Understanding how to draw a line of best fit through a scatter plot is crucial for modelling the relationship between two variables and making accurate predictions. This skill is foundational in statistics, offering a visual representation of the correlation between variables and serving as a basis for further analysis.

Introduction

The Line of Best Fit, also known as the trend line, is a straight line drawn through a scatter plot of two variables to best express their relationship. This technique is used extensively in statistics to summarise the direction and strength of a possible relationship between two variables. Drawing a line of best fit enables us to make predictions about one variable based on the known values of another.

Understanding Scatter Plots

• Definition: A scatter plot is a graphical representation of the relationship between two variables.
• Key Points:
  • Each dot represents an individual data point.
  • Scatter plots are used to observe relationships and correlations.

Drawing the Line of Best Fit

Steps:

1. Observation: Examine the scatter plot for a general trend.

2. Direction: Ascertain if the correlation is positive, negative, or none.

3. Line Drawing:

• Ensure the line has roughly equal numbers of points above and below.
• The line should mimic the data's trend.

Using the Line of Best Fit for Predictions

Once the line is drawn, it can be used to predict values:

• Interpolation: Predicting values within the range of the data points.
• Extrapolation: Predicting values outside the range of the data points, which carries more uncertainty.

Calculating the Line of Best Fit Mathematically

For more precise predictions, the Line of Best Fit can be calculated using the formula of a straight line: $y = mx + c$, where:

• $m$ is the slope of the line,
• $c$ is the y-intercept,
• $x$ and $y$ are the variables.

Finding the Slope $(m)$:

Finding the Y-intercept $(c)$:

Where:

• $n$ is the number of data points,
• $\sum$ denotes the summation symbol,
• $xy$ represents the product of the x and y values for each data point.

Example: Grade Prediction from Study Hours

Data Points:

Solution:

Calculate the Slope (m):

The formula for the slope (m) is:

First, we calculate each term in the formula:

• $\sum x = 2 + 4 + 6 + 8 = 20$
• $\sum y = 50 + 60 + 70 + 80 = 260$
• $\sum xy = (2 \times 50) + (4 \times 60) + (6 \times 70) + (8 \times 80) = 100 + 240 + 420 + 640 = 1400$
• $\sum x^2 = 2^2 + 4^2 + 6^2 + 8^2 = 4 + 16 + 36 + 64 = 120$

Now, substituting these into the slope formula:

$m = \dfrac{4(1400) - (20)(260)}{4(120) - (20)^2}$

$m = \dfrac{5600 - 5200}{480 - 400}$

$m = \dfrac{400}{80}$

$m = 5$

Calculate the Y-intercept (c):

The formula for the y-intercept (c) is:

Substitute the known values into the formula:

$c = \dfrac{260 - 5(20)}{4}$

$c = \dfrac{260 - 100}{4}$

$c = \dfrac{160}{4}$

$c = 40$

Graphical Representation:

Predict the Grade for a Student Who Studies for 7 Hours:

The equation of the line of best fit, based on our calculations, is:

$y = mx + c$

$y = 5x + 40$

For a student studying for 7 hours, we substitute $x$ with 7:

$y = 5(7) + 40$

$y = 35 + 40$

$y = 75$

Therefore, according to the line of best fit, a student who studies for 7 hours is predicted to score a grade of 75%.