Rules for determining uncertainty in results of arithmetic calculations

To calculate the uncertainty, below process should be used.

• Add a lowest amount of uncertainty in the original numbers. Example uncertainty for 3.2 will be ± 0.1 and for 3.22 will be ± 0.01. Calculate these in percentage also.
• After the calculations, the uncertainties get multiplied/divided/added/subtracted.
• Round off the decimal place in the uncertainty to get the final uncertainty result.

Example, for a rectangle, if length l = 16.2 cm and breadth b = 10.1 cm

Then, take l = 16.2 ± 0.1 cm or 16.2 cm ± 0.6% and breadth = 10.1 ± 0.1 cm or 10.1 cm ± 1%.

On Multiplication, area = length x breadth = 163.62 cm² ± 1.6% or 163.62 ± 2.6 cm².

Therefore after rounding off, area = 164 ± 3 cm².

Hence 3 cm² is the uncertainty or the error in estimation.

Rules

1. For a set experimental data of ‘n’ significant figures, the result will be valid to ‘n’ significant figures or less (only in case of subtraction).

Example 12.9 – 7.06 = 5.84 or 5.8 (rounding off to lowest number of decimal places of original number).

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

Example, accuracy for two numbers 1.02 and 9.89 is ±0.01. But relative errors will be:

For 1.02, (± 0.01/1.02) x 100% = ± 1%

For 9.89, (± 0.01/9.89) x 100% = ± 0.1%

Hence, the relative error depends upon number itself.

3. Intermediate results in multi-step computation should be calculated to one more significant figure in every measurement than the number of digits in the least precise measurement.

Example:1/9.58 = 0.1044

Now, 1/0.104 = 9.56 and 1/0.1044 = 9.58

Hence, taking one extra digit gives more precise results and reduces rounding off errors.