Propagation(Combination) of Errors | Error in Sum, Difference, Multiplication, Division and Raised to Power of Quantities

In experimental physics and chemistry, measurements are never perfectly precise. Small errors arise due to limitations in instruments and human observations. Understanding how these errors propagate when performing mathematical operations is essential for accurate calculations.

The following detailed explanation covers different types of error propagation, including sum, difference, product, quotient, and powers of measured quantities.

Types of Error Propagation

1. Error in Sum of Quantities

Formula: If x = a + b, then the absolute error in x is: \begin{array}{l} \Delta x = \Delta a + \Delta b \end{array}

• Let Δa be the absolute error in measurement of a
• Let Δb be the absolute error in measurement of b
• The total absolute error in x is the sum of absolute errors of a and b
• Percentage Error: \begin{array}{l} \%\text{ error in } x = \left( \frac{\Delta a + \Delta b}{a + b} \right) \times 100 \end{array}

2. Error in Difference of Quantities

Formula: If x = a – b, then the absolute error in x is: \begin{array}{l} \Delta x = \Delta a + \Delta b \end{array}

• Similar to addition, the absolute errors always add up.
• Percentage Error: \begin{array}{l} \%\text{ error in } x = \left( \frac{\Delta a + \Delta b}{a – b} \right) \times 100 \end{array}

3. Error in Product of Quantities

Formula: If x = a × b, then the maximum fractional error in x is: \begin{array}{l} \frac{\Delta x}{x} = \frac{\Delta a}{a} + \frac{\Delta b}{b} \end{array}

• Percentage Error: \begin{array}{l} \%\text{ error in } x = \%\text{ error in } a + \%\text{ error in } b \end{array}

4. Error in Division of Quantities

Formula: If x=a/b, then the maximum fractional error in x is: \begin{array}{l} \frac{\Delta x}{x} = \frac{\Delta a}{a} + \frac{\Delta b}{b} \end{array}

• Percentage Error: \begin{array}{l} \%\text{ error in } x = \%\text{ error in } a + \%\text{ error in } b \end{array}

5. Error in Quantity Raised to a Power

Formula: If x=a^mbⁿ, then the maximum fractional error in x is: \begin{array}{l} \frac{\Delta x}{x} = m \frac{\Delta a}{a} + n \frac{\Delta b}{b} \end{array}

• Percentage Error: \begin{array}{l} \%\text{ error in } x = m \times (\%\text{ error in } a) + n \times (\%\text{ error in } b) \end{array}
• Note: The variable with the highest power contributes the most to error.

FAQs

Q1. Why do we always add absolute errors in sums and differences?

Since errors can be positive or negative, taking their sum ensures that the total uncertainty accounts for the worst possible deviation.

Q2. How does power affect error propagation?

When a quantity is raised to a power, the percentage error gets multiplied by that power, making careful measurements crucial.

Q3. Why is error in division treated the same way as multiplication?

Fractional errors always add up whether quantities are multiplied or divided.

Multiple-Choice Questions (MCQs)

1. If two quantities A and B are added, their total absolute error is:

• (a) ΔA−ΔB
• (b) ΔA+ΔB ✅
• (c) ΔA/A+ΔB/B
• (d) None of these

2. The percentage error in x=a³b² is:

• (a) 3% error in a+2% error in b ✅
• (b) 2% error in a+3% error in b
• (c) Δa/3+Δb/2
• (d) None of these

Conceptual Questions

1. Explain why fractional errors are added in multiplication and division.

When quantities are multiplied or divided, small changes in one quantity proportionally affect the result, leading to the sum of fractional errors.

2. Why should we measure the variable with the highest power most accurately?

The percentage error contribution increases with power, so errors in such variables amplify the overall error.

Do You Know?

• The error in a sum or difference is independent of the actual values but depends only on individual errors.
• When using logarithms, error calculations become simpler, as fractional errors translate directly into additions.

Worksheet

1. Calculate the percentage error in x, given a = 2.5 ± 0.1 and b = 1.5 ± 0.05, if:
   • (a) x=a+b
   • (b) x=a−b
   • (c) x=a×b
   • (d) x=a/b
2. A length L=5.0 ± 0.2 cm and width W=3.0±0.1 cm. Find the percentage error in area A=L×W.

Important Points for Quick Revision

• Absolute errors always add up in sum and difference.
• Fractional errors add up in multiplication and division.
• Power raises error contribution by the factor of the exponent.
• The greater the power of a variable, the more precisely it must be measured.

Solved Problems

Problem 1: If all measurements in an experiment are performed up to the same number of times, then a maximum error occurs due to which measurement?

Solution:

The maximum error occurs due to the measurement of the quantity which appears with maximum power in the formula. If all the quantities in the formula have the same powers, then a maximum error occurs due to the measurement of the quantity whose magnitude is least.

Problem 2: If the length of the pencil is given by (4.16 ± 0.01) cm. What does it mean?

Solution:

It means that the true value of the length of the pencil is unlikely to be less than 4.15 cm or greater than 4.17 cm. 

Problem 3: Two resistances R₁=(100±5) ohm and R₂=(200±10) ohm are connected in series. Find the equivalent resistance of the series combination.

Solution:

Since, it is known that,

Equivalent resistance=R= R₁+R₂

Given that, the resistance is:

R₁ = (100 ± 5)

R₂ = (200 ±10)

Therefore,

R = (100 ± 5) + (200 ± 10)

   = (300 ± 15) ohm

Problem 4: A capacitor of capacitance C = (2.0 ± 0.1) µF is charged to a voltage V = (20 ± 0.2) V. What will be the charge Q on the capacitor?

Solution: 

Q = CV 

   = 2.0×10^-6 × 20 C 

   = 4.0×10^-5 Coulomb.

Proportional error in C = (ΔC/C) 

                                    = (0.1/2)

Percentage error in C = (0.1/2) ×100 

                                  =5 %

Proportional error in V = (ΔV/V) 

                                    = (0.2/20)

Percentage error in V = (0.2/20)×100

                                  =1%

Charge on capacitor,

(ΔQ/Q) = (ΔC/C) + (ΔV/V)

Percentage error in Q = 5%+1%

                                  = 6%

Charge = 4.0×10^-5 ± 6% Coulomb

            = (4.0±0.24)×10^-5 Coulomb

Problem 5: The centripetal force acting on a body of mass 50 kg moving in a circle of radius 4 m with a uniform speed of 10 m/s is calculated using the equation F = mv²/r. If the accuracies of measurement of m, v, and r are 0.5 kg, 0.02 m/s, and 0.01 m respectively, determine the percentage error in the force.

Solution: 

It is known that,

(ΔF/F) = (Δm/m) + 2(Δv/v) + (Δr/r)

(Δm/m) = (0.5/50) 

             = 0.01

(Δv/v) = (0.02/10)

          = 0.002

(Δr/r) = (0.01/4)

         = 0.0025

So, (ΔF/F) = 0.01 + 2(0.002) + (0.0025)

                = 0.0165

Thus, Percentage error in force = (0.0165) × 100% 

                                                = 1.65 %

Problem 6: The resistance R = V/I where V = (200 ± 5) V and I = (20 ± 0.2) A. Find the percentage error in R.

Solution: 

Proportional error in V = (ΔV/V) 

                                     = (5/200)

Percentage error in V = (5/200)×100% 

                                   = 2.5%

Proportional error in I = (ΔI/I) 

                                   = (0.2/20)

Percentage error in I  = (0.2/20) ×100% 

                                 = 1%

So, Percentage error in R = 2.5%+1%

                                        = 3.5%

Problem 7: The mass and the length of one side of a cube are measured and its density is calculated. If the percentage errors in the measurement of mass and length are 1% and 2% respectively, then what is the percentage error in the density?

Solution: 

If the mass of the cube is m and the length of its one side is l, then its density,

d = m/l³

So, (Δd/d) = (Δm/m) + 3(Δl/l)

Thus, Percentage error in density = (1+3×2)% 

                                                    = 7%

Test Your Knowledge (Quiz)

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