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Introduction to Statistics (STAT 101) · Module 2 of 7

The Normal Model & Linear Regression

The empirical rule and normal calculations, correlation, least-squares regression lines, residuals, and r².

In this module
  1. The Normal Distribution
  2. Scatterplots & Correlation
  3. Least-Squares Regression & Residuals
  4. Check yourself

The Normal Distribution

  • In a normal distribution, about 68% of values fall within 1 SD of the mean, 95% within 2 SD, and 99.7% within 3 SD (the empirical rule).
  • For other cutoffs, convert to a z-score and use a normal table or calculator (normalcdf).
The empirical rule
z-score68%95%−22
In a normal distribution about 68% of values fall within 1 standard deviation of the mean (dark shading) and about 95% within 2 (light shading).
Worked example

Heights are approximately normal with mean 66 in and SD 3 in. About what percent are between 60 and 72 inches?

Solution. 60 and 72 are each 2 SD from the mean, so about 95%.

Scatterplots & Correlation

  • Describe a scatterplot by direction, form (linear or curved), strength, and unusual points.
  • The correlation r (−1 to 1) measures the strength and direction of a LINEAR relationship only. It has no units, isn't resistant to outliers, and does not imply causation.
Worked example

Ice cream sales and drowning deaths have a strong positive correlation. Does ice cream cause drowning?

Solution. No. A lurking variable, hot weather, increases both. Correlation alone never establishes causation.

Least-Squares Regression & Residuals

  • The least-squares line ŷ = a + bx minimizes the sum of squared residuals. The slope b is the predicted change in y for each 1-unit increase in x.
  • Residual = observed y − predicted ŷ. A residual plot with no pattern means a linear model is appropriate.
  • r² is the fraction of the variation in y explained by the linear model with x. Avoid extrapolation: predicting far outside the range of the data is unreliable.
A regression line and one residual
xyŷ = 12 + 3xresidual +3
With the line ŷ = 12 + 3x, the point observed at x = 5, y = 30 is predicted to be 27, so its residual is 30 − 27 = +3: the line underpredicts it.
Worked example

The line is ŷ = 12 + 3x. For x = 5 the observed y is 30. Find the prediction and the residual.

Solution. ŷ = 12 + 3(5) = 27. Residual = 30 − 27 = +3: the model underpredicts this point by 3.

Common mistake: Interpret the slope with "predicted": "For each additional hour studied, the PREDICTED score increases by 3 points." Leaving out "predicted" loses AP credit.

Check yourself

2 of this module's 5 quiz questions. Pick your answers, then check them in the app, with an explanation for every question.

1. In a normal distribution, about what percent of values lie within 1 standard deviation of the mean?

2. A correlation of r = −0.9 indicates:

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