Practice Robustness in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick 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?

Showing a random 20 of 50 problems.

Example 1

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 2

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 3

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 4

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 5

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 6

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?

Example 7

medium

f(x)=2x+1

and

x=5\pm 0.1

, find the resulting range of

f(x)

and assess robustness.

Example 8

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 9

easy

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

Example 10

medium

True or false: a least-squares regression line is robust to outliers in

y

Example 11

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 12

easy

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

Example 13

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 14

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 15

hard

Why is the median absolute deviation (MAD) preferred over standard deviation for robust spread?

Example 16

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 17

easy

A poll of

1000

people is generally more robust to a few unusual respondents than a poll of

20

. True or false?

Example 18

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 19

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 20

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.

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Related Concepts

Sensitivity