Prerequisites
Before studying errors, you should understand:
- SI Units — Measurement standards
- Significant Figures — Precision representation
Why Study Errors?
No measurement is perfect! Understanding errors helps us:
- Know the reliability of measurements
- Express results with appropriate precision
- Improve experimental techniques
Accuracy vs Precision
These terms are often confused but mean different things:
| Accuracy | Precision |
|---|---|
| How close a measurement is to the true value | How close measurements are to each other |
| Related to systematic errors | Related to random errors |
| Can be improved by calibration | Can be improved by better instruments |
Imagine throwing darts:
High Accuracy, Low Precision: Darts scattered around the bullseye
Low Accuracy, High Precision: Darts clustered together but away from bullseye
High Accuracy, High Precision: Darts clustered tightly at the bullseye (IDEAL!)
Low Accuracy, Low Precision: Darts scattered everywhere (WORST!)
Types of Errors
1. Systematic Errors (Constant Errors)
Errors that occur in the same direction (always positive or always negative) with constant magnitude.
Characteristics:
- Same for all measurements
- Can be corrected if cause is known
- Affect accuracy
Common Sources:
Instrumental Errors:
- Zero error in vernier calipers
- Incorrect calibration
- Worn-out meter stick
Environmental Errors:
- Temperature affecting length measurements
- Air pressure affecting gas experiments
- Humidity affecting electrical measurements
Personal Errors:
- Parallax error in reading scales
- Reaction time in stopwatch measurements
- Always reading on one side of pointer
Theoretical Errors:
- Using simplified formulas
- Ignoring air resistance
- Approximations in calculations
A vernier caliper shows reading of 0.03 cm when jaws are closed (should be 0.00 cm).
This is a positive zero error of +0.03 cm.
Correction: Subtract 0.03 cm from all readings.
If measured value = 5.47 cm, True value = 5.47 - 0.03 = 5.44 cm
Interactive Demo: Visualize Error Analysis
Understand how errors propagate through calculations and measurements.
2. Random Errors (Accidental Errors)
Errors that vary in magnitude and sign from measurement to measurement.
Characteristics:
- Unpredictable
- Can be positive or negative
- Cannot be eliminated, only reduced
- Affect precision
Common Sources:
- Fluctuations in experimental conditions
- Limitations of measuring instruments
- Observer’s inability to read exactly
Reduction Method:
$$\boxed{\text{Take multiple measurements and find the MEAN}}$$The mean of many measurements is closer to true value than any single measurement.
5 measurements of time period: 2.1 s, 2.3 s, 2.2 s, 2.4 s, 2.0 s
Mean: $\bar{T} = \frac{2.1 + 2.3 + 2.2 + 2.4 + 2.0}{5} = \frac{11.0}{5} = 2.2$ s
The mean (2.2 s) is more reliable than any single reading.
3. Gross Errors (Blunders)
Large errors due to human mistakes.
Examples:
- Reading scale incorrectly (reading 5 instead of 6)
- Recording wrong value
- Using wrong formula
- Calculation mistakes
Solution: Careful observation and double-checking!
Absolute Error, Relative Error, and Percentage Error
Absolute Error (Δa)
The magnitude of difference between true value and measured value.
$$\boxed{\Delta a = |a_{\text{measured}} - a_{\text{true}}|}$$If true value is unknown, use mean of measurements:
For n measurements a₁, a₂, …, aₙ:
Mean value:
$$\boxed{\bar{a} = \frac{a_1 + a_2 + ... + a_n}{n}}$$Absolute errors:
$$\Delta a_1 = |a_1 - \bar{a}|, \quad \Delta a_2 = |a_2 - \bar{a}|, \quad ...$$Mean absolute error:
$$\boxed{\Delta a_{mean} = \frac{\Delta a_1 + \Delta a_2 + ... + \Delta a_n}{n}}$$Final result:
$$\boxed{a = \bar{a} \pm \Delta a_{mean}}$$Relative Error (δa)
The ratio of absolute error to the true value (or mean value).
$$\boxed{\delta a = \frac{\Delta a}{a}}$$Dimensionless quantity.
Percentage Error
Relative error expressed as percentage.
$$\boxed{\text{Percentage error} = \frac{\Delta a}{a} \times 100\%}$$Problem: Five measurements of length: 5.2 cm, 5.4 cm, 5.1 cm, 5.3 cm, 5.5 cm
Step 1: Mean value
$$\bar{l} = \frac{5.2 + 5.4 + 5.1 + 5.3 + 5.5}{5} = \frac{26.5}{5} = 5.3 \text{ cm}$$Step 2: Absolute errors
- Δl₁ = |5.2 - 5.3| = 0.1 cm
- Δl₂ = |5.4 - 5.3| = 0.1 cm
- Δl₃ = |5.1 - 5.3| = 0.2 cm
- Δl₄ = |5.3 - 5.3| = 0.0 cm
- Δl₅ = |5.5 - 5.3| = 0.2 cm
Step 3: Mean absolute error
$$\Delta l_{mean} = \frac{0.1 + 0.1 + 0.2 + 0.0 + 0.2}{5} = \frac{0.6}{5} = 0.12 \text{ cm}$$Step 4: Relative error
$$\delta l = \frac{0.12}{5.3} = 0.0226$$Step 5: Percentage error
$$\text{Percentage error} = 0.0226 \times 100\% = 2.26\%$$Final Result:
$$\boxed{l = (5.3 \pm 0.12) \text{ cm} \text{ or } 5.3 \text{ cm} \pm 2.26\%}$$Error Propagation (Combination of Errors)
When we calculate a quantity using measured values, errors propagate through the calculation.
Rule 1: Addition and Subtraction
$$\boxed{\text{If } Z = A + B \text{ or } Z = A - B}$$ $$\boxed{\text{then } \Delta Z = \Delta A + \Delta B}$$Maximum absolute error = Sum of absolute errors
- A = 50 ± 2 cm
- B = 30 ± 1 cm
- Z = A + B = ?
Answer:
- Z = 50 + 30 = 80 cm
- ΔZ = 2 + 1 = 3 cm
- Z = 80 ± 3 cm
Rule 2: Multiplication and Division
$$\boxed{\text{If } Z = A \times B \text{ or } Z = \frac{A}{B}}$$ $$\boxed{\text{then } \frac{\Delta Z}{Z} = \frac{\Delta A}{A} + \frac{\Delta B}{B}}$$Maximum relative error = Sum of relative errors
- A = 10 ± 0.2 cm
- B = 5 ± 0.1 cm
- Z = A × B = ?
Step 1: Z = 10 × 5 = 50 cm²
Step 2: Relative errors
- δA = 0.2/10 = 0.02
- δB = 0.1/5 = 0.02
Step 3: δZ = 0.02 + 0.02 = 0.04
Step 4: ΔZ = 0.04 × 50 = 2 cm²
Answer: Z = 50 ± 2 cm²
Rule 3: Power (Most Important for JEE)
$$\boxed{\text{If } Z = A^n}$$ $$\boxed{\text{then } \frac{\Delta Z}{Z} = n \cdot \frac{\Delta A}{A}}$$Relative error gets multiplied by the power!
Volume $V = \frac{4}{3}\pi r^3$
If radius measured: r = 5 ± 0.1 cm, find error in volume.
Solution:
Since V ∝ r³:
$$\frac{\Delta V}{V} = 3 \cdot \frac{\Delta r}{r} = 3 \times \frac{0.1}{5} = 3 \times 0.02 = 0.06$$V = (4/3)π(5)³ = 523.6 cm³
ΔV = 0.06 × 523.6 = 31.4 cm³
Answer: V = 524 ± 31 cm³ (or 6% error)
Rule 4: General Function
For $Z = \frac{A^a B^b}{C^c D^d}$:
$$\boxed{\frac{\Delta Z}{Z} = a\frac{\Delta A}{A} + b\frac{\Delta B}{B} + c\frac{\Delta C}{C} + d\frac{\Delta D}{D}}$$Density $\rho = \frac{m}{V}$ where V = l³
If m = 100 ± 1 g and l = 10 ± 0.1 cm:
Solution:
$$\rho = \frac{m}{l^3}$$ $$\frac{\Delta\rho}{\rho} = \frac{\Delta m}{m} + 3\frac{\Delta l}{l}$$ $$= \frac{1}{100} + 3 \times \frac{0.1}{10} = 0.01 + 0.03 = 0.04$$ρ = 100/1000 = 0.1 g/cm³
Δρ = 0.04 × 0.1 = 0.004 g/cm³
Answer: ρ = (0.100 ± 0.004) g/cm³ (or 4% error)
Summary of Error Propagation Rules
| Operation | Formula | Error Rule |
|---|---|---|
| Sum | Z = A + B | ΔZ = ΔA + ΔB |
| Difference | Z = A - B | ΔZ = ΔA + ΔB |
| Product | Z = A × B | ΔZ/Z = ΔA/A + ΔB/B |
| Quotient | Z = A / B | ΔZ/Z = ΔA/A + ΔB/B |
| Power | Z = Aⁿ | ΔZ/Z = n(ΔA/A) |
| General | Z = AᵃBᵇ/CᶜDᵈ | ΔZ/Z = a(ΔA/A) + b(ΔB/B) + c(ΔC/C) + d(ΔD/D) |
Addition/Subtraction: Add ABSOLUTE errors
Multiplication/Division: Add RELATIVE errors
Powers: MULTIPLY relative error by power
Least Count and Instrumental Error
Least Count (LC) = Smallest measurement possible with an instrument
| Instrument | Least Count |
|---|---|
| Meter scale | 1 mm = 0.1 cm |
| Vernier calipers | 0.01 cm = 0.1 mm |
| Screw gauge | 0.001 cm = 0.01 mm |
| Stopwatch (analog) | 0.1 s |
| Stopwatch (digital) | 0.01 s |
Instrumental error = ± (Least Count)
If least count is not given, take half the smallest division as the error.
Example: If ruler has mm markings, error = ± 0.5 mm
Common Mistakes to Avoid
Wrong: If A = 10 ± 2% and B = 5 ± 3%, then A×B has error 2% + 3% = 5%
Correct: Convert to relative errors (0.02 + 0.03 = 0.05), so 5% is correct!
Note: This “mistake” actually gives right answer for percentages, but conceptually we add relative errors, not percentage errors (they just happen to add the same way).
Wrong: If V = πr²h and r has 2% error, h has 3% error, then V has (2×2 + 3) = 7% error
Correct: V = πr²h → ΔV/V = 2(Δr/r) + (Δh/h) = 2(2%) + 3% = 7% ✓
(This happens to be correct!)
Wrong: If Z = A - B, then ΔZ = ΔA - ΔB
Correct: ΔZ = ΔA + ΔB (always ADD absolute errors!)
Errors always add, never subtract.
Practice Problems
Level 1: JEE Main Basics
If L = 10.0 ± 0.1 cm and B = 5.0 ± 0.1 cm, find area A = L × B with error.
Solution:
A = 10.0 × 5.0 = 50.0 cm²
$$\frac{\Delta A}{A} = \frac{\Delta L}{L} + \frac{\Delta B}{B} = \frac{0.1}{10.0} + \frac{0.1}{5.0}$$ $$= 0.01 + 0.02 = 0.03$$ΔA = 0.03 × 50 = 1.5 cm²
Answer: A = 50.0 ± 1.5 cm² (or 50 ± 2 cm² with proper sig figs)
Resistance R = V/I. If V = 100 ± 5 V and I = 10 ± 0.2 A, find R with error.
Solution:
R = 100/10 = 10 Ω
$$\frac{\Delta R}{R} = \frac{\Delta V}{V} + \frac{\Delta I}{I} = \frac{5}{100} + \frac{0.2}{10}$$ $$= 0.05 + 0.02 = 0.07$$ΔR = 0.07 × 10 = 0.7 Ω
Answer: R = 10.0 ± 0.7 Ω (or 7% error)
Level 2: JEE Main/Advanced
The period T of a simple pendulum is given by $T = 2\pi\sqrt{l/g}$. If error in measuring l is 2%, what is the error in T?
Solution:
$$T = 2\pi l^{1/2} g^{-1/2}$$ $$\frac{\Delta T}{T} = \frac{1}{2} \cdot \frac{\Delta l}{l} + \frac{1}{2} \cdot \frac{\Delta g}{g}$$If only l has error (g is constant):
$$\frac{\Delta T}{T} = \frac{1}{2} \times 2\% = \boxed{1\%}$$In an experiment, refractive index μ is calculated using μ = sin i / sin r. If maximum error in measuring angles is 1°, and i = 60°, r = 30°, find percentage error in μ.
Solution:
For small errors in angles (in radians):
$$\frac{\Delta(\sin\theta)}{\sin\theta} = \frac{\cos\theta \cdot \Delta\theta}{\sin\theta} = \cot\theta \cdot \Delta\theta$$Convert 1° to radians: Δi = Δr = 1° × π/180 = 0.0175 rad
$$\frac{\Delta\mu}{\mu} = \cot(60°) \times 0.0175 + \cot(30°) \times 0.0175$$ $$= \frac{1}{\sqrt{3}} \times 0.0175 + \sqrt{3} \times 0.0175$$ $$= 0.0101 + 0.0303 = 0.0404$$Percentage error = 4.04% ≈ 4%
Level 3: JEE Advanced
The density of a sphere is calculated using $\rho = \frac{6m}{\pi d^3}$ where m is mass and d is diameter. If errors in m and d are 3% and 1% respectively, find percentage error in ρ.
Solution:
$$\rho = 6\pi^{-1} m^1 d^{-3}$$ $$\frac{\Delta\rho}{\rho} = 1 \cdot \frac{\Delta m}{m} + 3 \cdot \frac{\Delta d}{d}$$ $$= 1 \times 3\% + 3 \times 1\% = 3\% + 3\% = \boxed{6\%}$$In a resonance tube experiment, two resonance lengths are l₁ = 25.0 ± 0.5 cm and l₂ = 75.0 ± 0.8 cm. Calculate wavelength λ = 2(l₂ - l₁) with absolute error.
Solution:
λ = 2(75.0 - 25.0) = 2 × 50.0 = 100.0 cm
For difference: Δ(l₂ - l₁) = Δl₂ + Δl₁ = 0.8 + 0.5 = 1.3 cm
For multiplication by constant: Δλ = 2 × Δ(l₂ - l₁) = 2 × 1.3 = 2.6 cm
Answer: λ = 100.0 ± 2.6 cm (or about 3% error)
Quick Reference Chart
| Operation | Formula | Error Rule |
|---|---|---|
| Sum | Z = A + B | $\Delta Z = \Delta A + \Delta B$ |
| Difference | Z = A - B | $\Delta Z = \Delta A + \Delta B$ |
| Product | Z = A × B | $\frac{\Delta Z}{Z} = \frac{\Delta A}{A} + \frac{\Delta B}{B}$ |
| Quotient | Z = A / B | $\frac{\Delta Z}{Z} = \frac{\Delta A}{A} + \frac{\Delta B}{B}$ |
| Power | Z = Aⁿ | $\frac{\Delta Z}{Z} = n \cdot \frac{\Delta A}{A}$ |
| Constant multiple | Z = kA | $\Delta Z = k \cdot \Delta A$ |
Related Topics
Within Units and Measurements
- SI Units — Standard measurement systems
- Dimensions — Dimensional correctness relates to errors
- Dimensional Analysis — Verifying formulas
- Significant Figures — Expressing precision
Connected Chapters
- Experimental Skills — Practical error reduction
- Kinematics — Measurement errors in motion
- Gravitation — Error in g calculation
Math Connections
- Statistics — Mean, standard deviation
- Calculus — Error in derivatives