Class 11 > Unit # 01: Physical Quantities & Measurements > Uncertainty Calculation with Eaxmples

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1. Core Concepts & Definitions

What is Uncertainty?

• Definition: Uncertainty in a measurement refers to the range of values within which the true value of a physical quantity is expected to lie.
• Origin: It reflects the inherent limitations of measuring instruments and the variability in the measurement process.
• Key Distinctions:
  • Certainty: Confirmed / Exact.
  • Uncertainty: Not confirmed / A range where the true value exists (e.g., stating a distance is "2 or 3 km").

Why Does Uncertainty Occur?

1. Instrument Limitations: Measuring devices have a minimum readable limit (Least Count).
2. Observational & Process Limits: Measurements often fall between two calibration marks on a scale (e.g., between 16.3 cm and 16.4 cm).

2. Types of Uncertainty

Type	Definition	Mathematical Representation	Example
Absolute Uncertainty	The explicit physical range of variation in a measured value.	Value ± Δx	25.0 cm ± 0.5 cm
Fractional Uncertainty	The ratio of absolute uncertainty to the measured (true) value.	Absolute Uncertainty / Measured Value	0.5 / 142 = 0.0035
Percentage Uncertainty	The fractional uncertainty expressed as a percentage.	(Absolute Uncertainty / Measured Value) × 100	0.0035 × 100 = 0.35%

3. Calculating Uncertainty in a Single Measurement

Rule for Single Instrument Readings

For a single measurement, the Absolute Uncertainty is equal to half of the instrument's Least Count (LC):

Absolute Uncertainty = Least Count / 2

Worked Example: Measuring Weight

• Observed Value: 142 N
• Least Count (LC): 1 N
• Absolute Uncertainty: 1 N / 2 = ± 0.5 N

Final Reading: 142 ± 0.5 N

• Fractional Uncertainty: 0.5 / 142 ≈ 0.0035
• Percentage Uncertainty: 0.0035 × 100 = 0.35%

4. Propagation of Uncertainty in Calculations

Case 1: Addition & Subtraction (Sum or Difference)

Golden Rule: When adding or subtracting quantities, always ADD their absolute uncertainties.

Worked Example: Extension of a Copper Wire

• Initial Length (L₁) at 30°C: 18.2 ± 0.04 cm
• Final Length (L₂) at 60°C: 19.7 ± 0.02 cm

1. Calculate Extension (ΔL):
   ΔL = L₂ - L₁ = 19.7 - 18.2 = 1.5 cm
2. Combine Absolute Uncertainties:
   Total Absolute Uncertainty = 0.04 + 0.02 = ±0.06 cm
3. Final Stated Result: 1.5 ± 0.06 cm

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Case 2: Multiplication & Division (Products or Ratios)

Golden Rule: When multiplying or dividing quantities, convert absolute uncertainties to percentage uncertainties first, and ADD the percentage uncertainties.

Worked Example: Pressure Calculation (P = F / A)

• Weight (F): 8.0 ± 0.3 N
• Base Area (A): 3.5 ± 0.2 m²

Step-by-Step Solution:

1. Find Percentage Uncertainty of each variable:
   • Weight % Uncertainty: (0.3 / 8.0) × 100 = 3.75%
   • Area % Uncertainty: (0.2 / 3.5) × 100 = 5.71%
2. Total Percentage Uncertainty in Pressure:
   Total % Uncertainty = 3.75% + 5.71% = 9.46%
3. Calculate Measured Value of Pressure:
   P = 8.0 / 3.5 = 2.3 Pa
4. Convert Total % Uncertainty back to Absolute Uncertainty:
   Absolute Uncertainty = (9.46 / 100) × 2.3 = 0.22 Pa
5. Final Result:
   • In Absolute Form: 2.3 ± 0.22 Pa
   • In Percentage Form: 2.3 Pa ± 9.46%

5. Quick Reference Summary Table

Mathematical Operation	Combination Rule for Uncertainties
Addition (A + B)	Add Absolute Uncertainties (ΔA + ΔB)
Subtraction (A - B)	Add Absolute Uncertainties (ΔA + ΔB)
Multiplication (A × B)	Add Percentage Uncertainties (%A + %B)
Division (A / B)	Add Percentage Uncertainties (%A + %B)