A residual is the difference between an observed data value and the value predicted by a statistical model, calculated as residual=yobserved−ypredicted. Positive residuals mean the model underestimated; negative residuals mean it overestimated.
If your model predicts 80 but the actual value is 85, the residual is +5. Residuals are 'leftovers' - what the model couldn't explain. Patterns in residuals reveal model problems.
Showing a random 20 of 76 problems.
Example 1
medium
Using y^=4x−1, find the residual at (3,9).Find the residual at (3, 9) on the line ŷ
Example 2
medium
y^=0.8x+4. Observed (20,22). Find the residual.
Example 3
medium
The sum of residuals for the least-squares regression line is always what number?
Example 4
medium
For y^=−x+10, find the residual at (2,5) and state over- or under-estimate.Find the residual at (2, 5) on ŷ
Example 5
medium
For y^=4x−1, find the residual at (2,6) and state whether the point lies above or below the line.Find the residual at (2, 6); is the point above or below the line?
Example 6
medium
If the largest residual is +12, what does this point look like on a scatterplot?
Example 7
medium
Using y^=0.5x+2, complete the residual table for (2,3),(4,5),(6,4),(8,7).Complete the residual table for points A–D on the line ŷ = 0.5x + 2
Example 8
medium
For a least-squares line, what is ∑(yi−y^i)?
Example 9
hard
Sum of squared residuals is 80 for n=10 points (regression with 1 predictor). Estimate the residual standard error s.
Example 10
medium
For y^=2x+3, find residuals at (1,6),(2,8),(3,10) and identify the worst-fit point.Find the residuals at A, B, C; all three are equal — identify the worst-fit point
Example 11
medium
Squaring residuals before summing produces what quantity used to fit the line?
Example 12
medium
Why do we square residuals when fitting the least-squares line?
Example 13
medium
A residual plot shows residuals randomly scattered around 0. What does this suggest?
Example 14
medium
How is a residual different from the model's 'error' term?
Example 15
easy
A point lies above the regression line. Is its residual positive or negative?
Example 16
medium
Two models fit the same point (4,20). Model A predicts 18, Model B predicts 23. Which model fits this point better?
Example 17
medium
Residual plot of a linear fit shows a clear funnel shape (spread grows with x). What does this reveal?
Example 18
medium
Regression line: y^=4x−1. Three observed points are (1,5),(2,6),(3,12). Find each residual.Compute the residual at each of the three points A, B, and C
Example 19
medium
A residual plot scatters randomly around zero with no pattern. What does this suggest about the linear model?
Example 20
hard
A regression model predicts y^=30 for a data point, but the actual value is y=35. Calculate the residual and interpret it.