Construct and interpret a residual plot

Which calculation defines a residual for one case?

At s=0 the line is the least-squares fit for the five original pairs. Predict whether shifting its intercept can remove the curved residual pattern. Move the slider, then compare the sum and sum of squares of residuals

At s=0 the residuals add to zero. Does that establish that a linear model is suitable?

Residual plots

A residual is the signed difference between an observed response and the response predicted by a fitted line. A residual plot uses explanatory values horizontally and residuals vertically to assess whether the departures have a clear pattern

1. 1Find the predicted response Substitute the explanatory value into the fitted line before calculating the residual \(\displaystyle \widehat{y}=12+5\times4=32\)
2. 2Subtract in the correct order Calculate the residual as the actual response minus the predicted response \(\displaystyle e=35-32=3\)
3. 3Keep the residual sign A point below the fitted line has a negative residual, while a point above it has a positive residual \(\displaystyle y<\widehat{y}\Rightarrow e<0,\quad y>\widehat{y}\Rightarrow e>0\)
4. 4Use the residual-plot axes Put the explanatory variable horizontally and the signed residual vertically, with zero as the reference line \(\displaystyle (x,e)=\left(4,3\right),\quad e=0\text{ is the reference line}\)

Given values

\(\displaystyle \hat y=24+3x,\quad x=7\)

Substitute x=7 into the fitted line. What response value is predicted?

Given values

\(\displaystyle y=39,\quad\hat y=45\)

Calculate the signed residual for the actual and predicted scores

The actual response is 39 and the predicted response is 45. Which line gives the signed residual?

Construct two residual plots and a model assessment

A: use the six original card sessions (minutes, cards): A(1,3), B(2,4), C(3,6), D(4,5), E(5,8), F(6,7). B: use the five original input/response pairs (1,1),(2,4),(3,9),(4,16),(5,25). The fitted lines are shown below. For each dataset produce a table of x, observed y, predicted ŷ and signed residual e. Keep exact values in the calculation; report A residuals to three decimal places in a separate display column

1. 1Construct, do not only recognise Draw the residual plots yourself. Label the explanatory axis, residual axis and units, include e=0 and plot each (x,e). Keep pairings intact, use a uniform scale and do not join observations into a fitted curve. Select and justify windows that show every residual and both sides of zero. Identify one overprediction and one underprediction from each dataset
2. 2Explain the evidence Compare the patterns in input order. For B calculate r² and explain whether a high value, both residual signs or a zero residual sum establishes linear suitability. Verify the displayed line using the means and least-squares slope. Construct a counterexample to “zero residual sum proves a suitable linear model” by adding 3 to every B response; explain which quantities change
3. 3Choose and verify technology Choose a table and XY-plot tool and justify the functions used. Record input columns, exact stored coefficients, row formulas for ŷ and e, and the selected windows. Check two residuals by hand, verify Σe=0 and Σxe=0 for each unrounded least-squares fit, and compare a plotted coordinate with its source row. Rounded displays need not sum exactly to zero

Model response: paired data to evidence

A uses the observed (minute, card) pairs (1,3), (2,4), (3,6), (4,5), (5,8), (6,7). B uses the observed (input, response) pairs (1,1), (2,4), (3,9), (4,16), (5,25). A: predictions are (115,146,177,208,239,270)/35 cards. Residuals are (−10,−6,33,−33,41,−25)/35 cards, displayed as −0.286,−0.171,0.943,−0.943,1.171,−0.714. Plot them at x=1,2,3,4,5,6 minutes. A and D have negative residuals, so the model overpredicts; C and E have positive residuals, so it underpredicts. The six residuals do not show an obvious smooth curve, providing limited support for a linear description in this setting, not proof of randomness, constant spread or individual accuracy

1. 1B exposes the limitation B predictions are −1,5,11,17,23 and residuals are 2,−1,−2,−1,2. Their plot has positive ends and a negative middle, so it questions the linear form. At x=1 the model underpredicts by 2 units; at x=3 it overpredicts by 2. The negative fitted prediction at x=1 is still used in e=1−(−1)=2; do not clamp it or remove its sign
2. 2Independent exact verification For B, x̄=3,ȳ=11, Sxx=10,Sxy=60,Syy=374. Thus b=6,a=−7 and r²=3600/(10×374)=180/187≈0.963. Residual SSE=14 and 1−14/374=180/187. Both Σe and Σxe are zero, yet the curve remains. A also has Σe=Σxe=0 and SSE=132/35. These balance identities require the unrounded least-squares fit with an intercept; they are not universal properties of every proposed line
3. 3Counterexample and reproducible plots Adding 3 to every B response gives (1,4),(2,7),(3,12),(4,19),(5,28). The new fitted line is ŷ=6x−4: predictions increase by 3, while residuals, their sum, SSE, r² and the curved pattern remain unchanged. One suitable residual window is x=0..6,e=−3..3 for B; for A use x=0..7,e=−1.5..1.5. Store each prediction as a+b*x and residual as observed−predicted, then plot the x and residual columns. This is a model workflow, not evidence of a learner operating a tool

Given values

\(\displaystyle \hat y=12/5+(31/35)x,\quad x=4\)

For session D at four minutes, give the exact model prediction in cards

Given values

\(\displaystyle x=4,\quad y=5,\quad\hat y=208/35\)

Session D records five cards. Find its exact signed residual

Which coordinate belongs on the residual plot for session D?

Given values

\(\displaystyle y=1,\quad\hat y=6(1)-7\)

Calculate the residual at x=1 when the fitted prediction is negative

Given values

\(\displaystyle e=(2,-1,-2,-1,2)\)

Calculate the sum of squared residuals for B.

At x=3, B has e=−2 response units. Which explanation is correct?

B has r²≈0.963 and residuals 2,−1,−2,−1,2. What is justified?

Distinguish a curved mean from changing spread

A separate original measurement series uses the reported model shown below. Its paired observations are (1,11),(1,13),(2,12),(2,16),(3,12),(3,20),(4,10),(4,26). Calculate and plot the residuals. Compare the average residual and the residual spread at each input. Explain whether the evidence establishes a curved mean, a constant prediction-error spread, or neither

1. 1Plan a responsible follow-up Suppose an additional observation lies far above a fitted line. Explain why a large residual alone neither proves that the point strongly changes the fitted slope nor justifies deleting it. Describe a data/context check and a fit comparison that would investigate its influence. Save your own explanation before reading the model

Model response: spread and unusual cases

The observed (input, response) pairs are (1,11), (1,13), (2,12), (2,16), (3,12), (3,20), (4,10), (4,26). The predictions at x=1,2,3,4 are 12,14,16,18. The paired residuals are ±1,±2,±4,±8. They average zero at every input, but their vertical spread widens. The evidence questions a constant-error-spread description and using one common error size for all inputs. It does not by itself establish a curved mean; distinguish that issue from the curvature in B.

1. 1A useful display Plot x horizontally and residual response units vertically, with x=0..5 and e=−9..9 so all points and zero are visible. Use separate unjoined points at repeated inputs. The table calculations and changing spread are objective components; a complete explanation still requires independent review
2. 2Investigate an unusual residual Check for recording errors, the measurement conditions and whether the observation belongs to the intended population. Compare the fitted line and justified predictions with and without the case, reporting both fits and the reason for comparison. Influence concerns the change in a fit; residual magnitude alone is insufficient, and automatic deletion is not justified. A clean residual plot also does not establish causation or validate extrapolation

What does this original residual plot establish?

One observation has a large residual. What should happen next?