the average change in Y’ for each change in X)
• X is any value of the independent variable

There are additional formulas for both a and b.

Let’s take a look at the following data-set, that compares the number of calls made for a product against the number of sales:

First we need to calculate the sum of X-squared, Y-squared and X*Y:

Returning to our formula, let’s start with b first:

The top of the equation looks like this: b = 10(10800) – 220 * 450 / n(∑X²)-(∑X)². We’ve simply filled in the values from our chart.

b = 10(10800) – 220 * 450
b = 108,000 – 99,000
b = 9,000 / n(∑X²)-( ∑X)²

Now we have to do the bottom half of the equation:

n(∑X²)-(∑X)²

=10(5600)-(220) ²
=56,000 – 48,400
=7,600

Returning to our equation:

b = 9,000 / 7,600
b = 1.1842

Now let’s move on to a:

a = 450 / 10 – 1.1842 * (220 / 10)
a = 45 – (1.1842 * 22)
a = 45 – 26.0524
a = 18.9476

So, going back to our original regression equation, Y’ = a + bX and plugging our numbers, we get:

Y’ = 18.9476 + (1.1842)X

To use this equation, we now put our desired value in for X. With an estimated 20 calls:

Y’ = 18.9476 + (1.1842)*20
Y’ = 18.9476 + 23.684
Y’ = 42.63

So, a salesperson who makes 20 calls will expect to make 42 sales.