Errors in Measurement of Physical Quantities
Senior 2 · Physics
Errors in Measurement of Physical Quantities
Lesson 1 of 4
20 min
Let's begin
Every reading you take has some doubt in it. That includes a length on a metre rule, a mass on a scale at Kimironko market and a time on a stopwatch. Physics is built on measurements, so you must be able to say where the doubt comes from and how large it is.
A nurse reading a thermometer needs this skill, and so does an engineer checking a bridge over the Nyabarongo or a coffee cooperative weighing a farmer's harvest. In national exams, questions on experiments almost always ask you to name an error and say how to reduce it.
The error in a measurement is the difference between the measured value and the true value. We almost never know the true value exactly, so instead we estimate the largest likely error. That estimate is called the uncertainty, and it is written with a ± sign, for example 46 cm³ ± 1 cm³.
A random error makes readings scatter unpredictably, some above the true value and some below. Reaction time on a stopwatch is the classic cause. Because the scatter goes both ways, repeating the measurement and taking the mean reduces the effect of random errors.
A systematic error pushes every reading in the same direction. Causes include a zero error (the instrument does not read zero when nothing is being measured), a wrongly calibrated scale, a slow clock, or parallax from always looking from the same side. Repeating the reading does not remove a systematic error. You must correct the instrument or correct each reading.
| Random error | Systematic error | |
|---|---|---|
| Effect on readings | Scatter above and below | All shifted the same way |
| Typical cause | Reaction time, reading a wobbling needle | Zero error, bad calibration |
| How to reduce it | Repeat and take the mean | Check the zero, recalibrate, correct the readings |
A spring balance reads 0.2 N with nothing hanging on it. Mutesi hangs a bag of beans on it and it reads 3.4 N. What is the true weight?
The balance adds 0.2 N to every reading, which is a systematic error. True weight = 3.4 N − 0.2 N = 3.2 N.
Gatete times 20 complete swings of a pendulum five times. He gets 31.2 s, 31.6 s, 31.5 s, 31.3 s and 31.4 s.
Timing 20 swings instead of one shares his reaction-time error across 20 swings, so the period is known 20 times better.
Tip
Random: Repeat. Systematic: Set it right
When an exam asks how to reduce an error, first name its type. Averaging cures random errors only. It never cures a zero error or a badly marked scale.
Keza reads the water level in a measuring cylinder with her eye above the surface. Name the error and say how to avoid it.
Solution: This is parallax error. If she always looks from above, it is systematic, because every reading comes out too high. She should put her eye level with the bottom of the curved water surface (the meniscus).
The first 3 mm of a metre rule have worn away. Habimana puts the worn end against one edge of a book and reads 45.0 cm at the other edge. What is the true length?
Solution: The rule's end is really at the 0.3 cm mark, so every reading is 0.3 cm too large. True length = 45.0 cm − 0.3 cm = 44.7 cm. This is a systematic error.
A ball is timed rolling down a ramp five times: 2.4 s, 2.6 s, 2.5 s, 2.3 s, 2.7 s. Find the mean and its uncertainty. What type of error causes the spread?
Solution: Mean = 12.5 s ÷ 5 = 2.5 s. Uncertainty ≈ half the range = (2.7 − 2.3) ÷ 2 = 0.2 s. The spread comes from random error, mostly reaction time.
A stack of 200 sheets of paper is 2.0 cm thick. Find the thickness of one sheet, and explain why this method is better than measuring one sheet.
Solution: 2.0 cm ÷ 200 = 0.010 cm (0.10 mm). One sheet is thinner than the smallest division of a ruler (1 mm), so you could not read it directly. Measuring the stack shares the reading uncertainty across 200 sheets.
Summary
Sources and credits
The Rwandan examples, worked examples and practice items are written by Ganzaa.
