IB MYP 3 Mathematics 9.4 Scatter Plots, Correlation and Lines of Best Fit Study Notes - New Syllabus

9.4 – Scatter Plots, Correlation and Lines of Best Fit

When two numerical variables are measured for the same set of individuals or objects, we can investigate whether there is a relationship between them. For example, we might investigate the relationship between:

• hours studied and test score;
• height and mass;
• age of a car and its value; or
• temperature and ice-cream sales.

Data involving two variables is called bivariate data. A useful way to display bivariate numerical data is a scatter plot.

Bivariate Data

Bivariate data consists of two values recorded for each observation. The two variables are usually called \(x\) and \(y\).

For example, a teacher records the number of hours students study and their test scores.

 Independent and Dependent Variables

When investigating two variables, it is useful to decide which variable is the explanatory variable and which is the response variable.

• The explanatory variable is placed on the horizontal \(x\)-axis.
• The response variable is placed on the vertical \(y\)-axis.

For example, when studying the relationship between study time and test score:

\(x=\text{study time}\)

\(y=\text{test score}\)

 Scatter Plots

A scatter plot displays pairs of numerical data as individual points on a coordinate grid. Each point represents one observation.

For example, the ordered pair \( (3,60) \) represents a student who studied for \(3\) hours and achieved a score of \(60\).

To construct a scatter plot:

1. Identify the two variables.
2. Choose suitable scales for the \(x\)- and \(y\)-axes.
3. Label both axes, including units.
4. Plot each ordered pair.
5. Add a clear title.

 Positive Correlation

A scatter plot shows positive correlation when larger values of one variable tend to be associated with larger values of the other variable.

The points generally rise from left to right.

For example, if students who study for more hours generally achieve higher test scores, the two variables have a positive correlation.

\(\text{Study time} \uparrow \quad \Rightarrow \quad \text{Test score tends to} \uparrow\)

 Negative Correlation

A scatter plot shows negative correlation when larger values of one variable tend to be associated with smaller values of the other variable.

The points generally fall from left to right.

For example, as the age of a car increases, its value may tend to decrease.

\(\text{Car age} \uparrow \quad \Rightarrow \quad \text{Car value tends to} \downarrow\)

No Correlation

There is no correlation when there is no clear relationship between the two variables. The points appear scattered without a clear upward or downward pattern.

For example, a person’s shoe size and the number of books they read in a month would not normally be expected to have a meaningful relationship.

Strength of Correlation

Correlation is not only about whether a relationship is positive or negative. We can also describe its strength.

A strong correlation occurs when the points lie close to a clear pattern. A weak correlation occurs when the points are more widely scattered.

 Describing Correlation

A complete description should state both the direction and strength of the relationship.

For example:

Avoid simply saying “the graph goes up”. A mathematical description should identify the direction and strength of the relationship.

Lines of Best Fit

When a scatter plot shows a clear linear trend, we can draw a line of best fit.

A line of best fit is a straight line that represents the overall trend of the data. It does not usually pass through every point.

Drawing a Line of Best Fit

A reasonable line of best fit should pass through the middle of the data. There should be approximately as many points above the line as below it, although the numbers do not need to be exactly equal.

The line should represent the overall trend rather than joining individual points.

1. Look at the overall pattern of the points.
2. Identify the direction of the relationship.
3. Draw a straight line through the centre of the data.
4. Try to balance the points above and below the line.
5. Do not force the line through every point.

Making Predictions Using a Line of Best Fit

Once a line of best fit has been drawn, it can be used to estimate the value of one variable when the other variable is known.

For example, suppose a line of best fit for study time and test score passes approximately through the point:

\( (3,60) \)

This suggests that a student who studies for approximately \(3\) hours might be expected to score about \(60\) marks.

 Interpolation and Extrapolation

Predictions can be made in two different ways.

• Interpolation means making a prediction within the range of the observed data.
• Extrapolation means making a prediction outside the range of the observed data.

 Equation of a Straight Line

A line of best fit can sometimes be represented using the equation of a straight line:

where:

• \(y\) is the predicted value;
• \(x\) is the explanatory variable;
• \(m\) is the gradient of the line; and
• \(b\) is the \(y\)-intercept.

For example, suppose a line of best fit is:

\(y=6x+45\)

If \(x=5\):

\(y=6(5)+45\)

\(y=30+45=75\)

The predicted value of \(y\) is 75.

Gradient of a Line of Best Fit

The gradient tells us how much the predicted \(y\)-value changes when \(x\) increases by one unit.

For:

\(y=4x+20\)

the gradient is \(4\). This means that for every increase of \(1\) in \(x\), the predicted \(y\)-value increases by \(4\).

A positive gradient represents an upward trend, while a negative gradient represents a downward trend.

 Correlation Does Not Mean Causation

If two variables are correlated, this does not automatically mean that one variable causes the other.

For example, ice-cream sales and the number of people swimming at a pool may both increase during warmer weather. The two variables are related, but increased ice-cream sales do not necessarily cause more people to swim. A third variable, such as temperature, may influence both.

 Interpreting a Scatter Plot Step by Step

1. Identify the two variables.
2. Identify which variable is on the \(x\)-axis and which is on the \(y\)-axis.
3. Look for the general direction of the points.
4. Decide whether the correlation is positive, negative or absent.
5. Decide whether the relationship is strong or weak.
6. Look for unusual points or possible outliers.
7. If a line of best fit is present, use it to make reasonable estimates.

 Comparing Types of Correlation