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

addition-and-subtraction

For your information

sensitive-thermometersThese are not decoration pieces of glass but are the earliest known exquisite and sensitive thermometers, built by the academia del cimento (1657-1667)in Florence. They contained slcohol, some times coloured red for easier reading.

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

multiplication-and-division

Hence total uncertainty in the value of resistance R when V is divided by I is 8%. The result is thus quoted as

value-of-resistance

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

Interesting information

Some-specific-temperaturesSome specific temperatures

Multiply the percentage uncertainty by that power. For example, in the calculation of the volume of a sphere using

power-factor%age uncertainty in V = 3 x % age uncertainty in radius r.

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

equation

 

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

power-factor-equation

Thus the result should be recorded as

power-factor-result-equation

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.

uncertainty-average-value

= 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,

deviations-equation

= 0.01 mm

Thus, likely uncertainty in the mean diameter 1.21 mm is 0.01 mm recorded as 1.21 ± 0.01 mm.

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