Rules for Determining the Uncertainty in the Results of Arithmatic Calculations

The rules for determining the uncertainty or error in the number/measured quantity in arithmetic operations can be understood from the following examples. 

(1) If the length and breadth of a thin rectangular sheet are measured, using a metre scale as 16.2 cm and, 10.1 cm respectively, there are three significant figures in each measurement. It means that the length l may be written as

 l = 16.2 ± 0.1 cm 

 = 16.2 cm ± 0.6 %. 

Similarly, the breadth b may be written as 

b = 10.1 ± 0.1 cm 

 = 10.1 cm ± 1 % 

Then, the error of the product of two (or more) experimental values, using the combination of errors rule, will be 

 $l_b$ = 163.62 $cm^2$ + 1.6% 

= (163.62 + 2.6) $cm^2$ 

This leads us to quote the final result as 

$l_b$ = (164 + 3) $cm^2$ 

Here 3 $cm^2$ is the uncertainty or error in the estimation of area of rectangular sheet. 

(2) If a set of experimental data is specified to n significant figures, a result obtained by combining the data will also be valid to n significant figures. 

However, if data are subtracted, the number of significant figures can be reduced.

For example, 12.9 g – 7.06 g, both specified to three significant figures, cannot properly be evaluated as 5.84 g but only as 5.8 g, as uncertainties in subtraction or addition combine in a different fashion (smallest number of decimal places rather than the number of significant figures in any of the number added or subtracted). 

(3) The relative error of a value of number specified to significant figures depends not only on n but also on the number itself. 

For example, the accuracy in measurement of mass 1.02 g is ± 0.01 g whereas another measurement 9.89 g is also accurate to ± 0.01 g. The relative error in 1.02 g is 

= (± 0.01/1.02) × 100 % 

= ± 1% 

Similarly, the relative error in 9.89 g is

 = (± 0.01/9.89) × 100 %

 = ± 0.1 % 

Finally, remember that intermediate results in a multi-step computation should be calculated to one more significant figure in every measurement than the number of digits in the least precise measurement. 

These should be justified by the data and then the arithmetic operations may be carried out; otherwise rounding errors can build up. For example, the reciprocal of 9.58, calculated (after rounding off) to the same number of significant figures (three) is 0.104, but the reciprocal of 0.104 calculated to three significant figures is 9.62. However, if we had written 1/9.58 = 0.1044 and then taken the reciprocal to three significant figures, we would have retrieved the original value of 9.58. 

This example justifies the idea to retain one more extra digit (than the number of digits in the least precise measurement) in intermediate steps of the complex multi-step calculations in order to avoid additional errors in the process of rounding off the numbers.