Line of Best Fit (HSC SSCE Mathematics Standard): Revision Notes

Line of Best Fit

Introduction

When we examine bivariate data on a scatterplot, we often see patterns that suggest a relationship between the two variables. If the points tend to follow a straight line pattern, we can draw a line through the data to represent this relationship. This line is called the line of best fit.

A line of best fit is a straight line that best approximates the linear relationship between data points on a scatterplot. The process of finding this line is called linear regression.

The equation of a line of best fit follows the familiar gradient-intercept form:

$y = mx + c$

where:

• $m$ is the gradient (or slope) of the line
• $c$ is the y-intercept

Using the line of best fit for predictions

Once we have established a line of best fit, we can use it to make predictions about one variable based on the other. There are two types of predictions:

Interpolation occurs when we make a prediction within the existing data range. This is generally more reliable because we are working within the boundaries of our observed data.

Extrapolation occurs when we make a prediction outside the existing data range. We must use this cautiously because the linear relationship we observed may not continue beyond our data.

Method of least squares

Why we need a systematic method

If we simply tried to draw a line that appears to balance the points above and below it, different people would likely draw slightly different lines. This subjective approach is not reliable for scientific analysis. We need a systematic mathematical method that produces the same result every time.

Understanding residuals

For any line drawn through a scatterplot, we can measure how well it fits the data by calculating the vertical distance from each point to the line. This vertical distance is called a residual.

A residual measures how far a data point is from the line:

• If the line passes exactly through a point, the residual is zero
• The larger the residual, the worse the fit for that particular point

The least-squares approach

The least-squares line of best fit is the line that minimizes the sum of all squared residuals. Here's how it works:

1. Calculate the residual (vertical distance) for each data point
2. Square each residual value
3. Add all the squared values together
4. The best line is the one that makes this total as small as possible

By squaring the residuals, we ensure that:

• Negative and positive distances don't cancel out
• Larger errors are penalized more heavily than smaller ones

Calculating the equation of the least-squares line

To find the equation of the least-squares line of best fit, we need five statistical measures from our data:

• $\bar{x}$ = mean of the $x$ values
• $\bar{y}$ = mean of the $y$ values
• $s_x$ = standard deviation of the $x$ values
• $s_y$ = standard deviation of the $y$ values
• $r$ = Pearson's correlation coefficient

Formula for the gradient

The gradient (slope) of the line is calculated using:

$m = r \times \frac{s_y}{s_x}$

This formula shows that:

• The gradient depends on the correlation strength ($r$)
• It's adjusted by the ratio of standard deviations
• A stronger correlation produces a steeper gradient

Formula for the y-intercept

Once we have the gradient, we calculate the y-intercept using:

$c = \bar{y} - m\bar{x}$

This ensures the line passes through the point $(\bar{x}, \bar{y})$, which is the centre of the data.

Complete equation

Combining these, the equation of the least-squares line of best fit is:

$y = mx + c$

where $m = r \times \dfrac{s_y}{s_x}$ and $c = \bar{y} - m\bar{x}$

Worked example 1: Calculating from given statistics

Worked example 2: Complete analysis with data

Exam tips

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