Scatter Graphs and Correlation for the TMUA

Bivariate Data and Scatter Graphs

Bivariate data consists of pairs of measurements taken from the same subject, denoted as $(x, y)$. To investigate the relationship between these two variables, we use a scatter graph. In a scatter graph, the independent variable (or the variable we suspect might influence the other) is typically plotted on the $x$-axis, while the dependent variable is plotted on the $y$-axis. Each individual data point is represented by a single dot at its corresponding $(x, y)$ coordinate.

The primary purpose of a scatter graph is to look for a pattern or trend. If the points appear to follow a certain path, we say there is a relationship between the variables. If the points are scattered randomly with no discernible pattern, there is no relationship.

Interpreting Correlation

Correlation refers to the nature of the linear relationship between the two variables. It is described in terms of its direction and its strength.

1. Positive Correlation: As the $x$ values increase, the $y$ values also tend to increase. The points on the scatter graph generally move from the bottom-left to the top-right.
2. Negative Correlation: As the $x$ values increase, the $y$ values tend to decrease. The points on the scatter graph move from the top-left to the bottom-right.
3. No Correlation: There is no apparent linear relationship between the variables. The points are distributed randomly across the plot.

The strength of the correlation is determined by how closely the data points follow a straight line. If the points lie very close to a single line, the correlation is strong. If the points are widely dispersed but still follow a general direction, the correlation is weak.

Correlation and Causation

A critical distinction in statistics is that correlation does not indicate causation. Just because two variables are correlated, it does not mean that a change in $x$ causes a change in $y$.

There are several reasons why correlation might exist without causation:

• Common Response: A third, unseen variable (a confounding variable) may be affecting both $x$ and $y$. For example, ice cream sales and shark attacks are positively correlated, but neither causes the other. Both are influenced by the common factor of warm weather.
• Coincidence: In small datasets, a trend may appear simply by chance.
• Reverse Causality: It may be that $y$ causes $x$, rather than the other way around.

Lines of Best Fit

A line of best fit is a straight line drawn through the center of the data points on a scatter graph to represent the general trend. When drawing a line of best fit by eye, the aim is to minimize the total distance between the points and the line, ensuring that roughly half the points lie above the line and half below it.

A useful rule of thumb for drawing an estimated line of best fit is that it should pass through the mean point $(\bar{x}, \bar{y})$, where $\bar{x}$ is the mean of all $x$ values and $\bar{y}$ is the mean of all $y$ values. The line should only be drawn if there is a clear linear correlation.

Prediction: Interpolation and Extrapolation

Once a line of best fit is established, it can be used to predict values of one variable based on the other. This is done by identifying a value on one axis, moving to the line of best fit, and reading the corresponding value on the other axis.

• Interpolation: This is the process of predicting a value within the range of the original data. If the data set ranges from $x = 10$ to $x = 50$, predicting a value for $x = 30$ is interpolation. This is generally considered reliable, provided the correlation is strong.
• Extrapolation: This is the process of predicting a value outside the range of the original data. Predicting a value for $x = 100$ when the data only goes up to $x = 50$ is extrapolation.

Extrapolation is inherently dangerous because we have no evidence that the trend continues beyond the observed data range. The relationship might become non-linear, it might level off, or the variables may simply cease to be related. For example, a child's height is positively correlated with their age, but extrapolating this trend to age 50 would result in an impossible prediction.