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: Robustness is whether a result stays approximately correct when its assumptions are slightly violated.

Common stuck point: The procedure for robustness is the easy part; the trap is calling a method robust because it's accurate on clean data. Asking "If the assumptions are slightly violated, does the result stay approximately correct?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: If the assumptions are slightly violated, does the result stay approximately correct?

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.

Answer

\text{Relative error} \approx 0.05\%

First step

Exact:

A = \pi(25) \approx 78.5398

cm².

Full solution

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

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\}

Example 3

medium

Given

\{4,5,6,7,8\}

, compute the mean. Then replace

8

with

80

and recompute. By how much did the mean shift?

Example 4

medium

Algorithm A solves a problem in

0.1

s on clean input but

30

s when input is noisy. Algorithm B takes

0.5

s on clean input and

0.6

s on noisy. Which is more robust?

Example 5

medium

Round

\pi=3.14159\dots

3.14

and compute the relative error.

Example 6

hard

The dataset

\{1,2,3,4,5,6,7,8,9,10\}

has mean

5.5

and median

5.5

. Replace the value

10

with

1000

. Compute the new mean and median.

Example 7

hard

In computer arithmetic, computing

(a+b)+c

a+(b+c)

can give different results due to rounding. What property does this break, and what makes the algorithm 'non-robust'?

Example 8

hard

Compare the breakdown points of the mean and median: roughly what fraction of contaminated data values can each tolerate before producing arbitrary results?

Example 9

challenge

The condition number of a matrix measures how much output error can amplify input error in solving

Ax=b

. If the condition number is

10^6

, by what factor can a relative input error be amplified in the worst case?

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?

Example 3

easy

The data set 4, 5, 6, 7, 8 has a mean of 6. Replace 8 with 100. Which is more robust to this outlier, the mean or the median?

Example 4

easy

A bridge is designed to hold exactly the expected load with no margin. Is this design robust to small overloads?

Example 5

easy

A sorting algorithm gives correct results even if a few input values are duplicated or out of expected range. What property does this describe?

Example 6

easy

Two thermometers: one reads 20.0 C reliably within +/-0.5 C in any weather; another is perfectly accurate only at sea level. For field use, which is more robust?

Example 7

easy

A formula assumes friction is exactly zero. Real surfaces have small friction. A robust result would still be approximately correct. True or false?

Example 8

easy

A poll of 5 people predicts an election; a poll of 5000 predicts the same election. Which prediction is more robust to a few unusual respondents?

Example 9

easy

Estimating a sum, you round 4.9 to 5 and 3.1 to 3. The estimate 8 is close to 8.0. Is rounding here a robust shortcut?

Example 10

easy

A recipe still tastes good whether you use 1.9 or 2.1 cups of flour. Is the recipe robust to small measurement errors?

Example 11

medium

To estimate a typical income, dataset incomes (in thousands) are 30, 35, 40, 45, 5000. Compute mean and median, and state which is the robust estimate of 'typical'.

Example 12

medium

Method A solves a problem with error proportional to input noise (error = noise). Method B has error proportional to (noise)^2. For small noise = 0.01, which method is more robust, and by what factor smaller is its error?

Example 13

medium

A control system stays stable if a parameter k is in [2, 8]. The current design uses k = 5. By how much can k drift in either direction before instability, and is k = 5 a robust choice?

Example 14

medium

A least-squares fit and a least-absolute-deviations fit both model points, but one point is a gross outlier. Which fit is more robust to that outlier and why?

Example 15

medium

A formula divides by (x - 3). Tested only at x = 10 it works. Why might this not be robust, and at what input does it fail?

Example 16

medium

Two route-planning algorithms: A is optimal on perfectly known travel times; B is within 10% of optimal but unaffected by traffic surprises. In real, uncertain traffic, which is the robust choice?

Example 17

challenge

A quadratic root computation x = (-b + sqrt(b^2 - 4ac))/(2a) suffers catastrophic cancellation when b > 0 and 4ac is tiny. Give a robust algebraically-equivalent formula for that root and explain why it is robust.

Example 18

challenge

A model's prediction p(x) = 1/(1 + (x - 2)^2) is evaluated near x = 2. Show whether the prediction is robust (insensitive) to a small input change there, using the derivative.

Example 19

challenge

A voting rule must elect the same winner even if up to 1 of 5 ballots is corrupted. Candidate X has 4 votes, Y has 1. Show the rule's outcome is robust to one corrupted ballot, and find the smallest X-lead that guarantees robustness.

Example 20

medium

A measurement process gives 10.0 whether the room is 18 C or 25 C, but drifts badly outside 15-30 C. Within what range is the process robust, and is a lab kept at 22 C safe?

Example 21

medium

An estimate uses the approximation (1+x) approx 1 + x for small x. For x = 0.02 the true (1+x)^1 is exact, but for (1+x)^10 the approximation 1+10x ignores higher terms. Is the linear approximation robust at x = 0.02 for the 10th power (true value approx 1.219)?

Example 22

medium

A clustering algorithm gives the same groups whether you start from random seed A or seed B. What robustness property does this demonstrate?

Example 23

easy

Which is more robust to a single huge outlier: the mean or the median?

Example 24

easy

Estimating

\pi\approx 3.14

vs the true value

3.14159\dots

: is the small error a robust approximation for computing the area of a moderate circle?

Example 25

easy

Which is more robust to outliers, the standard deviation or the interquartile range (IQR)?

Example 26

easy

A weather-prediction model is accurate to within

\pm 2

°C in

90\%

of cases but is off by

10

°C on rare extreme days. Is it robust to extremes?

Example 27

medium

f(x)=2x+1

and

x=5\pm 0.1

, find the resulting range of

f(x)

and assess robustness.

Example 28

medium

In statistics, a trimmed mean drops the top and bottom

10\%

of values before averaging. Why is this more robust than the ordinary mean?

Example 29

medium

A formula assumes inputs are positive integers. Is using it for

x=0

a robust extension?

Example 30

hard

A function

f(x)=1/(x-3)

is asked to predict outputs near

x=3

. Is it robust to small input errors there?

Example 31

hard

A theorem assumes the input function is continuous. The proof goes through if 'continuous' is weakened to 'piecewise continuous'. Is the theorem robust under this weakening?

Example 32

hard

A recipe calls for

2

cups of flour. Cooks A measure

1.95

and

2.05

. Both cakes turn out fine. Is the recipe robust to this variation?

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Related Concepts

Sensitivity

Background Knowledge

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

sensitivity