Introduction to the Line of Best Fit

Once you have plotted a scatter diagram, you will often see a pattern or correlation between the two variables. To make sense of this pattern and, more importantly, to make predictions, we draw a Line of Best Fit (LOBF).

Think of the Line of Best Fit as a "trend line." It is a straight line that best represents the data on a scatter diagram. It doesn't have to touch every point (in fact, it rarely does!), but it should follow the general path of the data points.

Important Note: In your GCSE Statistics exam, you are only required to draw linear (straight) lines of best fit. Non-linear models (curves) are not tested here!


The "Double Mean Point" – Your Golden Rule

To draw an accurate line of best fit by eye, there is one mathematical rule you must follow: your line must pass through the double mean point.

The double mean point is the "average" position of all your data points. It acts like a balance point or the "centre of gravity" for your scatter diagram. It is written as the coordinate \((\bar{x}, \bar{y})\).

How to calculate the double mean point:

1. Find the mean of all the \(x\)-values (the horizontal axis). This is called \(\bar{x}\).
2. Find the mean of all the \(y\)-values (the vertical axis). This is called \(\bar{y}\).
3. The coordinate \((\bar{x}, \bar{y})\) is your double mean point.

The formula for the mean you should remember is:
Mean of \(x\) (\(\bar{x}\)) = \(\frac{\sum x}{n}\) (The sum of all \(x\) values divided by how many there are).
Mean of \(y\) (\(\bar{y}\)) = \(\frac{\sum y}{n}\) (The sum of all \(y\) values divided by how many there are).

(Note: If the data is in a frequency table, use \(\bar{x} = \frac{\sum fx}{\sum f}\) as seen in other chapters.)


Step-by-Step: Drawing the Line

Don't worry if this seems tricky; just follow these four steps to get full marks on a drawing question:

Step 1: Calculate the means. Work out \(\bar{x}\) and \(\bar{y}\) from the data provided in the question.

Step 2: Plot the double mean point. Mark the coordinate \((\bar{x}, \bar{y})\) on your scatter diagram with a small, clear 'x' or a circled dot. Label it clearly so the examiner knows you've used it!

Step 3: Position your ruler. Place your ruler so that it passes exactly through your double mean point.

Step 4: Balance the points. Rotate your ruler (keeping it on the mean point) until there is a roughly equal number of points above and below the line, and the line follows the same direction as the correlation. Draw a single, thin, straight line.

Quick Tip: If there is zero correlation (points are scattered randomly like a cloud), you generally cannot draw a reliable line of best fit!


Interpolation vs. Extrapolation

The whole point of a Line of Best Fit is to make estimates. However, the reliability of your estimate depends on where you are looking on the line.

Interpolation (Inside the data)

Interpolation is when you use the line to predict a value within the range of the data you already have. For example, if your data covers students aged 11 to 16, predicting a value for a 14-year-old is interpolation.

Reliability: This is usually very reliable because you have evidence of the trend on both sides of the point.

Extrapolation (Outside the data)

Extrapolation is when you extend the line to predict a value outside the range of your collected data. For example, using that same data to predict the height of a 40-year-old.

Reliability: This is dangerous and often unreliable. You have no evidence that the trend continues beyond your data points. In real life, trends often change, slow down, or stop.

"Did you know?" If you measured the growth of a puppy for three months and used extrapolation to predict its size in 10 years, your line might suggest the dog will be the size of a house! This is why extrapolation is risky.


Common Mistakes to Avoid

  • Ignoring the double mean point: If you draw a line that looks "okay" but misses the \((\bar{x}, \bar{y})\) point, you will lose marks.
  • The "Origin Trap": Do not force your line to go through \((0,0)\) unless the data and the double mean point naturally lead it there.
  • Using a "dot-to-dot" approach: A line of best fit must be one single straight line, not a series of zig-zags between points.
  • Over-extending: Only draw your line slightly past the last data point. Drawing it across the whole page can lead to accidental extrapolation.

Key Takeaways

1. The Line of Best Fit shows the trend of bivariate data.
2. It must pass through the double mean point \((\bar{x}, \bar{y})\).
3. Interpolation (predicting inside the data range) is generally reliable.
4. Extrapolation (predicting outside the data range) is unreliable and should be treated with caution.
5. While you draw the line "by eye," the Higher Tier curriculum also uses the Linear Regression Line (a line calculated by a computer/calculator), which we cover in a later chapter.