Bivariate Data (HSC SSCE Mathematics Advanced): Revision Notes

Bivariate Data

Introduction to bivariate data

When we collect data about two related variables, we call this bivariate data. This type of data consists of ordered pairs of measurements, where we measure two things about each item in our sample.

For example, we might measure both the height and weight of a group of people. Each person gives us one ordered pair: (height, weight).

In bivariate data analysis, we need to identify:

• The independent variable ($x$): the variable we think might influence the other
• The dependent variable ($y$): the variable that might be influenced

These pairs of measurements can be displayed on a scatterplot, which is a graph showing all the data points on a coordinate plane with $x$ on the horizontal axis and $y$ on the vertical axis.

The two key tools for analysing bivariate data are:

• Correlation: measures how closely the variables are related
• Line of best fit: a line that best represents the trend in the data

Understanding correlation vs functional relationships

In most of mathematics, we work with functions where $y$ is completely determined by $x$. For example, if an electrician charges $100 for a visit plus $40 per power point, the total fee $y$ for installing $x$ power points is exactly:

$y = 100 + 40x$

This is a perfect functional relationship with positive gradient. Every point lies exactly on the line.

Similarly, if 100 old cars are being removed from a park at 7 per day, the number remaining after $x$ days is exactly:

$y = 100 - 7x$

This is a perfect functional relationship with negative gradient.

However, many real-world relationships are not perfect functions. Consider height and weight in people. While taller people tend to be heavier, people of the same height don't all weigh the same amount. The relationship exists, but it's not exact.

Pearson's correlation coefficient

The strength and direction of linear correlation is measured by Pearson's correlation coefficient, denoted by $r$.

The scale looks like this:

$r = -1 \quad \quad \quad \quad r = 0 \quad \quad \quad \quad r = 1$

Correlations of $1$ and $-1$ are called perfect correlations because every point lies exactly on a straight line.

Positive correlation: heights and weights

Let's look at a real example. When we plot the heights ($x$ in cm) against weights ($y$ in kg) of 50 people, we get a scatterplot like this:

Understanding correlation strength

Correlation can be classified as strong, moderate, weak, or none. Here are visual guides:

Positive correlations

• Strong positive: Points cluster tightly around an upward-sloping line
• Moderate positive: Points show an upward trend but with more scatter
• Weak positive: Points are loosely scattered with slight upward tendency
• None: Points are randomly scattered with no pattern

Negative correlations

• Strong negative: Points cluster tightly around a downward-sloping line
• Moderate negative: Points show a downward trend but with more scatter
• Weak negative: Points are loosely scattered with slight downward tendency
• None: Points are randomly scattered with no pattern

Types of correlation

Positive correlation

When $0 < r \leq 1$, we have positive correlation:

• The cluster slopes upwards from left to right
• As $x$ increases, $y$ tends to increase
• Example: height and weight (taller people tend to be heavier)

Negative correlation

When $-1 \leq r < 0$, we have negative correlation:

• The cluster slopes downwards from left to right
• As $x$ increases, $y$ tends to decrease
• Example: waiting time and customer satisfaction (longer waits lead to lower satisfaction)

Zero correlation

When $r = 0$:

• There is no linear relationship between the variables
• The points are scattered randomly
• Knowing $x$ tells us nothing about $y$

Example: no correlation

Example: negative correlation

The line of best fit

When we see correlation in our data, we can draw a line of best fit (also called the regression line) through the points. This line:

• Best represents the overall trend in the data
• Can be used to make predictions
• Shows the average relationship between the variables

The most common method for calculating this line is called the least squares regression line. This line minimises the sum of the squared vertical distances from all points to the line.

Drawing the line by eye

For now, we can estimate the line of best fit by eye. Here's the heights and weights example with a line of best fit:

By estimating visually:

• The gradient is approximately $0.65$
• The $x$-intercept is approximately $85$

This gives us the equation:

$$y - 0 = 0.65(x - 85) \\ y = 0.65x - 55.25$$

This means: for every extra centimetre of height, we expect weight to increase by about $0.65$ kg.

Lines of best fit for weak correlation

When correlation is weak, it's harder to draw the line by eye. Here are the customer service examples:

The top graph shows all data points (weak negative correlation). The bottom graph has the outlier removed (moderate negative correlation). Notice how:

• The line is clearer when correlation is stronger
• Removing the outlier changes both the correlation strength and the line of best fit
• Decisions about outliers should be based on understanding the context, not just mathematics

Repeated data points

Sometimes the same ordered pair appears more than once in your data. These are called multiple points. For example, two different people might both be 170 cm tall and weigh 68 kg.

Making predictions: interpolation and extrapolation

Once we have a line of best fit, we can use it to make predictions. However, we need to distinguish between two types of prediction:

Interpolation

Interpolation means predicting values within the range of our data. For example:

• If we have height data from 150 cm to 190 cm
• We can reasonably predict the weight of someone who is 170 cm tall
• This is relatively safe because we're working within our observed range

Interpolation is justified provided our sample is not biased and represents the population well.

Extrapolation

Extrapolation means predicting values outside the range of our data. For example:

• Using our height-weight data to predict the weight of someone 85 cm tall
• Or someone 220 cm tall

Understanding causation

When two variables are correlated, students often ask: "Does one cause the other?" This is a complex question with several possibilities.

If events $A$ and $B$ are correlated, four scenarios are possible:

1. $A$ causes $B$
2. $B$ causes $A$
3. Both $A$ and $B$ are caused by some third factor $C$
4. The correlation is coincidental (a fluke)

Additionally, many real phenomena have multiple causes, particularly in:

• Medicine (health outcomes have many contributing factors)
• Weather (many variables interact)
• Human behaviour (complex motivations)

Non-linear correlation

Not all relationships follow a straight line. Sometimes data cluster around a curve rather than a line. This is called non-linear correlation.

Key points to remember

Remember!