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Combination of Errors

If we do an experiment involving several measurements, we must know how the errors in all the measurements combine. For example, density is the ratio of the mass to the volume of the substance.

If we have errors in the measurement of mass and of the sizes or dimensions, we must know what the error will be in the density of the substance. To make such estimates, we should learn how errors combine in various mathematical operations. For this, we use the following procedure:
  1. Error of a Sum or a Difference
Suppose two physical quantities A and B have measured values A ± Δ A, B ± Δ B respectively where Δ A and Δ B are their absolute errors. We wish to find the error Δ Z in the sum

Z = A + B.
We have by addition,
Z ± Δ Z = (A ± Δ A) + (B ± Δ B).
The maximum possible error in Z
Δ Z = Δ A + Δ B
For the difference Z = A – B, we have
Z ± Δ Z = (A ± Δ A) (B ± Δ B)
= (A – B) ± Δ A ± Δ B
or, ± Δ Z = ± Δ A ± Δ B
The maximum value of the error Δ Z is again
Δ A + Δ B.

When two quantities are added or subtracted, the absolute error in the final result is the sum of the absolute errors in the individual quantities.
  1. Error of a Product or a Quotient
Suppose Z = AB and the measured values of A and B are A ± Δ A and B ± Δ B. Then
Z ± Δ Z = (A ± Δ A) (B ± Δ B)
= AB ± B Δ A ± A Δ B ± Δ A Δ B.
Dividing LHS by Z and RHS by AB we have,
1±(Δ Z/Z) = 1 ± (Δ A/A) ± (Δ B/B) ± (Δ A/A) ( Δ B/B).
Since Δ A and Δ B are small, we shall ignore their product.
Hence the maximum relative error
Δ Z/ Z = (Δ A/A) + (Δ B/B).
You can easily verify that this is true for division also.

When two quantities are multiplied or divided, the relative error in the result is the sum of the relative errors in the multipliers.
  1. Error in Case of a Measured Quantity Raised to a Power
Suppose Z = A2, Then,

Δ Z/Z = (Δ A/A) + (Δ A/A) = 2 (Δ A/A).
Hence, the relative error in A2 is two times the error in A.
In general, if Z = Ap Bq/Cr
Δ Z/Z = p (Δ A/A) + q (Δ B/B) + r (Δ C/C).

The relative error in a physical quantity raised to the power k is the k times the relative error in the individual quantity.

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