Question

Could someone please help me find the answer for this?

Answer 1

Measurement is the foundation of all experimental science and technology. The result of every measurement by any measuring instrument contains some uncertainty. This uncertainty is called error. Every calculated quantity which is based on measured values, also has an error. We shall distinguish between two terms: accuracy and precision. The accuracy of a measurement is a measure of how close the measured value is to the true value of the quantity. Precision tells us to what resolution or limit the quantity is measured. The accuracy in measurement may depend on several factors, including the limit or the resolution of the measuring instrument. For example, suppose the true value of a certain length is near 3.678 cm. In one experiment, using a measuring instrument of resolution 0.1 cm, the measured value is found to be 3.5 cm, while in another experiment using a measuring device of greater resolution, say 0.01 cm, the length is determined to be 3.38 cm. The first measurement has more accuracy (because it is closer to the true value) but less precision (its resolution is only 0.1 cm), while the second measurement is less accurate but more precise.
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 cm2 + 1.6% = 163.62 + 2.6 cm2
This leads us to quote the final result as l b = 164 + 3 cm2
Here 3 cm2 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.