Robustness Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Robustness.

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

Concept Recap

The property of a result, algorithm, or model remaining valid or approximately correct even when its assumptions are slightly violated.

Is this answer fragile, or does it survive small errors and changes?

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: Robust results are more reliable; fragile results need more scrutiny.

Common stuck point: Robustness is relative to a specific type of perturbation — a method can be robust to outliers but fragile to model misspecification, or vice versa.

Sense of Study hint: Change one input slightly (add 1, round a value, introduce a small error) and recompute. If the answer changes drastically, the method is fragile and you may need a more robust approach.

Worked Examples

Example 1

easy
You estimate \pi \approx 3.14 instead of 3.14159\ldots in the formula A = \pi r^2 with r = 5 cm. Compute the relative error.

Solution

  1. 1
    Exact: A = \pi(25) \approx 78.5398 cm².
  2. 2
    Approximate: A \approx 3.14 \times 25 = 78.5 cm².
  3. 3
    Absolute error: |78.5398 - 78.5| = 0.0398 cm².
  4. 4
    Relative error: \frac{0.0398}{78.5398} \approx 0.051\%.

Answer

\text{Relative error} \approx 0.05\%
Robustness measures how sensitive a result is to approximations in its inputs. A tiny relative error in \pi produces an equally tiny error in the computed area — the formula is robust to this approximation.

Example 2

medium
Show that the median is more robust than the mean as a measure of centre when outliers are present. Use the data set \{2, 3, 4, 5, 100\}.

Practice Problems

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

Example 1

easy
If an input x = 10 has a measurement error of \pm 1, find the range of f(x) = 2x+3 and assess whether f is robust to this error.

Example 2

medium
Prove that the statement 'n^2 > n for all n > 1' is robust to replacing > with \ge: does 'n^2 \ge n for all n \ge 1' still hold?
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Related Concepts

Sensitivity

Background Knowledge

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

sensitivity