Robustness Math Example 1

Follow the full solution, then compare it with the other examples linked below.

Example 1

easy
You estimate π≈3.14\pi \approx 3.14 instead of 3.14159…3.14159\ldots in the formula A=πr2A = \pi r^2 with r=5r = 5 cm. Compute the relative error.

Solution

  1. 1
    Exact: A=π(25)≈78.5398A = \pi(25) \approx 78.5398 cm².
  2. 2
    Approximate: A≈3.14×25=78.5A \approx 3.14 \times 25 = 78.5 cm².
  3. 3
    Absolute error: ∣78.5398−78.5∣=0.0398|78.5398 - 78.5| = 0.0398 cm².
  4. 4
    Relative error: 0.039878.5398≈0.051%\frac{0.0398}{78.5398} \approx 0.051\%.

Answer

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

About Robustness

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

Learn more about Robustness →

More Robustness Examples

Example 2 medium

Show that the median is more robust than the mean as a measure of centre when outliers are present.

Example 3 easy

If an input [formula] has a measurement error of [formula], find the range of [formula] and assess w

Example 4 medium

Prove that the statement '[formula] for all [formula]' is robust to replacing [formula] with [formul

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Related Concepts

Sensitivity