Detailed Notes on Random Errors and Measurement Uncertainty

Measurement Errors

• Definition of Measurement Errors:

  • Measurement errors will always be present regardless of measurement care and instrument accuracy.

  • True value is generally unknown or unknowable; thus, expected value is used for error assessments.

• Accuracy of Instruments:

  • Accuracy reflects how closely the measured value aligns with the true value.

  • Without error indication, a measurement's utility is limited.

  • True Value vs. Measured Value.

Types of Measurement Errors

• Absolute Error:

  • \text{Absolute Error} = \text{Measured Value} - \text{True (Expected) Value}

  • Example: For a temperature of 20.6°C (measured) vs 20.0°C (expected):

  • Absolute Error = 20.6 - 20.0 = 0.6°C

• Relative Error:

  • \text{Relative Error} = \frac{\text{Absolute Error}}{\text{Expected Value}} \times 100\%

  • Is dimensionless and represents instrument accuracy better than absolute error.

  • Example: For a temperature measurement:

  • Relative Error = \frac{0.6°C}{20.0°C} \times 100\% = 3\%

  • Another example: \text{Relative Error} = \frac{36°C}{1800°C} \times 100\% = 2\%

Full Scale Error

• Concept:

  • \text{Full Scale Error} = \frac{\text{Absolute Error}}{\text{Full Scale Deflection}} \times 100\%

  • Examples:

  • Voltmeter giving ±0.6V over a 0-30V range.

  • Thermometer with a 0-500°C range showing ±1% error:

    • Absolute Error = \text{Full Scale Error} \times \text{Full Scale Deflection} = ±1\% × 500 = ±5°C

Systematic Errors

• Definition:

  • Errors that are predictable and remain constant.

• Examples:

  • A thermometer consistently reading 1.5°C high.

  • Common sources include failure to zero the instrument and loading effect of measurements.

Random Errors

• Definition:

  • Cannot be predicted due to unknown factors such as noise or human error.

• Minimization:

  • Reducing these errors is possible through careful instrument usage and repeated measurements.

  • Example: Measurements like 10.1Ω, 9.9Ω, 10.0Ω, etc.

Statistical Analysis for Random Errors

• Mean Value (Average):

  • \text{Mean} = \frac{x1 + x2 + … + x_n}{n} where n = number of readings (usually 25+).

• Standard Deviation (s):

  • s = \sqrt{\frac{\sum (x_i - \text{mean})^2}{n-1}}

  • Measurement spread around the mean.

• Example Calculation:

  • For the readings 20.2°C, 20.1°C, …, standard deviation found through steps: 1) Calculate mean, 2) Find deviations, 3) Compute standard deviation.

Normal Distribution

• Normal Distribution Characteristics:

  • Reflects measurement distributions, with standard deviations representing the spread:

  • About 68% of readings fall within 1 standard deviation (s),

  • About 95% within 2s, and

  • About 99.7% within 3s.

Measurement Uncertainty

• Definition:

  • Represents the margin of doubt in a measurement estimate, requiring:

  1. Measured value (mean value).

  2. Measurement uncertainty.

  3. Confidence level/limit for uncertainty.

• Confidence Levels:

  • Commonly default to 95%.

  • For example: A mean temperature of 20.3°C with standard deviation 0.2°C presented as:

    • Measurement result: 20.3°C ± 0.2°C (95% confidence level).