To assess the total uncertainty or error, it is necessary to evaluate the likely uncertainties in all the factors involved in that calculation. The maximum possible uncertainty or error in the final result can be found as follows. The proofs of these rules are given in appendix 2.
1. For addition and subtraction
Absolute uncertainties are added: for example, the distance x determined by the difference between two separate position measurements
2. For multiplication and division
Percentage uncertainties are added. For example the maximum possible uncertainty in the value of resistance R of a conductor determined from the measurements of potential difference V and resulting current flow I by using R = V/ I is found as follows:
V = 5.2 ± 0.1 V
I = 0.84 ± 0.05A
Hence total uncertainty in the value of resistance R when V is divided by I is 8%. The result is thus quoted as
Uncertainty of 8%
That is R = 6.2 ± 0.5 ohms
The result is rounded off to two significant digits because both V and R have two significant figures and uncertainty, being an estimate only, is recorded only, is recorded by one significant figure.
3. For power factor
Multiply the percentage uncertainty by that power. For example, in the calculation of the volume of a sphere using
As uncertainty is multiplied by power factor, it increases the precision demand of measurement. If the radius of a small sphere is measured as 2.25 cm by a vernier calipers with least count 0.01 cm, then
The radius r is recorded as
R= 2.25 ± 0.01 cm
Absolute uncertainty = least count = ± 0.01 cm
Total percentage uncertainty in V = 3 x 0.4 = 1.2 %
Thus the result should be recorded as
4. For uncertainty in the average value of many measurements.
(i) Find the average value of measured values.
(ii) Find deviation of each measured value from the average value.
(iii) The mean deviation is the uncertainty in the average value
For example, the six readings of the micrometer screw gauge to measure the diameter of a wire in mm are
1.20, 1.22, 1.23, 1.19, 1.22, 1.21.
= 1.21 mm
The deviation of the readings, which are the difference without regards to the sign, between each reading and average value are 0.01, 0.01, 0.02,0.02, 0.01, 0,
= 0.01 mm
Thus, likely uncertainty in the mean diameter 1.21 mm is 0.01 mm recorded as 1.21 ± 0.01 mm.