Errors in Measurement of Physical Quantities
Senior 2 · Physics
Errors in Measurement of Physical Quantities
Lesson 2 of 4
25 min
Let's begin
Two shopkeepers weigh the same standard 1.000 kg mass. The first scale gives 0.98 kg, 1.02 kg, 0.99 kg and 1.01 kg. The second gives 1.21 kg, 1.20 kg, 1.21 kg and 1.20 kg. Which scale is better? The answer depends on two different ideas, accuracy and precision, and on how many digits a reading deserves.
The first scale is accurate, because its readings cluster around the true 1.000 kg, but it is less precise. The second scale is precise but not accurate: its readings agree closely, but all of them are about 0.2 kg too high. A precise but inaccurate instrument usually has a systematic error, such as a zero error.
An instrument with smaller divisions can measure more precisely.
| Instrument | Smallest division | A sensible reading |
|---|---|---|
| Tape measure | 1 cm | 146 cm |
| Metre rule | 1 mm (0.1 cm) | 36.7 cm |
| Vernier caliper | 0.1 mm (0.01 cm) | 2.46 cm |
| Micrometer screw gauge | 0.01 mm | 5.34 mm |
Write only the digits your instrument can give you. A metre rule cannot give 36.71 cm, because it cannot resolve a hundredth of a centimetre.
The significant figures (s.f.) in a measurement are all the digits you are sure of, plus the last digit, which is estimated. The rules:
Multiplying or dividing: give the answer to the fewest significant figures in the data.
Worked example. A classroom floor is 7.45 m long (3 s.f.) and 6.2 m wide (2 s.f.). Area = 7.45 × 6.2 = 46.19 m², which is written as 46 m² (2 s.f.).
Adding or subtracting: give the answer to the fewest decimal places in the data.
Worked example. A 12.52 kg bag of rice and a 3.1 kg bag of sugar together: 12.52 + 3.1 = 15.62 kg, which is written as 15.6 kg (1 decimal place).
Look at the first digit you are going to drop. If it is 5 or more, add 1 to the digit before it. If it is less than 5, leave that digit alone.
Round only the final answer. If you round at every step, the small rounding errors add up.
Tip
Zeros in front never count. Zeros at the back count only when there is a decimal point
For × and ÷, count significant figures. For + and −, count decimal places. Exam answers with far too many digits (such as 46.19 m² from 6.2 m) often lose a mark.
How many significant figures are there in (a) 0.0305, (b) 1.200, (c) 5.0 × 10⁴?
Solution: (a) 3, because the leading zeros do not count and the zero between 3 and 5 does. (b) 4, because the trailing zeros after the decimal point count. (c) 2.
Round 0.008736 to 2 s.f., and 25.451 to 1 decimal place.
Solution: 0.008736 becomes 0.0087, because the dropped digit 3 is less than 5. 25.451 becomes 25.5, because the dropped digit 5 rounds up.
A garden plot measures 12.5 m by 8.0 m. Give its area to the correct number of significant figures.
Solution: 12.5 × 8.0 = 100 m². The data have at least 2 s.f., so write 1.0 × 10² m². Standard form shows that exactly 2 figures are significant.
Add 250.4 g and 12.36 g.
Solution: 262.76 g, which rounds to 262.8 g, because 250.4 has only 1 decimal place.
A standard 50.0 g mass is weighed four times on a balance: 52.1 g, 52.0 g, 52.1 g, 52.0 g. Describe the readings.
Solution: They are precise, because they agree to within 0.1 g, but not accurate, because they are all about 2 g too high. This points to a systematic error, probably a zero error.
Summary
Sources and credits
The Rwandan examples, worked examples and practice items are written by Ganzaa.
