22
Relative error
Textbook: pp. 119–120
GoalKnow the definition of the relative error and compare the accuracy of different measurements using it.
New words
relative error · nisbiy xatolikabsolute error · absolut xatolikpercent · foizmeasurement accuracy · o‘lchash aniqligi
Explanation
To compare different approximations of the same quantity the absolute error is enough, but to compare different quantities it is not. For example, an error of 2 km in 500 km versus an error of 1 mm in 12 cm — which is more accurate? So the relative error is introduced: if a is the approximate value of x, the relative error is the ratio of the absolute error to |a|, that is |x - a|/|a|. It is usually expressed in percent. The measurement with the smaller relative error is more accurate. If the absolute error is not known exactly but is at most h, the relative error does not exceed h/|a|.
Worked examples
Length 20 ± 0.5 cm: the relative error is at most 0.5/20 = 0.025 = 2.5 %.
Distance 400 ± 2 km: 2/400 = 0.005 = 0.5 %. So the second measurement (0.5 %) is more accurate than the first (2.5 %).
Class activity
“The more accurate measurement”: groups compare two measurement results (a length, a mass) by relative error and prove which is more accurate.
Practice
1
For a = 150 and absolute error 3, find the relative error in percent.
2
2
Which is more accurate: (5 ± 0.05) kg or (200 ± 1) kg?
The second: 1/200 = 0.5 %, the first 0.05/5 = 1 %.
3
The approximate value is 80 and the absolute error is 4. What is the relative error in percent?
5
4
Why is the relative error convenient for comparing different quantities?
Because it compares the error with the quantity itself and does not depend on units.