Errors in measurement · Ages 13–14
Absolute, relative and percentage errors
Greatest absolute error = unit ÷ 2; relative error = that ÷ value.
How to do it
Method
Greatest absolute error = unit ÷ 2; relative error = that ÷ value
25 cm: 0.5 ÷ 25 = 0.02 = 2%
Another way
From the upper limit
25.5 − 25 = 0.5
In full
The greatest absolute error is the most the measured value and the true value can differ by: half the unit. Measured as 25 cm to the nearest 1 cm, the greatest absolute error = 1 ÷ 2 = 0.5 cm, which is also upper limit − measured value = 25.5 − 25. Relative error = greatest absolute error ÷ measured value = 0.5 ÷ 25 = 0.02 (or 150); percentage error = relative error × 100% = 2%. A relative error has no unit, so it compares how accurate different measurements are: the smaller, the more accurate. A 25 cm book and a 400 m track, each to the nearest unit: the track's relative error, 0.5 ÷ 400, is far smaller, so it is the more accurate.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
A pen is 5 cm long, to the nearest 0.1 cm. The greatest absolute error is cm, the relative error , the percentage error %.
Fill in each blank
Show the answer and working
Answer: 0.05, 0.01, 1
0.1 ÷ 2 = 0.05 cm; 0.05 ÷ 5 = 0.01; 0.01 × 100% = 1%.
Example 2
A plant is 4 cm tall, to the nearest 1 cm. What is the percentage error of this measurement?
Show the answer and working
Answer: 12.5%
Relative error 0.5 ÷ 4 = 0.125, percentage error = 0.125 × 100% = 12.5%.
Example 3
Of these four measurements, which is the most accurate (the smallest relative error)?
- A. 7 mm (to the nearest 1 mm)
- B. 35 mm (to the nearest 10 mm)
- C. 300 mm (to the nearest 1 mm)
- D. 170 mm (to the nearest 10 mm)
Choose the right answer
Show the answer and working
Answer: C. 300 mm (to the nearest 1 mm)
Work out each relative error = half the unit ÷ the value: the smallest is 1600. A small absolute error alone doesn't make a measurement the most accurate.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.A towel's length is measured as 116 cm to the nearest 1 cm. What is the relative error of this measurement, as a fraction?
Answer in simplest form
Answer:
2.A parcel is weighed as 3.7 kg to the nearest 0.1 kg. What is the greatest absolute error of this measurement, in kg?
Answer:
3.Of these four measurements, which is the most accurate (the smallest relative error)?
- A. 152 mm (to the nearest 1 mm)
- B. 27 mm (to the nearest 1 mm)
- C. 34 mm (to the nearest 10 mm)
- D. 475 mm (to the nearest 100 mm)
Choose the right answer
Answer:
4.Measurement A: 43 cm, to the nearest 0.1 cm; measurement B: 351 cm, to the nearest 1 cm. A is more accurate than B.
True or false?
Answer:
5.A strip of wood is measured to the nearest 1 cm with a percentage error of 0.5%. What length was measured, in cm?
Answer:
All the answers
- 1. 1232
- 2. 0.05 kg
- 3. A. 152 mm (to the nearest 1 mm)
- 4. True
- 5. 100 cm
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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