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Question 5: What is uncertainty in measurement? Explain the propagation of uncertainty in addition, subtraction, multiplication and division.


Uncertainty in Measurement

“It is the magnitude of doubt in the measurement.”
Uncertainty reports errors from all possible sources (i-e, both systematic and random), therefore, it is the most appropriate mean of expressing accuracy of the result.
There is uncertainty in all measurements, therefore, every measurement needs to be written in the form; measurement = best estimate ± uncertainty.

Types of uncertainties

There are two types of uncertainties.

Absolute uncertainty

If the length of a line is 3.4 cm ± 0.1 cm, then 3.4 cm is called the reported value and 0.1 cm is called the absolute uncertainty. Similarly, the range of true values is from 3.3 cm to 3.5 cm.
“Thus we define the absolute uncertainty as the number which, when combined with a reported value, gives the range of true values. It is denoted by Δ and has the same unit as the quantity.”

Relative or percent uncertainty

Relative uncertainty is the ratio of absolute uncertainty to the reported value of the measurement. For example, in the above case, the relative uncertainty = (0.1/3.4). The percent uncertainty is the percentage of the relative uncertainty. In the above example, percentage uncertainty is = (0.1/3.4) × 100. It is denoted by ϵ and has no units.

(Note: In a problem, when you are asked to find the uncertainty, you have to find the absolute uncertainty.)

Indicating uncertainty in the final result

When we do mathematical operations (i-e, addition, subtraction, division, multiplication, squaring and other) including uncertainties in the quantities, the uncertainty part follow some rules in these operations. Using these rules to find the correct answer is called indicating (i-e, showing) uncertainties in the final results.

Indicating uncertainty in sum or difference

The rule for determining uncertainty in addition and subtraction is that the absolute uncertainties are added both in the case of addition or subtraction. If the measured values of two quantities A and B are A ± ΔA and B ± ΔB to give Z = Z ± ΔZ, then,
Addition: Z ± ΔZ = A ± ΔA + B ± ΔB ⇒ Z ± ΔZ = A + B ± ΔA ±ΔB = A + B ± (ΔA + ΔB) Therefore, the absolute uncertainties (ΔA and ΔB, respectively) are added.
Subtraction: Z ± ΔZ = A ± ΔA – (B ± ΔB) = A – B ±ΔA – (±ΔB) = A – B ± ΔA ∓ΔB which implies that Z ± ΔZ = A – B + (±ΔA ±ΔB) = A – B ± (ΔA + ΔB).

Therefore, the absolute uncertainties are added in the case of subtraction, too.

Indicating uncertainty in multiplication and division:

The rule for determining uncertainty in multiplication and division is that the percentage uncertainties are added.

Let the measured values of two quantities A and B are A ± ΔA and B ± ΔB, respectively.

Multiplication or Product

Let quantities A and B are multiplied to give quantity Z, then Z ± ΔZ = (A ± ΔA) (B ± ΔB).
(1) Convert the uncertainties in the values of the quantities to be multiplied to percentage uncertainties so that ΔA% and ΔB% are the respective percentage uncertainties. (How? ΔA% = (ΔA/A) × 100 and ΔB% = (ΔB/B) × 100).
(2) Calculate the product as follow; Z ± ΔZ = AB ± (ΔA + ΔB)%.
(3) Convert back to fractional uncertainties. [How? Fractional uncertainty = ΔZ = {(ΔA + ΔB)%/100} × AB.]

Division or Quotient

Let the measured values of quantities A and B are A ± ΔA and B ± ΔB, respectively and we have to find the quotient, Z = A/B. Then, Z ± ΔZ = (A ± ΔA)/(B ± ΔB).
(1) Convert the fractional uncertainties to percentage uncertainties so that Z ± ΔZ = (A ± ΔA%)/(B ± ΔB%).
(2) Divide the confident values and add the percentage uncertainties. Z + ΔZ = (A/B) ± (ΔA% + ΔB%).
(3) Convert back to fractional uncertainties. [How? ΔZ = {(ΔA%+ΔB%)/100 × (A/B)}.


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