Lesson 6 of 10 · 11 min
Uncertainty in calculated results
NCERT §1.3.3
Riya's card says g = 9.81 m s⁻². A judge asks: plus or minus how much? Every reading had a small uncertainty, and the question is how those uncertainties travel into g.
The lesson in notes
In short
A reading from an instrument carries an uncertainty of about its least count: a length of 16.2 cm read on a metre scale is 16.2 ± 0.1 cm.
The relative (percentage) error is the uncertainty divided by the value: ±0.1 cm on 16.2 cm is about ±0.6%, and ±0.1 cm on 10.1 cm is about ±1%.
When measured values are multiplied or divided, their percentage errors add: the area l × b = 163.62 cm² carries ±1.6%, about ±2.6 cm², so it is quoted as 164 ± 3 cm².
The same absolute uncertainty is a larger relative error on a smaller value: ±0.01 g is ±1% of 1.02 g but only ±0.1% of 9.89 g.
Uncertainties in a sum or difference follow decimal places, not significant figures, so a difference can lose figures: 12.9 g − 7.06 g is 5.8 g, not 5.84 g.
If the data are good to n significant figures, a result obtained by combining them is valid to about n significant figures, fewer if data are subtracted.