7.3 Scatter Plots, Lines of Fit, and Correlation
Key Takeaways
- A scatter plot displays bivariate data as ordered pairs; the pattern of points reveals whether the association is positive, negative, or roughly none.
- A line of fit is drawn to follow the trend of the data; its slope predicts how much the response variable changes per unit increase in the explanatory variable, and its intercept gives the predicted response when the explanatory variable is zero (when meaningful).
- The correlation coefficient r ranges from −1 to 1; |r| near 1 is a strong linear association, near 0 is a weak or no linear association, and the sign matches the direction of the slope.
- Correlation does not imply causation; a strong r only means the two variables move together, not that one causes the other, because confounding variables may drive both.
- A residual is the actual value minus the predicted value from the line of fit; patterns in residuals suggest the linear model is inappropriate (DP.2.4, DP.2.6, also DP.1.3).
Quick Answer: A scatter plot shows bivariate data as points; a line of fit summarizes the trend. The slope of the line predicts the change in the response variable per unit increase in the explanatory variable. The correlation coefficient r measures the strength and direction of the linear association on a scale from −1 to 1—but correlation never proves causation without a controlled study.
Scatter Plots and Association (MA.912.DP.2.4, also DP.1.3)
A scatter plot graphs bivariate (two-variable) data as ordered pairs (x, y), where the explanatory variable x is the input and the response variable y is the output. Each point represents one observation.
The overall pattern of points reveals the association:
| Pattern | Direction | Description |
|---|---|---|
| Points trend upward left-to-right | Positive | As x increases, y tends to increase |
| Points trend downward left-to-right | Negative | As x increases, y tends to decrease |
| No clear trend | No association | y does not systematically change with x |
A scatter plot also reveals whether the association is linear (points follow a straight-line trend) or non-linear (points follow a curve). A linear model is appropriate only when the scatter plot shows a roughly straight trend.
Fitting a Line to Bivariate Data
A line of fit (also called a line of best fit) is a straight line that follows the trend of the data. Draw it by eye so that about half the points lie above and half below. The equation has the form ŷ = mx + b, where:
- m (slope) predicts how much the response variable changes for each one-unit increase in the explanatory variable.
- b (y-intercept) predicts the response variable when the explanatory variable is 0. This may not be meaningful if x = 0 is outside the observed data range.
- ŷ (y-hat) is the predicted value—distinguish it from the actual observed y.
Interpreting Slope and Intercept in Context
Worked example: A line of fit for the relationship between hours studied (x) and exam score (y) is ŷ = 5.2x + 52.
- Slope 5.2: For each additional hour studied, the predicted exam score increases by about 5.2 points.
- Intercept 52: A student who studies 0 hours is predicted to score about 52 points. This may be meaningful as a baseline but can also be an extrapolation if no student in the data studied 0 hours.
- Prediction: A student who studies 6 hours is predicted to score
5.2(6) + 52 = 31.2 + 52 = 83.2points.
The Correlation Coefficient r (MA.912.DP.2.6)
The correlation coefficient, denoted r, is a number between −1 and 1 that measures the strength and direction of a linear association.
| Value of r | Strength | Direction |
|---|---|---|
| r | ≥ 0.7 | |
| 0.3 ≤ | r | < 0.7 |
| 0 < | r | < 0.3 |
| r = 0 | No linear association | None |
The sign of r always matches the sign of the slope of the line of fit: positive r means the line slopes up, negative r means the line slopes down. The value r is unitless and must satisfy −1 ≤ r ≤ 1; any computed value outside this range signals an arithmetic error. A value of r near 0 means no linear relationship, not 'no relationship'—a perfect quadratic curve can have r = 0.
Correlation vs. Causation
A strong correlation between two variables does not prove that one causes the other. Three explanations for an observed correlation are:
- Causation: x directly causes y (verified only by a controlled experiment).
- Confounding variable: a third variable z causes both x and y (ice cream sales and drowning both increase in summer; heat is the confounder).
- Coincidence: the correlation is accidental in small data sets.
Florida EOC items often present a strong r and ask whether 'x causes y' is justified. The correct answer is 'No—correlation does not establish causation; a controlled study is required.'
Worked example: A study finds r = 0.85 between the number of fire trucks at a fire and the dollar damage. It is wrong to conclude that more trucks cause more damage. The confounding variable is fire size—larger fires attract more trucks and cause more damage.
Residuals and Prediction
A residual is the difference between the actual y-value and the predicted value: residual = actual y − predicted ŷ. A positive residual means the line underpredicts (the point lies above the line). A negative residual means the line overpredicts (the point lies below). A residual of 0 means the point lies on the line.
Using Residuals to Assess a Linear Model
- If residuals are scattered randomly above and below 0 with no pattern, a linear model is appropriate.
- If residuals show a curved pattern, the relationship is non-linear and a linear model is inappropriate.
- The sum of squared residuals measures total prediction error; the line of best fit minimizes this sum.
Prediction: Interpolation vs Extrapolation
- Interpolation (predicting within the observed range of x) is generally reliable.
- Extrapolation (predicting outside the observed range) is risky—the trend may not continue.
Example: If the line of fit is based on data for x between 2 and 12 hours of study, predicting for 8 hours (interpolation) is safe; predicting for 25 hours (extrapolation) is not—the score may plateau at 100.
These skills anchor roughly 8–12% of the EOC and appear as extended-response items combining interpretation, calculation, and a written conclusion.
A line of fit for the relationship between hours of practice (x) and free-throw percentage (y) is ŷ = 3.5x + 40. What does the slope represent in context?
A study reports a correlation coefficient of r = −0.92 between daily screen time and hours of sleep. Which conclusion is valid?
For a data point with actual y-value 18 and a line of fit that predicts ŷ = 22 at that x-value, what is the residual, and where does the point lie relative to the line?