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Error Propagation

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All physical quantities that are measured have some uncertainty (error) associated to them. Error propagation (or Propagation of Uncertainty) refers to the fact that whenever you use measurements (with some error) to calculate new numerical values, the resulting values have more error associated to them. The error “propagates” – that is the resulting error (uncertainty) increases.

What this means is that suppose you were to add two quantities that have been measured. Then the resulting sum would also have some error (uncertainty) associated with it which would be greater than the individual errors.

In this article we will understand how to calculate the associated error when quantities are added, subtracted, divided, multiplied, etc.

Throughout this article if X is any quantity we shall use ∆X to denote the error associated to it.

Error Propagation Rule for Addition/Subtraction:

If two quantities are being added or subtracted then we simply add up the absolute error to find the resulting uncertainty. Notice that the absolute errors are added irrespective of whether the quantities are being added or subtracted.

So if Z=X+Y or if Z=X-Y then,

∆Z=|∆X|+|∆Y|

Example: If the height of a father is 1.7m with uncertainty 0.01m and the height of the mother is 1.6m with error 0.02m then what is the total height? Also find the error associated with it.

Solution: The total height is 1.7m+1.6m = 3.3m

To calculate the error in the total height we use the formula,

∆Z=|∆X|+|∆Y|= 0.01+0.02 = 0.03m which is the final uncertainty of the sum obtained.

Error Propagation Rule for Multiplication:

If two quantities are being multiplied then we find the error in both cases by using the below formula:

So if Z=X*Y then,

∆Z=X*|∆Y|+Y*|∆X|

Error Propagation Rule for Division:

If two quantities are being divided then we find the error in both cases by using the below formula:

So if Z=X/Y then,

∆Z={(|∆X|/X)+(|∆Y|/Y)}*Z

that is, we add the relative errors and then multiply by Z

Error Propagation Rule for Raising to Power:

If a quantity is being raised to a power then we find the error in both cases by using the below formula:

So if Z= Xn then,

∆Z= n*|∆X|*Xn-1

Example: If the value of a quantity is X=3.5 metres with error of 0.2 metres then what is the error in the quantity X3.

Solution: We use the formula ∆Z= n*|∆X|*Xn-1

So ∆Z= 3*0.2*(3.5)2 = 7.35 metres is the required error

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