ASSESSMENT OF TOTAL UNCERTAINTY IN THE FINAL RESULT

To assess the total uncertainty or error, it is necessary to evaluate the likely uncertainty in all the factors included in that calculation. The total uncertainty in the final result can be found as follows:

1- For Addition and subtraction:

Absolute uncertainties are added. For example, the distance “X” found by the difference between two separate position measurements.

X₁ = 10.5 ± 0.1 cm   and X₂ = 26.8 ± 0.1 cm. The difference “X” between them is recorded as

X = X₂-X₁ = (26.8 ± 0.1) – (10.5 ± 0.1)

So X = 16.3 ± 0.2 cm.

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 ± 0.05 A

The %age uncertainty for V =0.1v/5.2v x 100/100 = about 2% 

The %age uncertainty for I = 0.05A/0.84A x 100/100 = about 6% 

Hence Total uncertainty in the value of resistance ‘R’ when ‘V’ is divided by ‘I’ is 8% The result is thus given as R = 5.2v/0.84A = 6.19A^-1 = 6.19 ohms with a %age. uncertainty of 8% because %age uncertainty for V = 2% and for I = 6% so

Total Uncertainty = º2%+6%=8%

Hence R = 6.2 ± 0.5 Ohms (Here result is rounded off to two significant figures)

3- For Power Factor

If absolute uncertainty of a measurement is known and that measurement occurs in power in a formula. Then total percentage uncertainty is calculated by multiplying the power and absolute uncertainty i.e. multiply the %age uncertainty by that power.

For Example:

For the calculation of the volume of a sphere, we use the formula V= 4/3pr³. Percentage uncertainty in volume = 3 x 5age uncertainty in radius ‘r’. When uncertainty is multiplied by power factor, then it increases the precision demand of measurement. If the radius of a small sphere is measured as 2.25 cm by 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

%age uncertainty in r = 0.01cm/2.25cm x 100/100 = 0.4%

Total percentage uncertainty in V = 3 x 0.4 = 1.2%

Thus volume V= 4/3pr3= 4/3 x 3.14 x (2.25)³ = 47.689 cm³ with 1.2% uncertainty

Hence the result should be recorded as V = 47.7 0.6 cm³.