Uncertainty of Measurement

Introduction

Measurement is the process of determining the magnitude of a quantity, such as temperature, weight, or height. It provides a numerical value that describes a property. Counting is not considered measurement, nor is comparing two objects based on length or test results that conclude with yes/no or pass/fail.

Uncertainty of Measurement

Uncertainty refers to the quality of a measurement. Even the most precise measurements have a range of error. It is essential to understand the size of this range, which is quantified as the measurement uncertainty.

Definitions and Terms

• True Value: The value consistent with the definition of a specific quantity. True values are inherently indeterminable and are represented as XtXt.
• Reference Value: An organized set of continuous or discrete values agreed upon as a reference. It is often represented as XtXt and is typically derived from multiple measurements using a precise instrument.
• Accuracy of Measurement: The closeness of agreement between a measurement result and the true value. Accuracy is a qualitative concept and should not be confused with precision.
• Bias of Measurement: The closeness of agreement between a measurement result and the reference value.
• Error: The difference between the measured value and the true value.
• Deviation: The difference between the obtained value and the reference value.

Types of Errors

• Random Error: The difference between a measurement result and the mean of results from an infinite number of measurements under repeatable conditions. Random errors vary unpredictably and can be positive or negative.
• Systematic Error: The difference between the mean of results from an infinite number of measurements and the true value under repeatable conditions. Systematic errors are usually correctable and include errors from equipment, reference standards, reading errors, and environmental conditions.
• Correction Factor: A number used to compensate for systematic errors in an uncorrected measurement result.

Uncertainty of Measurement

Uncertainty is a parameter associated with the result of a measurement that characterizes the dispersion of values that could reasonably be attributed to the measurand. It is typically composed of multiple components, some of which can be evaluated from statistical distributions of measurement results, while others are based on assumed probability distributions.

Result of a Measurement

The value attributed to a measurand obtained through measurement. A complete measurement result includes information about the measurement uncertainty.

Resolution

The smallest interval between gradations on a measuring instrument, expressed in the specified unit.

Methods for Selecting Measuring Instruments

• Resolution Method: The measuring instrument should be able to divide the tolerance into 10 parts.
• Uncertainty Method: The measurement uncertainty should be ≤ 0.1 – 0.2 of the tolerance. This method ensures that no good part is rejected and no defective part is accepted.

What is Uncertainty?

Uncertainty is a factor associated with the measurement result that defines the range of values the result can have. It indicates the confidence level that the true value lies within the specified range.

Why is Uncertainty Important?

Uncertainty quantifies the quality of the measurement result, indicating how far the result might be from the true value. It is expressed as a ± value around the measurement result and becomes crucial when results are close to acceptable limits.

Confidence Level

The range within which the true value is expected to lie with a specified level of confidence, such as 95%.

Standard Deviation

A common measure of precision, where ±σ covers 68% of measurements, ±2σ covers 95%, and ±3σ covers 99.73% of measurements.

Standard Uncertainty

If the uncertainty of measurement results is expressed as a standard deviation, it is called standard uncertainty. It is the average uncertainty of a standard uncertainty.

Type A Standard Uncertainty

Evaluated by statistical analysis of a series of observations.

Type B Standard Uncertainty

Evaluated based on information other than statistical analysis of observations.

Combined Standard Uncertainty

The square root of the sum of the squares of the uncertainties affecting the final measurement result.

Expanded Uncertainty

A quantity that defines an interval around the measurement result, within which the true value is expected to lie with a high level of confidence.

Coverage Factor

A numerical factor used to obtain the expanded uncertainty from the combined standard uncertainty.

Probability Distribution

A function that describes the likelihood of a random variable taking on a particular value.

Error and Uncertainty

• Error: The difference between the measured value and the true value.
• Uncertainty: A range that indicates the quality of the measurement.

Factors Contributing to Uncertainty

• Random factors that appear in repeated measurements.
• Systematic factors.
• Environmental factors such as temperature, pressure, and dust.
• Calibration uncertainty related to reference standards.
• Instrument-related factors such as bias and resolution.
• Measurement method-related factors.

What is Not Measurement Uncertainty?

• Operator errors and lack of precision, which can be controlled through MSA.
• Specifications and appearance of the product.
• Accuracy and error, which are components of uncertainty.

Calculation of Uncertainty

• Random Factors: Calculated using statistical formulas such as standard deviation.
• Systematic Factors: Estimated based on probability distributions.

Examples of Uncertainty Calculation

• Example 1: Digital caliper with a resolution of 0.01 mm.
• Example 2: Three-point internal micrometer with a resolution of 0.001 mm.
• Example 3: Load cell with a resolution of 0.001 N.

Reducing Measurement Uncertainty

• Regular calibration of measuring instruments.
• Correcting instrument errors using the manual.
• Ensuring instruments are traceable to primary standards.
• Selecting the best and most appropriate measuring instruments.
• Conducting daily checks and repeated measurements.

Uncertainty for Multiple Quantities

The uncertainty for multiple quantities is calculated using partial derivatives of the measurement function with respect to each variable.

Conclusion

Understanding and accurately calculating measurement uncertainty is crucial for ensuring the reliability and quality of measurement results. By identifying and addressing the sources of uncertainty, we can improve the precision and accuracy of our measurements.

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