Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Quick answer: Regression describes the linear relationship between an explanatory and a response variable. Correlation r measures the strength and direction of a linear pattern, the least-squares line ŷ = a + bx predicts the response, r² gives the percent of variation explained, and residual plots check whether a line is appropriate.
Standards: AP Statistics Unit 5 (2026-27) · Level: Grade 12 AP Statistics
How do you describe a scatterplot?
A scatterplot shows two quantitative variables measured on the same individuals. The explanatory variable goes on the x-axis and the response variable on the y-axis. Describe the direction (positive or negative), form (linear or curved), strength, and unusual features such as outliers.
The smoothie stand data show a strong, positive, linear association between temperature and cups sold, with no outliers.
What does the correlation r measure?
r = 1n − 1 ∑ xi − x̄sx · yi − ȳsy
−1 ≤ r ≤ 1. The sign gives the direction and the size gives the strength of a linear pattern. Changing units does not change r, and r is not resistant to outliers.
Here r = 0.9933. Correlation does not show causation: a lurking variable can drive both variables.
How do you find and use the least-squares line?
ŷ = a + bx, b = r sysx, a = ȳ − bx̄
The line always passes through (x̄, ȳ). Residual = y − ŷ (actual minus predicted).
For the smoothie data, x̄ = 78.44, sx = 10.561, ȳ = 59.89, sy = 15.592. Then b = 0.9933 × 15.592/10.561 = 1.467 and a = 59.89 − 1.467(78.44) = −55.15. The line is ŷ = −55.15 + 1.47x. Slope: each additional degree is associated with about 1.47 more cups sold, on average. Intercept: at 0°F the model predicts −55.1 cups, which is outside the data and not meaningful. For 85°F, ŷ = 69.5 cups.
How do you judge whether the line fits?
The coefficient of determination r2 = 0.9866 says that about 98.7% of the variation in cups sold is explained by the linear relationship with temperature. A residual plot should show random scatter around 0; a curved pattern means a line is not appropriate.
An influential point is one whose removal changes the line noticeably, typically a point far from the others in the x direction. Extrapolation, predicting outside the range of the data, is unreliable.
Interpret slope with the words on average, and name both variables with units. Never say that x causes y unless the data come from a randomized experiment.
- Describe a scatterplot by direction, form, strength and unusual features.
- Slope b = r·s_y/s_x; the line passes through (x̄, ȳ).
- Residual = actual y − predicted ŷ.
- r² is the proportion of variation in y explained by the line.
- Extrapolation and influential points can make a model misleading.
Regression Analysis: Lines, Residuals and r: frequently asked questions
What does the slope of a regression line mean?
The slope is the predicted change in the response for each one-unit increase in the explanatory variable. Use the words predicted or on average, and include units, for example: each extra degree is associated with about 1.4 more cups sold.
What does r² tell you?
It is the proportion of the variation in the response that is accounted for by the linear relationship with the explanatory variable. An r² of 0.96 means about 96% of the variation in y is explained by the line; the rest is unexplained.
What is a residual and what does a residual plot show?
A residual is the actual y minus the predicted y. A residual plot graphs residuals against x or ŷ. Random scatter around 0 means a line is appropriate; a curve or a fan shape shows the linear model is not a good fit.
What is the difference between an outlier and an influential point?
An outlier has an unusual y value given its x, giving a large residual. An influential point is one whose removal would substantially change the slope or intercept. Points far away in the x direction are often influential even with a small residual.
Why is extrapolation risky?
The linear pattern is only known for the range of x-values in the data. Beyond that range the relationship may change, so predictions can be far from the truth. Interpolating within the range is much safer.
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