Correlation and linear regression

Correlation describes linear association, while a regression line predicts a response from an explanatory variable. The least-squares line minimises squared vertical residuals. A strong association does not establish causation. Prediction beyond the observed input range is extrapolation; transformations can reveal a relationship hidden in the original coordinates.

Begin with paired observations

The notes ask about links between GenAI use and examination scores, and between temperature and rainfall intensity. These are questions to investigate, not established conclusions. In the source’s controlled experiment, vary the slope angle through $10^\circ,20^\circ,30^\circ,40^\circ,50^\circ$ and record a ball’s travel time. The angle is the independent (explanatory) variable, and the travel time is the dependent (response) variable.

When control is impractical, the source proposes observing temperature and sea level each day over ten years. This is observational data collection. The notes use “natural experiments” broadly here; an observational study is not automatically a natural experiment with an externally imposed change. Controlled studies are possible in some settings even in fields mentioned by the source; identify what is actually controlled in the particular investigation.

Bivariate data consist of ordered pairs $(x_1,y_1),\ldots,(x_n,y_n)$, such as each person’s age and salary. Each pair must refer to the same observation. The explanatory variable supplies the basis for prediction or study; the response is the outcome observed or predicted. In an observational study the explanatory variable need not be manipulated. Calling a variable “independent” in this role does not assert statistical independence of X and Y.

Try the five source situations before opening their working.

Inspect a scatter diagram

A scatter diagram plots the paired numerical observations, usually with the explanatory variable on the horizontal x-axis and the response on the vertical y-axis. The notes also call it a scatter plot, scatter graph or scatter chart. Inspect:

• Shape: approximately linear, curved (such as the source’s inverted U), or another pattern (such as its loop).
• Direction for a linear pattern: positive/direct means y tends to rise with x; negative/inverse means y tends to fall with x. “Inverse” here describes direction, not a proved reciprocal formula $y=k/x$.
• Strength of the linear pattern: tight clustering about a line is stronger than a widely scattered cloud. A strong curved pattern need not have strong linear correlation.

All eight schematic panels from the opening notes are reconstructed in the accompanying figure: three shape examples, two direction examples and three strength examples. They are illustrations, not measured numerical data.

The source uses numerical variables such as height, mass, rates and counts. It contrasts beauty and happiness as not directly numerical. Such concepts can be studied using explicitly defined numerical scores, but a score’s meaning and measurement quality need justification; merely assigning category numbers does not establish a quantitative scale.

For the linear correlation studied here, the source’s sequence is sketch the scatter diagram, then find the correlation coefficient. Plotting first can reveal curvature or unusual observations hidden by a single summary number.

Transfer a calculator plot to a clear sketch

Retain all five source pointers:

1. Check the window scale: the origin may not be visible.
2. Label both axes and put the explanatory variable horizontally, response vertically.
3. Mark the minimum and maximum x and y values.
4. Match the number of observations with the number of plotted dots or crosses. Coincident pairs can overlap, so record their multiplicity if needed.
5. Preserve relative positions accurately. Graph paper is unnecessary unless requested.

Example 1 retains every data pair and the complete six-step calculator route before describing the observed pattern.