Measurement Principles and Uncertainty in Metrology

Posted on Jan 13, 2025 in Mathematics

Measurement Principles and Uncertainty

1. Definition of Measurement

Measurement is the process of experimentally obtaining one or more quantity values that can reasonably be attributed to a quantity.

2. Uncorrected, Corrected, and Final Results

  • Uncorrected Result: The result of a measurement before correction for systematic error.
  • Corrected Result: The result of a measurement after correction for systematic error.
  • Complete Result: The result of a measurement after correction for systematic error, accompanied by estimated uncertainty from random errors.

3. Principle of Measurement and Measurement Method

  • Principle of Measurement: The scientific basis of a measurement.
  • Measurement Method: A generic description of a logical organization of operations used in a measurement.
    • Direct Measurement Method: The value of a quantity is obtained directly by comparing the unknown with the standard.
    • Indirect Measurement Method: Several parameters Xi; i = 1,…,n (to which the quantity Xp to be measured is linked) are measured directly, and then the value Xp is determined by a mathematical relationship.

4. True Value and Conventional True Value

  • True Value (of a Quantity): The value consistent with the definition of a given particular quantity. This is a value that would be obtained by a perfect measurement.
  • Conventional True Value (of Quantity): The value attributed to a particular quantity and accepted, sometimes by convention, as having an uncertainty appropriate for a given purpose.

5. Measurement Error

Measurement Error results from the fact that all measurement results, including those obtained with very high precision instruments and with high experimental accuracy, are not accurate but approximate. ΔX=X–X0

6. Classification of Measuring Errors

  • Random Measurement Error: The component of measurement error that varies randomly (in sign and magnitude) for repeated measurements of one and the same quantity.
  • Systematic Measurement Error: The component of measurement error that, in repeated measurements, remains constant or varies in a predictable manner.
  • Gross Measurement Error (Failure): Measurement error significantly exceeding the error expected under the given conditions.

7. Probability Density Function (PDF)

The Probability Density Function (PDF) specifies the probability of a random variable falling within a particular range of values. This probability is given by the integral of this variable’s PDF over that range—that is, it is given by the area under the density function but above the horizontal axis and between the lowest and greatest values of the range.

8. Main Gauss Theory Assumptions

  • If we measure the same value in unchanged conditions, the most probable value is the mean.
  • The accuracy of measurement is affected by many factors at the same time.
  • The probability of errors decreases with increasing values.

9. Confidence Interval

Confidence Interval: A range of values so defined that there is a specified probability P=1-α that the value of a measurement lies within it.

10. Measurement Uncertainty

Measurement Uncertainty: An interval distributed symmetrically with respect to the measurement result (mean value) in which the true value of the measured quantity is included with the specified probability P.

  • Standard Uncertainty u: Uncertainty expressed by standard deviation.
  • Expanded Uncertainty U: The product of a standard uncertainty u and a factor larger than the number one, k.

U=ku

11. Definitions in Dimensional Metrology

  • Basic Size: A number expressing a numerical value of length or angle.
  • Limits of Size: The maximum and minimum sizes permissible for a specific dimension.
  • Tolerance: The total permissible variation in the size of a dimension.
  • Maximum Material Size (MMC): The limit of size of a feature that results in the part containing the maximum amount of material.
  • Fit: The relationship between two mating parts with respect to the amount of clearance.
  • Three Basic Types of Fits: Clearance, interference, and transition.
  • Allowance: Fit parameter defined as the difference between the basic dimensions of the mating parts.
  • Upper Deviation: Maximum dimension minus nominal dimension. ES=Gu – N
  • Lower Deviation: Minimum dimension minus nominal dimension. EI=Gu – N
  • Fundamental Deviation (h): Deviation to define the position of the tolerance zone in relation to the zero line (nominal size).