Underfitting (Intuition) Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Underfitting (Intuition).

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Underfitting occurs when a model is too simple to capture the true pattern in the data, performing poorly on both training data and new data.

The model misses important structure—it's not learning enough.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Underfitting: too simple. The model can't capture real patterns.

Common stuck point: Underfitting is not always obvious from training error alone — it shows up most clearly when the model cannot follow known patterns even in training data.

Worked Examples

Example 1

easy
A scatter plot shows a clear U-shaped relationship between x and y. A linear model is fitted and gets R^2 = 0.05. Explain what underfitting is and how to fix this model.

Solution

  1. 1
    Underfitting: the model is too simple to capture the U-shaped pattern — a line cannot represent a curve
  2. 2
    R^2 = 0.05: only 5% of variation explained — the model is nearly useless
  3. 3
    Diagnosis: both training error and test error are high (model fails everywhere)
  4. 4
    Fix: use a quadratic model \hat{y} = ax^2 + bx + c which can capture U-shapes; check that R^2 improves significantly

Answer

Underfitting: linear model can't capture U-shape. Fix: use quadratic model to match the true curved relationship.
Underfitting (high bias) occurs when the model's functional form is too restrictive for the true relationship. Unlike overfitting (which needs simplification), underfitting requires a more flexible model class to capture the real pattern in the data.

Example 2

medium
Compare two regression models on the same data: Model 1 (linear): training R^2=0.40, test R^2=0.38. Model 2 (cubic): training R^2=0.95, test R^2=0.35. Diagnose each model.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
A student uses \hat{y} = 50 (the mean) to predict all exam scores. Training R^2 = 0. Explain why this represents maximum underfitting.

Example 2

hard
A linear model for time-series stock prices has high residuals for all time periods. Identify this as underfitting and explain why stock prices might inherently have an upper bound on R^2 regardless of model complexity.
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Related Concepts

Model Fit (Intuition)

Background Knowledge

These ideas may be useful before you work through the harder examples.

model fit intuition