Measurements as an interval

A transparent way of reporting an uncertain measurement is to write it as an interval. Instead of stating a single value, the extremes within which the true value is certainly contained are stated:

$b = [\,6{,}9\;;\;7{,}2\,]\;\text{cm}$

means exactly this — the real value of the length $b$ lies somewhere between $6{,}9\;\text{cm}$ and $7{,}2\;\text{cm}$, and we cannot say more.

Equivalently one writes $b = 7{,}0(1)\;\text{cm}$: here the digit in brackets indicates the first uncertain digit, i.e. the one affected by error. The two notations say exactly the same thing: the first makes the interval explicit, the second is more compact and highlights where the “safe” part of the measurement ends and the doubtful part begins.

The advantage of interval notation is that it lends itself to simple arithmetic: when several uncertain measurements are combined, the uncertainty can be propagated by reasoning directly on the extremes. A typical case is the estimate of the area of an irregular figure.

The grid-square method

To estimate the area of an irregular figure — for example a stain, a leaf, or a hand-drawn circle — a grid of squares of known side is overlaid (say $0{,}5\;\text{cm}$, i.e. $0{,}25\;\text{cm}^2$ per square) and one counts:

• the squares entirely inside the figure: say 39;
• those at least partly inside (including the previous ones): say 65.

Then the minimum and maximum areas are $A_\text{min} = 39\cdot 0{,}25 = 9{,}75\;\text{cm}^2 \qquad A_\text{max} = 65\cdot 0{,}25 = 16{,}25\;\text{cm}^2,$ so the area is known to lie in the interval $[\,9{,}75\;;\;16{,}25\,]\;\text{cm}^2$. From here, if the figure is roughly circular, the radius can also be derived from $r = \sqrt{A/\pi}$, giving $[\,1{,}76\;;\;2{,}27\,]\;\text{cm}$.

The grid-square method. A grid of known side is overlaid on the figure: by counting the squares entirely contained and those merely touched, one obtains the minimum and maximum area, i.e. an interval for the area.

Collegamenti

Argomenti: Grandezze fisiche e misura Concetti: Cifre significative e incertezza Metodi: Metodo dei quadretti

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