Propagation of errors

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:

i. Error in the sum of two quantities

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

ii. Error in the difference of two quantities

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)

(iii) Error in the product of two quantities

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

(iv) Error in the division or quotient of two quantities

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.

(v) Error in the power of a quantity

Consider the n^th 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

(i) Example Problem for Error in the sum of two quantities

Example 1.5

Two resistances R1  = (100 ± 3) Ω, R2  = (150 ± 2) Ω, are connected in series. What is their equivalent resistance?

Solution

Equivalent resistance R = ?

Equivalent resistance R = R1 + R2

(ii) Example Problem for Error in the difference of two quantities

Example 1.6

The temperatures of two bodies measured by a thermometer are t1 = (20 + 0.5)°C, t2 = (50 ± 0.5)°C. Calculate the temperature difference and the error therein.

Solution

(iii) Example Problem for Error in the product of two quantities

Example 1.7

The length and breadth of a rectangle are (5.7 ± 0.1) cm and (3.4 ± 0.2) cm respectively. Calculate the area of the rectangle with error limits.

Solution

(iv) Example Problem for Error in the division or quotient of two quantities

Example 1.8

The voltage across a wire is (100 ± 5)V and the current passing through it is (10±0.2) A. Find the resistance of the wire.

Solution

(v) Example Problem for Error in the power of a quantity

Example 1.9

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.

Solution

The percentage error in x is given by

The percentage error is x = 17.5%