Rules for Determining Uncertainty in Arithmetic Results

Rules for Determining Uncertainty in Arithmetic Results

Product of Measured Quantities

When you multiply measured values, the percentage uncertainties add.

Example: A rectangular sheet has length $l = 16.2 \pm 0.1$ cm and breadth $b = 10.1 \pm 0.1$ cm.

The length has three significant figures. The uncertainty of 0.1 cm in 16.2 cm gives a percentage uncertainty of:

Similarly, the breadth has percentage uncertainty:

The area $l \times b = 163.62$ cm². The total percentage uncertainty is $0.6\% + 1\% = 1.6\%$.

The final result should be reported as:

Subtraction Can Reduce Significant Figures

When data are subtracted, the number of significant figures can decrease dramatically.

Example: 12.9 g − 7.06 g. Both numbers are specified to three significant figures. The arithmetic difference is 5.84 g. But the uncertainties combine differently in subtraction — the rule is about decimal places, not significant figures. 12.9 g is correct to one decimal place, so the result should be:

Not 5.84 g. The subtraction has reduced the precision.

Relative Error Depends on the Number Itself

The relative error of a value specified to $n$ significant figures depends not only on $n$ but also on the magnitude of the number.

Example: A mass of 1.02 g measured to $\pm 0.01$ g has a relative error of:

A mass of 9.89 g measured to $\pm 0.01$ g has a relative error of:

$\frac{0.01}{9.89} \times 100\% = \pm 0.1\%$ …