Let's learn about bivariate data and linear relationship in A-Level Maths!

**Bivariate Data**

Consider the below results table, for the variables x and y.

Data that consists of **pairs of values of two random variables**, like the above table, is called **Bivariate data**.

The explanatory variable is usually plotted as x, and, in the context of an experiment, is the variable that is changed. The response variable changes as a result, and is dependent on the explanatory variable.

Plotting the above data on a **scatter graph** is a good way of determining linear relationships.

We would say that this data is positively correlated. Correlation is the focus later on in this segment.

Also, we could plot a line of **best fit** on this data so as to evaluate the linear trend.

The above diagram shows the same data, but with a line of best fit. When working on paper, we would judge a line of best fit on eye, whilst a computer will use a mathematical method to calculate the perfect line of best fit based on the whole data set. This mathematical method will form the focus on linear regression in this chapter.

A line of best fit, called a least squares regression line in the Statistics world, will be a linear line. Linear lines have the following formula:

**y = mx + c**

The gradient, m, needs to be calculated, as well as the y-intercept, c.

If the data follows a negative relationship, then the gradient of the least squares regression line will be negative, and if the data follows a positive relationship, the gradient of the least squares regression line will be positive.

A linear regression line will always pass through the point ( x̄ , ȳ), the mean of the explanatory variable and the mean of the response variable.

The regression coefficient of y on x (the way to calculate the value of b, the gradient) is given by the expression:

Which means the covariance of the Bivariate data divided by the variance of the explanatory variable. This is expressed in the below formula, for b:

Where n is the number of data pairs, so, for example, in the table presented on the previous page, n would equal 9. To calculate the value of a, the y-intercept, we use the below formula:

Drafted by Eunice (Maths)

Reference

https://mathtec.weebly.com/bivariate-data-2020.html