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2.7 Residuals

Part of Exploring Two-Variable Data in AP® Statistics. Read the opening section, then create a free account for the full interactive lesson, practice, and progress tracking.

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What is a Residual?

A residual

e=y−y^=observed−predictede = y - \hat{y} = \text{observed} - \text{predicted}

Interpretation:

  • Positive residual → actual value is above the regression line (underprediction)
  • Negative residual → actual value is below the regression line (overprediction)
  • Residual = 0 → point is exactly on the regression line

Example: A student studies 4 hours. The regression line predicts a score of 78. If the actual score is 85: e=85−78=+7{e = 85 - 78 = +7} (overprediction by 7 points... wait, underprediction: actual > predicted → positive residual → the model underestimated this student's score)

Key property of LSRL:
∑ei=0(residuals always sum to zero)\sum e_i = 0 \quad \text{(residuals always sum to zero)}

In this unit

9 lessons

  1. 2.1Introducing Statistics: Are Variables Related?Preview lesson
  2. 2.2Representing Two Categorical VariablesPreview lesson
  3. 2.3Statistics for Two Categorical VariablesPreview lesson
  4. 2.4Representing the Relationship Between Two Quantitative VariablesPreview lesson
  5. 2.5CorrelationPreview lesson
  6. 2.6Linear Regression ModelsPreview lesson
  7. 2.7ResidualsThis lesson
  8. 2.8Least Squares RegressionPreview lesson
  9. 2.9Analyzing Departures from LinearityPreview lesson

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