Error Intervals

OCR

GCSE

Error intervals quantify the range of possible values for a variable given a specific degree of accuracy, distinguishing rigorously between rounding and truncation. Candidates must calculate precise lower and upper bounds and express these ranges using strict inequality notation, typically in the form $a \leq x < b$ for continuous data. The topic extends to error propagation, requiring the determination of maximum and minimum possible values for calculated quantities derived from imprecise measurements. Mastery of this concept is critical for analyzing uncertainty in mathematical modeling and engineering contexts.

Objectives

Exam Tips

Pitfalls

Key Terms

Mark Points

What You Need to Demonstrate

Key skills and knowledge for this topic

Award 1 mark for identifying the correct lower bound (e.g., 2.45 for 2.5 rounded to 1 d.p.)
Award 1 mark for identifying the correct upper bound (e.g., 2.55 for 2.5 rounded to 1 d.p.)
Award 1 mark for correct inequality notation, specifically using ≤ for the lower bound and < for the upper bound
Credit responses that correctly derive error intervals from truncated data (e.g., 8.3 truncated to 1 d.p. implies 8.3 ≤ x < 8.4)

Example Examiner Feedback

Real feedback patterns examiners use when marking

"Check your upper bound inequality sign — why must it be strictly less than (<) rather than less than or equal to (≤)?"
"You treated this as a rounding problem, but the question specifies 'truncated'. How does this change the upper bound?"
"Good identification of the bounds. Now, to find the maximum value of the fraction, which bound should you use for the denominator?"
"Remember that for significant figures (e.g., 200 to 1 s.f.), the interval is determined by the place value of the significant digit."

Marking Points

Key points examiners look for in your answers

Award 1 mark for identifying the correct lower bound (e.g., 2.45 for 2.5 rounded to 1 d.p.)
Award 1 mark for identifying the correct upper bound (e.g., 2.55 for 2.5 rounded to 1 d.p.)
Award 1 mark for correct inequality notation, specifically using ≤ for the lower bound and < for the upper bound
Credit responses that correctly derive error intervals from truncated data (e.g., 8.3 truncated to 1 d.p. implies 8.3 ≤ x < 8.4)

Examiner Tips

Expert advice for maximising your marks

💡For truncation questions, the upper bound is always the 'face value' plus one unit of the last specified digit (e.g., 3.4 truncated → UB is 3.5)
💡When calculating bounds for derived quantities (e.g., speed = distance/time), write out the error interval for each variable explicitly before substituting
💡Ensure your inequality signs match the nature of the data; continuous data requires x < UB, while discrete data might allow x ≤ UB if adjusted

Common Mistakes

Pitfalls to avoid in your exam answers

Using '≤' for the upper bound (e.g., x ≤ 5.5), failing to recognise that 5.5 would round up to 6
Confusing truncation with rounding, incorrectly stating the upper bound of '4.7 truncated' is 4.75 instead of 4.8
Incorrectly determining the interval width for significant figures (e.g., for 300 to 1 s.f., using bounds 295-305 instead of 250-350)

Study Guide Available

Comprehensive revision notes & examples

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Key Terminology

Essential terms to know

Upper and Lower Bounds (Limits of Accuracy)

Inequality Notation for Continuous vs Discrete Data

Differentiation between Rounding and Truncation

Error Propagation in Compound Calculations

Likely Command Words

How questions on this topic are typically asked

Write down

State

Work out

Calculate

Find

Practical Links

Related required practicals

•{"code":"Engineering","title":"Tolerance Intervals","relevance":"Determining if machined parts will fit within specified safety limits"}
•{"code":"Kinematics","title":"Speed Calculations","relevance":"Calculating maximum possible speed given rounded distance and time measurements"}

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