The equation of a line of best fit Revision Notes for AQA GCSE Statistics

Regression line

What is a regression line?

A regression line is also known as the line of best fit. This is a straight line drawn through the middle of the points on a scatter diagram to show the general trend or pattern in the data. The regression line helps us understand the relationship between two variables and allows us to make predictions about one variable based on the value of another.

infoNote

When you have the equation of a regression line, you can use it to find specific points that lie on the line and draw it accurately on a scatter diagram.

The equation of a regression line

Regression lines follow the standard straight line equation format:

$y = mx + c$

Where:

m is the gradient (slope) of the line
c is the y-intercept (where the line crosses the y-axis)
x and y are the coordinates of points on the line

Using the equation to find points on the line

To draw a regression line on a scatter diagram, you need to find at least two points that lie on the line using the equation. Here's how to do it step by step:

lightbulbExample

Worked Example: Academic Performance

Let's say we have data about students' performance in maths and science tests, and the regression line has the equation:

$y = 1.05x - 3$

Where $x$ = maths mark (%) and $y$ = science mark (%)

Step 1: Choose suitable x-values Pick x-values that are easy to work with and spread across the range of your data. Good choices might be $x = 30$ and $x = 80$ .

Step 2: Calculate the corresponding y-values

When $x = 30$ : $y = 1.05 \times 30 - 3$ $y = 31.5 - 3$ $y = 28.5$

When $x = 80$ : $y = 1.05 \times 80 - 3$ $y = 84 - 3$ $y = 81$

Step 3: Plot and join the points Plot the points $(30, 28.5)$ and $(80, 81)$ on your scatter diagram, then draw a straight line through them.

Interpreting the components of the equation

The y-intercept (c-value)

The y-intercept tells you where the regression line crosses the y-axis (when $x = 0$ ). In our example, $c = -3$ , which means the line crosses the y-axis at $-3$ .

infoNote

What this means in context: If a student scored 0% on the maths test, the regression line predicts they would score -3% on the science test. However, this is impossible since you can't score negative marks, which shows that the regression line may not be reliable for predicting values outside the range of the original data.

The gradient (m-value)

The gradient shows how much y increases for every 1-unit increase in x. In our example, $m = 1.05$ .

What this means in context: For every extra 1% scored on the maths test, a student would be expected to score 1.05% more on the science test. This positive gradient indicates a positive correlation between maths and science performance.

Making predictions using regression lines

Regression lines can be used to make predictions, but you must consider their reliability:

lightbulbExample

Worked Example: Car Depreciation

If we have a regression line with equation $y = 20000 - 1500x$ , where $x$ = age of car (years) and $y$ = value (£):

The y-intercept (20000) represents the car's initial value when new
The gradient (-1500) shows the car loses £1500 in value each year
We can predict the car's value after any number of years

However, predictions become less reliable the further they are from the original data range.

Key points about reliability

chatImportant

When using regression lines for predictions:

Stay within the data range: Predictions are most reliable when made within the range of the original data
Consider the context: Sometimes predictions may give impossible or unrealistic results
Correlation doesn't mean causation: A strong correlation doesn't necessarily mean one variable causes changes in the other
Check the scatter: If the points are very spread out from the line, predictions will be less reliable

Drawing regression lines step by step

Identify the equation of the regression line
Choose two x-values that are easy to substitute and cover a good range
Calculate the corresponding y-values using the equation
Plot both points on the scatter diagram
Draw a straight line through both points, extending it across the diagram
Check your line passes close to the middle of the scattered data points

bookmarkSummary

Key Points to Remember:

The regression line is the same as the line of best fit and shows the general trend in scattered data
Use the equation $y = mx + c$ to find points on the line by substituting x-values
The gradient (m) tells you how much y changes for each unit increase in x
The y-intercept (c) shows where the line crosses the y-axis when $x = 0$
Predictions are most reliable within the range of the original data and may become unrealistic outside this range

The equation of a line of best fit (AQA GCSE Statistics): Revision Notes

Regression line

What is a regression line?

The equation of a regression line

Using the equation to find points on the line

Interpreting the components of the equation

The y-intercept (c-value)

The gradient (m-value)

Making predictions using regression lines

Key points about reliability

Drawing regression lines step by step

Explore AQA GCSE Statistics Model Answers by Topics

Scatter diagrams

Correlation

Causal relationships

Line of best fit

Interpolation and extrapolation