Errors in Measurement

1. Introduction

Measurement is never perfectly exact. Every measured value contains some degree of uncertainty or error due to limitations of instruments, environment, and human factors.

Error is defined as the difference between the true value and the measured value.

$$\text{Error} = \text{True Value} – \text{Measured Value}$$

2. Types of Errors in Measurement

Errors are broadly classified into:

• Gross Errors
• Systematic Errors
• Random Errors

3. Gross Errors

Definition

Errors caused by human mistakes.

Examples

• Incorrect reading of scale
• Recording wrong data
• Improper instrument usage

Minimization

• Proper training
• Repeating measurements
• Using digital instruments

4. Systematic Errors

These errors occur due to consistent and predictable causes.

4.1 Instrumental Errors

Causes

• Poor calibration
• Wear and tear
• Zero error

Example

• Vernier caliper showing non-zero reading when closed

4.2 Environmental Errors

Causes

• Temperature variation
• Humidity
• Pressure changes

Example

• Expansion of metal due to heat affecting measurement

4.3 Observational Errors

Cause

• Improper reading by observer

Example

• Parallax error

4.4 Theoretical Errors

Cause

• Imperfect assumptions or formulas

5. Random Errors

Definition

Errors that occur due to unpredictable variations.

Causes

• Fluctuations in environment
• Electrical noise
• Human estimation

Characteristics

• Cannot be eliminated completely
• Reduced by repeated measurements

6. Classification Based on Nature

6.1 Absolute Error

$$\text{Absolute Error} = | \text{True Value} – \text{Measured Value} |$$

6.2 Relative Error

$$\text{Relative Error} = \frac{\text{Absolute Error}}{\text{True Value}}$$

6.3 Percentage Error

$$\text{Percentage Error} = \text{Relative Error} \times 100$$

7. Accuracy and Precision

Accuracy

• Closeness of measured value to true value

Precision

• Repeatability of measurements

8. Sources of Errors

• Instrument defects
• Environmental conditions
• Human error
• Improper calibration
• External disturbances

9. Minimization of Errors

• Calibration of instruments
• Controlled environment
• Proper training
• Use of high-precision instruments
• Taking multiple readings and averaging

10. Propagation of Errors

When multiple measurements are involved, errors combine.

Addition/Subtraction

• Absolute errors are added

Multiplication/Division

• Relative errors are added

11. Significant Figures

• Indicate precision of measurement
• More significant figures → higher precision

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