Welcome to Scatter Diagrams, Correlation, and Lines of Best Fit!
Have you ever wondered if studying more hours actually leads to higher exam marks? Or whether taller basketball players score more points? In statistics, whenever we compare two different measurements from the same person or item, we are working with bivariate data.
In this chapter, you will learn how to turn pairs of numbers into visual stories using scatter diagrams, describe how closely they are linked using correlation, and make sensible future predictions using a line of best fit.
Don't worry if this seems tricky at first! Step by step, we will break down all the rules and methods so you can tackle any exam question with confidence.
Did you know? The word bivariate simply comes from bi (meaning two, like a bicycle) and variate (meaning variable or measurement).
1. Bivariate Data and Scatter Diagrams
Bivariate data involves pairs of related measurements. For example, recording both the temperature and the number of ice creams sold on \(10\) different days.
Explanatory vs Response Variables
When we plot these two variables on a graph, we assign them to specific axes:
• Explanatory Variable (Independent Variable): This is the variable that we control or think is causing a change. It is always plotted on the horizontal \(x\)-axis.
• Response Variable (Dependent Variable): This is the variable that responds or changes as a result. It is always plotted on the vertical \(y\)-axis.
Analogy: Think of sunshine and plant growth. The amount of sunshine is the explanatory variable (\(x\)-axis), and how tall the plant grows is the response variable (\(y\)-axis).
How to Draw a Scatter Diagram
1. Choose sensible, linear scales for both the \(x\)-axis and \(y\)-axis.
2. Label both axes clearly with variable names and their units (e.g., Temperature (\(^\circ\text{C}\)) and Ice cream sales (\pounds)).
3. Plot each pair of data points \((x, y)\) accurately using a neat small cross (\(\times\)) or dot.
4. Give your scatter diagram a clear title.
Key Takeaway: A scatter diagram is simply a visual display of pairs of data plotted on coordinate axes to help us spot connections between two variables.
2. Understanding Correlation
Correlation describes the nature and strength of a statistical relationship between two variables.
Types of Correlation
• Positive Correlation: As the \(x\) variable increases, the \(y\) variable also tends to increase. The points trend upwards from bottom-left to top-right.
Example: Revision time and exam score.
• Negative Correlation: As the \(x\) variable increases, the \(y\) variable tends to decrease. The points trend downwards from top-left to bottom-right.
Example: The age of a second-hand car and its market value.
• Zero / No Correlation: There is no linear pattern or connection between the two variables. The points are scattered randomly across the diagram.
Example: Shoe size and GCSE Maths score.
Strength of Correlation
We also describe how tightly clustered the points are around a straight line:
• Strong Correlation: Points lie very close to a straight line path.
• Moderate Correlation: Points follow a clear trend, but are more spread out.
• Weak Correlation: Points are widely scattered, but a vague upward or downward trend can still be seen.
Linear vs Non-Linear Relationships
• Linear: The points follow a straight-line pattern.
• Non-Linear (Curvilinear): The points show a clear pattern, but it curves (for instance, like an upside-down \(U\)-shape). An example is the speed of a car and its fuel efficiency.
Memory Trick: Think of a staircase. Going up the stairs = Positive. Going down the slide = Negative. Flat floor with toys scattered everywhere = Zero correlation.
Key Takeaway: When asked to describe correlation in an exam, always give both the strength and the direction (e.g., "Strong positive linear correlation").
3. Correlation vs Causation
The Golden Rule of Statistics
Correlation does NOT imply causation!
Just because two variables show a strong correlation, it does not automatically mean that one variable directly causes the other to change.
Real-World Example: In summer, ice cream sales and the number of sunburn cases both increase strongly. Does eating ice cream cause sunburn? Of course not! Both are caused by a third factor: hot, sunny weather.
Lurking (Confounding) Variables
A lurking variable (or confounding variable) is a hidden third variable that influences both of the variables you are studying, creating an apparent link between them.
Common Mistake to Avoid: In exam explanations, never write "Variable \(A\) causes Variable \(B\)" simply because the graph has a strong positive or negative correlation. Always look out for underlying common causes.
Key Takeaway: Correlation shows a connection, but further experimental investigation is needed to prove a direct cause-and-effect link.
4. The Line of Best Fit
A line of best fit is a straight line drawn through the data on a scatter diagram that shows the general trend.
The Mean Point \((\bar{x}, \bar{y})\)
In your exam, your line of best fit must pass through the mean point \((\bar{x}, \bar{y})\).
To calculate the mean point:
1. Calculate the mean of all the \(x\)-values: \(\bar{x} = \frac{\sum x}{n}\)
2. Calculate the mean of all the \(y\)-values: \(\bar{y} = \frac{\sum y}{n}\)
3. Plot the point \((\bar{x}, \bar{y})\) on your scatter diagram (often marked with a distinct circle or label).
Step-by-Step: Drawing the Line of Best Fit
1. Calculate and plot the mean point \((\bar{x}, \bar{y})\).
2. Place your ruler so that it passes directly through \((\bar{x}, \bar{y})\).
3. Rotate the ruler until it matches the overall direction of the points, ensuring an roughly equal number of points lie above and below the line.
4. Draw a single, clean straight line extending across the full range of the plotted points.
Finding the Equation of the Line: \(y = mx + c\)
The equation of any straight line of best fit is given by:
\(y = mx + c\)
• \(m\) is the gradient (slope): \(m = \frac{\text{change in } y}{\text{change in } x} = \frac{y_2 - y_1}{x_2 - x_1}\)
• \(c\) is the \(y\)-intercept (the value of \(y\) where the line crosses the vertical axis, at \(x = 0\)).
Interpreting the Gradient and Intercept in Context
Examiners love asking what the gradient and intercept mean in real-world scenarios:
• Gradient (\(m\)): Represents the estimated change in the response variable (\(y\)) for every \(1\) unit increase in the explanatory variable (\(x\)).
Example: A gradient of \(+2.5\) means "for every extra hour of revision, the test score increases by an estimated \(2.5\) marks."
• Intercept (\(c\)): Represents the baseline value of \(y\) when \(x = 0\).
Example: A \(y\)-intercept of \(15\) means "a student who did \(0\) hours of revision is predicted to score \(15\) marks."
Key Takeaway: Always draw your line of best fit straight through the calculated mean point \((\bar{x}, \bar{y})\) using a clear ruler.
5. Making Predictions: Interpolation vs Extrapolation
Once you have a line of best fit, you can use it to estimate unknown values.
Interpolation (Reliable)
Interpolation means estimating a value within the range of the existing data points.
• Reliability: Generally reliable because we have observed data supporting the trend in this region.
• Example: If your data covers temperatures between \(10^\circ\text{C}\) and \(30^\circ\text{C}\), estimating sales at \(20^\circ\text{C}\) is interpolation.
Extrapolation (Unreliable)
Extrapolation means estimating a value outside the range of the existing data points.
• Reliability: Highly unreliable and risky because the trend may change, flatten out, or stop completely beyond our observed values.
• Example: Using the same data to estimate sales at \(45^\circ\text{C}\) is extrapolation. People might stay inside due to extreme heat, causing ice cream sales to drop.
Quick Review Box:
• Inside data range = Interpolation = Reliable
• Outside data range = Extrapolation = Unreliable / Risky
Key Takeaway: In exam questions, if asked whether a prediction is reliable, check if the \(x\)-value falls inside the data range (interpolation) or outside it (extrapolation).
6. Outliers in Scatter Diagrams
An outlier (or anomaly) is a data point that does not fit the general pattern or trend shown by the rest of the data.
How to Handle Outliers
• On a scatter diagram, an outlier sits far away from the main cluster of points or the line of best fit.
• Outliers can be caused by measurement errors, data entry mistakes, or rare genuine events.
• When calculating the mean point \((\bar{x}, \bar{y})\) and drawing a line of best fit, genuine errors should usually be identified and investigated before drawing the final line.
Summary Checklist: Exam Essentials
Before sitting your exam, make sure you can:
• Correctly plot bivariate coordinates \((x, y)\) on a scatter diagram.
• Clearly describe correlation using both strength (strong/moderate/weak) and type (positive/negative/zero).
• Explain why correlation does not necessarily mean causation.
• Calculate the mean point \((\bar{x}, \bar{y})\) and draw the line of best fit through it.
• Determine and interpret the equation \(y = mx + c\) in context.
• Distinguish between interpolation (reliable) and extrapolation (unreliable).