Propagation of errors
A number of measured quantities may be involved in the final calculation of an experiment. Different types of instruments might have been used for taking readings. Then we may have to look at the errors in measuring various quantities, collectively.
The error in the final result depends on
i. The errors in the individual measurements
ii. On the nature of mathematical operations performed to get the final result. So we should know the rules to combine the errors.
The various possibilities of the propagation or combination of errors in different mathematical operations are discussed below:
Let ∆A and ∆B be the absolute errors in the two quantities A and B respectively. Then,
Measured value of A = A ± ∆A
Measured value of B = B ± ∆B
Consider the sum, Z = A + B
The error ∆Z in Z is then given by
Let ΔA and ΔB be the absolute errors in the two quantities, A and B, respectively. Then,
Measured value of A = A ± ΔA
Measured value of B = B ± ΔB
Consider the difference, Z = A – B
The error ΔZ in Z is then given by Z ± ΔZ = (A ± ΔA) – (B ± ΔB)
Let ΔA and ΔB be the absolute errors in the two quantities A, and B, respectively. Consider the product Z = AB
The error ΔZ in Z is given by Z ± ΔZ = (A ± ΔA) (B ± ΔB)
Dividing L.H.S by Z and R.H.S by AB, we get,
As ΔA /A, ΔB / B are both small quantities, their product term ΔA/A . ΔB/B can be neglected. The maximum fractional error in Z is
Let ΔA and ΔB be the absolute errors in the two quantities A and B respectively.
As the terms ΔA/A and
ΔB/B are small, their product term can be neglected.
Consider the nth power of A, Z = An The error ΔZ in Z is given by
We get [(1+x)n ≈1+nx, when x<<1] neglecting remaining terms, Dividing both sides by Z
A physical quantity x is given by x
If the percentage errors of measurement in a, b, c and d are 4%, 2%, 3% and 1% respectively then calculate the percentage error in the calculation of x.
The percentage error in x is given by
The percentage error is x = 17.5%