4.2 Correlation, Causation, and Regression

Why This Section Matters

Praxis 5165 expects you to connect graphical models to statistical reasoning. Scatterplots, correlation language, and lines of best fit show up in pure math items and in task-of-teaching questions where a student claims that a strong association proves causation. You must know the mathematics and the instructional correction.

Scatterplots and Correlation

A scatterplot displays paired (x, y) data. Before calculating anything, describe:

• Direction: positive (upward trend), negative (downward), or no clear linear trend
• Form: linear, curved, or scattered
• Strength: tight clustering near a line versus wide scatter
• Outliers: points far from the overall pattern

Correlation (often r) quantifies linear association on a scale from −1 to +1.

• r near +1: strong positive linear association
• r near −1: strong negative linear association
• r near 0: little or no linear association (a curved pattern can still exist)

Correlation is not causation. Ice cream sales and sunscreen sales may rise together in summer because temperature is a lurking variable affecting both.

Causation Requires Stronger Evidence

Association alone does not establish that changing x will change y. Common Praxis wrong answers claim causation from a scatterplot. Stronger evidence includes:

1. A well-designed experiment with random assignment
2. A plausible mechanism supported by context
3. Control for confounding variables

Good teacher questions push students to ask: Could something else explain this pattern? That is the heart of distinguishing correlation from causation on the exam.

Line of Best Fit (Linear Regression)

The line of best fit (least-squares regression line) has form y = mx + b, where:

• m = slope = predicted change in y per 1-unit increase in x
• b = y-intercept = predicted y when x = 0 (only interpret in context if x = 0 is realistic)

The line minimizes the total squared vertical distances from points to the line. On Praxis items you may be given the equation and asked for a prediction or a residual.

Worked Example: Prediction and Residual

A regression line for hours studied (x) and quiz score (y) is y = 5x + 70. For x = 6:

Predicted score: y = 5(6) + 70 = 100.

Suppose the student's actual score is 92.

Residual = actual − predicted = 92 − 100 = −8.

A negative residual means the point lies below the line — the student scored 8 points lower than the model predicted for 6 hours of study. If another student scored 104 at x = 6, the residual would be +4 (above the line).

Worked Example: Interpreting Slope in Context

If the line is y = 1.5x + 12 where x is hours of practice and y is free-throw percentage, the slope 1.5 means: for each additional hour of practice, the model predicts free-throw percentage increases by 1.5 percentage points, on average. The intercept 12 is only meaningful if 0 hours of practice is a realistic setting for the model.

Correlation versus Regression Mistakes

Regression on the Calculator

The on-screen graphing calculator on computer-delivered Praxis tests can compute regression equations. Still decide first whether a linear model is appropriate from the scatterplot shape. Forcing a line through curved data produces misleading predictions.

Worked Example: Correlation Strength in Context

A teacher compares hours of sleep (x) and quiz scores (y) for 20 students. The scatterplot shows a moderate upward trend; points are scattered but generally rise together. A correlation of r ≈ 0.55 indicates a moderate positive linear association — not proof that sleeping more causes higher scores, but evidence that students who sleep more tend to score higher in this sample.

A second variable such as overall study habits could influence both sleep and scores. The best instructional question asks students what other factors might explain the pattern before discussing causation.

Using Technology Wisely

Regression on a calculator returns y = ax + b quickly. Before accepting the model, verify that:

1. Both variables are quantitative.
2. The relationship looks approximately linear on the scatterplot.
3. There are no extreme outliers unduly pulling the line.

If the pattern is curved, a linear model may still give a positive slope while r is only moderate — predictions outside the data range can be especially unreliable.

Worked Example: Reading Slope and Intercept Together

Suppose a regression line for years of experience (x) and annual salary in thousands (y) is y = 3.2x + 42.

• Slope 3.2: each additional year of experience is associated with a predicted salary increase of $3,200 on average.
• Intercept 42: a teacher with 0 years of experience is predicted to earn $42,000 — only meaningful if x = 0 occurs in the real hiring context.

If x = 12, predicted y = 3.2(12) + 42 = 80.4, or $80,400. An actual salary of $76,000 gives residual 76 − 80.4 = −4.4 thousand dollars — below the line.

Exam Checklist

1. Describe the scatterplot in words before choosing an answer.
2. Separate prediction (use the equation) from causation (needs design/context).
3. Compute residual = actual − predicted carefully; sign matters.
4. In teaching items, pick the response that corrects the statistical reasoning error, not just the arithmetic.