Errors and Uncertainty Analysis
Systematic versus random errors, absolute error, fractional and percentage uncertainty, error combination rules (addition, subtraction, multiplication, division, powers), and significant figure rules.
Marking a lesson complete represents finishing your study of the topic; it does not denote tested mastery.
Learning Objectives
- Distinguish between systematic errors (constant shift) and random uncertainties (statistical spread).
- Calculate absolute, fractional, and percentage uncertainties for experimental values.
- Apply error propagation rules to compute uncertainties in combined formulas.
Core Theory & Concepts
Every physical measurement carries uncertainty due to instrument limitations and experimental variations. Scientific communication demands reporting not just a number, but its uncertainty and appropriate significant figures.
Systematic vs Random Errors
Systematic errors shift measurements predictably in one direction (e.g. zero error, miscalibration, parallax). Repeating readings does NOT remove them. Random errors fluctuate unpredictably both above and below the true value; averaging multiple readings minimizes them.
Uncertainty Definitions
If measurement is x ± Δx: Absolute uncertainty = Δx; Fractional uncertainty = Δx / x; Percentage uncertainty = (Δx / x) × 100%.
Propagation in Addition & Subtraction
When adding or subtracting quantities Z = X ± Y: Absolute uncertainties always accumulate: ΔZ = ΔX + ΔY. Errors NEVER cancel out.
Propagation in Products, Quotients & Powers
For Z = X^a × Y^b / W^c: Fractional uncertainties combine: (ΔZ / Z) = a(ΔX / X) + b(ΔY / Y) + c(ΔW / W). Powers have the largest multiplier effect.
Common Misconceptions
When subtracting two numbers (Z = X - Y), you subtract their uncertainties: ΔZ = ΔX - ΔY.
Syllabus Topic Outline
- 1Systematic errors: calibration, zero error, end correction
- 2Random errors: parallax, thermal fluctuations, timing reaction
- 3Absolute uncertainty (Δx), fractional uncertainty (Δx/x), percentage uncertainty
- 4Propagation in addition and subtraction: ΔZ = ΔX + ΔY
- 5Propagation in multiplication, division, and powers: ΔZ/Z = a(ΔX/X) + b(ΔY/Y)
- 6Significant figures in experimental reporting