Unit 2 · Topic 2.6

Competing Function Model Validation

Residuals (actual minus predicted) measure how far a model misses each data point.

Practice 3: Communication and Reasoning

Part 1 · Hook

Why this matters

Two analysts model a channel's followers. One fits a line, the other an exponential function, and both graphs look close to the data. The residuals, the misses each model makes, decide which one to trust for next month.

Part 2 · Before you start

What this builds on

Part 3 · Prerequisite check

Quick check before you start

1. Over equal steps, outputs go $100, 125, 156.25$ . What fits?

Exponential, ratio $1.25$
Linear, slope $25$
Linear, slope $1.25$
Quadratic

Show the answer

$\frac{125}{100}$ $= \frac{156.25}{125}$ $= 1.25$ : constant ratio.

Correct: Exponential, ratio $1.25$ :
Linear, slope $25$ :
Linear, slope $1.25$ :
Quadratic:

2. What does a calculator regression do?

Finds the constants of a chosen model type that fit all the data closest
Finds a model through exactly two points
Chooses the model type for you
Rounds the data

Show the answer

You choose the type; the regression chooses the constants.

Correct: Finds the constants of a chosen model type that fit all the data closest:
Finds a model through exactly two points:
Chooses the model type for you:
Rounds the data:

Part 4 · See it

See it first

Residual plot for seven weekly follower counts: the linear model's residuals (squares) form a U, positive at both ends and negative in the middle; the exponential model's residuals (circles) stay near 0 with no pattern. — Residuals (actual minus predicted) of two models of the same data. The linear model's residuals form a U; the exponential model's scatter near 0 with no pattern, so it is the more appropriate model. LevlPrep original diagram.

Part 5 · Step by step

How it works, step by step

For each data point, subtract the prediction from the actual valuethe residuals: positive where the model underestimates, negative where it overestimates
Plot the residuals against the inputa residual plot
Small residuals with no patternthe model is appropriate; a curve or trend means it misses the shape of the data

Part 6 · Key ideas

Key ideas

Residual $=$ actual $-$ predicted. Positive: the model underestimates. Negative: it overestimates.
An appropriate model has a residual plot scattered around 0 with no pattern; a U or a trend shows the model has the wrong shape.
Error can also be stated as predicted $-$ actual, or as an absolute error $|\text{predicted} - \text{actual}|$ . Compare competing models by their residuals.

Part 7 · Misconception

A common mistake

The wrong idea: If a model's residuals are both positive and negative, the model fits well.

What actually happens: Mixed signs are expected; what matters is the pattern. Residuals that go positive, negative, positive in order show a curve the model misses.

Part 8 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

Graph

Residual plots for two models of follower counts

A new video channel had

210, 262, 330, 410, 515, 640

and

805

followers at the end of weeks

t = 0

6

. A linear regression

L

and an exponential regression

E

were fit to the data. The plot shows each model's residuals, actual minus predicted: open circles for

L

, filled dots for

E

1. Based on the residual plots, which model is more appropriate for the data, and why?

$L$ , because its residuals are both positive and negative
$E$ , because its residuals are small and show no pattern
$L$ , because its residuals form a smooth curve
$E$ , because its residuals are mostly above $0$

Show the answer

An appropriate model leaves residuals scattered around $0$ with no pattern. The residuals of $L$ form a U: positive, then negative, then positive, which means the data bend away from the line. The residuals of $E$ stay near $0$ with no pattern.

$L$ , because its residuals are both positive and negative: Mixed signs are fine only without a pattern; these form a U.
Correct: $E$ , because its residuals are small and show no pattern: Small, patternless residuals.
$L$ , because its residuals form a smooth curve: A curved pattern in the residuals is the sign of a poor model.
$E$ , because its residuals are mostly above $0$ : The residuals of $E$ are positive and negative.

2. The residual of $L$ at $t = 0$ is positive. What does that mean?

$L$ predicts more followers than the channel really had at $t = 0$
The channel gained followers at $t = 0$
$L$ predicts fewer followers than the channel had at $t = 0$
$L$ is increasing at $t = 0$

Show the answer

Residual $=$ actual $-$ predicted. A positive residual means actual $>$ predicted: the model underestimates there.

$L$ predicts more followers than the channel really had at $t = 0$ : That is a negative residual, an overestimate.
The channel gained followers at $t = 0$ : A residual compares the model to the data; it is not a change in followers.
Correct: $L$ predicts fewer followers than the channel had at $t = 0$ : Actual minus predicted is positive: an underestimate.
$L$ is increasing at $t = 0$ : The sign of a residual says nothing about whether the model is increasing.

Table

Two models of a growing plant

Height of a plant, in centimeters, at week t, with the predictions of a linear model A and an exponential model B
$t$ (weeks)	0	1	2	3
Actual height (cm)	50	62	78	97
Model $A$ (cm)	48	63	78	93
Model $B$ (cm)	50	62.5	78.1	97.7

3. What is the residual of model $A$ at $t = 3$ ?

No calculator. Type the exact value (a fraction, π, e or a root is fine), in cm.

Show the answer

Residual $=$ actual $-$ predicted $= 97 - 93 = 4$ cm. Model $A$ underestimates at $t = 3$ .

Answer: 4 cm

4. Which model fits the data in the table better, and why?

$B$ , because its residuals are all closer to $0$
$A$ , because it matches the data exactly at $t = 2$
$A$ , because its residuals are whole numbers
$B$ , because its predictions are larger

Show the answer

The residuals of $A$ are $2, -1, 0, 4$ ; those of $B$ are $0, -0.5, -0.1, -0.7$ . Every residual of $B$ is at least as close to $0$ , and the largest error of $B$ ( $0.7$ cm) is smaller than the largest of $A$ ( $4$ cm).

Correct: $B$ , because its residuals are all closer to $0$ : Smaller errors at each time.
$A$ , because it matches the data exactly at $t = 2$ : One exact match does not outweigh larger errors elsewhere.
$A$ , because its residuals are whole numbers: Whole numbers are not a measure of fit.
$B$ , because its predictions are larger: Larger predictions are better only where the model underestimates.

5. How is a residual defined in this course?

Predicted value minus actual value
Actual value divided by predicted value
Actual value minus predicted value
The difference between two predictions

Show the answer

A residual is actual minus predicted. The error of a prediction can be stated as predicted minus actual, or as the absolute value of the difference.

Predicted value minus actual value: This is the opposite sign; it is the error of the prediction, not the residual.
Actual value divided by predicted value: A residual is a difference, not a ratio.
Correct: Actual value minus predicted value: Positive when the model underestimates.
The difference between two predictions: A residual compares a prediction with data.

6. A model predicts $142$ visitors, and $150$ visitors come. What is the absolute error of the prediction?

No calculator. Type the exact value (a fraction, π, e or a root is fine), in visitors.

Show the answer

The absolute error is $|\text{predicted} - \text{actual}|$ $= |142 - 150|$ $= 8$ visitors.

Answer: 8 visitors

Part 9 · Summary

Summary

Residuals (actual minus predicted) measure how far a model misses each data point. Choose the model whose residuals are small and patternless, and read the sign of the residuals to say where a model underestimates or overestimates.

Part 10 · Up next

What comes next

Part 11 · Connections

Connections

Question set for this topic