Significant figures and uncertainty

No measurement in physics is ever perfectly exact: every instrument has a sensitivity, i.e. the smallest variation it can still distinguish. A ruler graduated in millimetres can say nothing about hundredths of a millimetre; a kitchen scale cannot tell a gram from half a gram. The sensitivity of the instrument therefore sets a natural limit on the precision with which we can know a quantity.

When we report a measurement, the digits we write implicitly convey how much we trust the result. Writing one extra digit means claiming to have actually measured it: for this reason the number of digits reported is never arbitrary, but is itself a piece of information about the uncertainty.

Principle — Rule of significant figures

To count the significant figures of a number obtained as the result of a measurement:

1. Ignore all leading zeros (those between the decimal point and the first non-zero digit).
2. All remaining digits are significant, including zeros in the middle and trailing ones.

Counting significant figures

Measurement	Significant figures	Comment
$33\,240$	5	all non-zero or intermediate digits
$0{,}001080$	4	leading zeros do not count
$3030{,}32$	6	internal zeros count
$0{,}50000$	5	trailing zeros after the decimal point count
$0{,}0505$	3	the internal zero counts, the leading ones do not
$50{,}00$	4	trailing zeros count

Tip

“If you write $2{,}3250$ with the trailing zero, that zero must have been measured. Otherwise it should not be written.”

A concrete example

Suppose you measure the height of a door. By eye: $2{,}3\;\text{m}$ (2 significant figures, you are sure of the metre and the decimetres). With a mason’s tape measure: $2{,}325\;\text{m}$ (4 figures: sure down to the millimetre). With a laser: $2{,}3250\;\text{m}$ (5 figures: you have really measured even the tenth of a millimetre). More significant figures $\Leftrightarrow$ more precise measurement.

Collegamenti

Argomenti: Grandezze fisiche e misura Concetti: Cifre significative e incertezza Competenze: Uso delle cifre significative

Esercizi collegati: Esercizio svolto — Area di un rettangolo misurato · Esercizio svolto — Cane o gatto: il sondaggio della classe · Problema — Ranking di incertezze relative