Why this matters
A lot of analysis is about two variables at once: does more study time mean higher scores? Does a longer commute mean lower scores? Scatter plots, correlation, and regression let you describe and even predict these relationships, while staying alert to the trap of mistaking coincidence for cause.
The idea
A scatter plot plots each (x, y) pair as a point. The cloud of points shows a tendency:
- Positive correlation: y tends to rise as x rises.
- Negative correlation: y tends to fall as x rises.
- No correlation: no clear tendency either way.
The correlation coefficient r puts a number on it: −1 ≤ r ≤ 1. The closer to 1, the
stronger the positive correlation; the closer to −1, the stronger the negative; near 0
means little or none. A spreadsheet's CORREL function computes it.
Correlation is not cause. A causal relationship is when one variable really drives the other (hotter days → more shaved-ice sales). A pseudo-correlation is when two things move together with no cause between them. Shaved-ice sales and heatstroke cases both rise with temperature, but neither causes the other.
Cross tabulation is a different summary: a table that counts data across two or more categories at once (for example agree/oppose/neither, broken down by male/female).
Regression analysis turns a correlation into a prediction. Simple regression fits a regression line y = ax + b that best follows the points, found by the least squares method. That method chooses the line that makes the residuals (the vertical gaps between points and line) as small as possible. Plug an x into the equation to predict y.
Picture it
flowchart TD
S[Scatter plot of x, y pairs] --> C{What is the trend?}
C -- y rises as x rises --> P[Positive correlation, r near 1]
C -- y falls as x rises --> N[Negative correlation, r near -1]
C -- no clear trend --> Z[No correlation, r near 0]
P --> R[Fit regression line y = ax + b by least squares]
N --> R
R --> PR[Predict y for a new x]Worked example
A regression line for sleep duration versus daily steps is y = 0.0287x + 297.86, where x is steps and y is minutes of sleep. To predict sleep for 5,000 steps, substitute x = 5,000:
y = 0.0287 × 5,000 + 297.86 = 143.5 + 297.86 = 441.36 ≈ 441 minutes.
The line came from the least squares method, which picked the slope and intercept that minimize the residuals across all the data points. Notice the difference from a cause claim: more steps correlating with more sleep does not prove steps cause sleep. That would need separate evidence.
Your turn
Practise reading scatter-plot trends, matching correlation strengths to r values, telling causation from pseudo-correlation, and predicting from a regression line.
Recap
- A scatter plot shows positive, negative, or no correlation between two variables.
- The correlation coefficient r runs from −1 to 1; magnitude near 1 means a strong relationship.
- Causation means one variable drives the other; pseudo-correlation only looks that way.
- Regression fits y = ax + b by the least squares method (minimizing residuals) so you can predict y from x.