Scatter graphs and correlation

Introduction to Scatter Graphs

Welcome to the world of Scatter Graphs! Have you ever wondered if there is a link between two different things? For example, does the amount of time you spend gaming affect your exam scores? Or does the temperature outside change how many ice creams a shop sells?

In Statistics, we use scatter graphs to look for these relationships. They are a visual way to see how two different sets of data (called bivariate data) interact with each other.

1. What is Bivariate Data?

Don't let the name scare you! Bivariate data simply means data that involves two variables.

• Bi means two (like a bicycle has two wheels).
• Variate refers to variables (the things we are measuring).

On a scatter graph, we plot one variable on the horizontal \(x\)-axis and the other on the vertical \(y\)-axis. Each person or object is represented by a single "cross" \((x, y)\) on the graph.

2. Understanding Correlation

Once the points are plotted, we look for a pattern. This pattern is called correlation. It tells us how strongly the two variables are related.

Types of Correlation

There are three main types you need to recognize:

1. Positive Correlation: As one variable increases, the other also increases. The points generally go "upwards" from left to right.
   Example: The taller a person is, the longer their arms tend to be.
2. Negative Correlation: As one variable increases, the other decreases. The points generally go "downwards" from left to right.
   Example: The more miles a car has driven, the lower its value becomes.
3. No Correlation: There is no obvious pattern. The points are scattered randomly.
   Example: A person's height and their favorite color.

Strength of Correlation

We also describe how "strong" the relationship is:

• Strong Correlation: The points are very close to forming a straight line.
• Weak Correlation: The points follow a general direction but are more spread out.

Quick Tip: Think of positive correlation like a hill you are climbing up, and negative correlation like a slide you are going down!

3. Correlation vs. Causation

This is a very important point for your exams! Just because two things have a correlation, it does not mean that one causes the other. This is known as the "correlation does not imply causation" rule.

Example: Imagine a graph shows that ice cream sales and shark attacks both go up at the same time. This is a positive correlation. However, eating ice cream doesn't cause shark attacks! Both are actually caused by a third factor: hot weather, which makes people buy ice cream and go swimming in the sea.

4. The Line of Best Fit

To help us make sense of the data and make predictions, we draw a Line of Best Fit. This is a straight line that follows the general trend of the points.

How to draw a good Line of Best Fit:

1. Use a ruler and a sharp pencil.
2. The line should follow the direction of the points.
3. Try to have an equal number of points above and below the line.
4. The line does not have to go through the origin \((0, 0)\) unless the data suggests it should.
5. The line does not have to touch any of the actual points, though it often passes through some.

Outliers: Sometimes, you might see a point that is far away from all the others. This is called an outlier. When drawing your line of best fit, you should generally ignore outliers as they don't follow the main trend.

5. Making Predictions (Interpolation and Extrapolation)

We can use our Line of Best Fit to estimate values that aren't in our original data set. If you are given an \(x\) value, go up to the line and then across to find the \(y\) value (and vice-versa).

Interpolation (Safe)

Interpolation is when we predict a value inside the range of data we have already plotted.
Example: If we have data for children aged 5 to 10, predicting a value for a 7-year-old is interpolation. This is usually quite reliable.

Extrapolation (Risky)

Extrapolation is when we extend our Line of Best Fit to predict a value outside the range of our data.
Example: If we have data for children aged 5 to 10, using the line to predict the height of a 40-year-old is extrapolation.
The Danger: Extrapolation is unreliable because we don't know if the trend continues forever. (A 40-year-old won't keep growing at the same rate as a child!)

6. Common Mistakes to Avoid

• Connecting the dots: Never join the points like a dot-to-dot puzzle! A scatter graph needs a single straight Line of Best Fit.
• Ignoring the trend: Don't just draw a line from the first point to the last point. It must represent the average trend of all points.
• Forcing \((0, 0)\): Only start your line at \((0, 0)\) if it makes sense for the data.

Summary Checklist

- Bivariate data: Data with two variables \((x \text{ and } y)\).
- Positive Correlation: Points go up.
- Negative Correlation: Points go down.
- Line of Best Fit: A straight line showing the trend (use a ruler!).
- Correlation \(\neq\) Causation: A link doesn't mean one thing causes the other.
- Interpolation: Predicting inside the data range (reliable).
- Extrapolation: Predicting outside the data range (risky).