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I. Basic Concepts of Error:
1. Definition of Error:
Error = Measured Value − True Value;
Therefore, error is a value—mathematically, a point on the coordinate axis, a numerical value with a positive or negative sign.
2. Methods of Expressing Error:
① Absolute Error:
Absolute Error = Measured Value − True Value (Conventional True Value)
In verification work, a standard of one grade higher accuracy is commonly used as the true value to obtain the absolute error.
For example: When calibrating a second-class piston pressure gauge with a first-class piston pressure gauge, the first-class piston pressure gauge reads 100.5N/cm2 and the second-class piston pressure gauge reads 100.2N/cm2, so the measurement error of the second-class piston pressure gauge is -0.3N/cm2.
② Relative Error:
Relative Error = Absolute Error / True Value × 100%
Relative error has no unit but does have a positive or negative sign.
For example: When calibrating a second-class standard mercury thermometer with a first-class standard mercury thermometer, the first-class standard mercury thermometer reads 20.2℃ and the second-class standard mercury thermometer reads 20.3℃, so the relative error of the second-class standard mercury thermometer is 0.5%.
③ Fiducial Error:
Fiducial Error = Indication Error / Upper Limit of Measuring Range (or Specified Value) × 100%
Fiducial error is a simplified and practical form of relative error for instrument indications.
For example, for a working force gauge with an upper measuring range limit of 3000N, the indication at the calibrated point of 2400N is 2392.8N, so its fiducial error is -0.3%.
3. Classification of Errors:
① Systematic Error: Under repeatability conditions, the difference between the mean of an infinite number of measurement results for the same measurand and the true value of the measurand.
② Random Error: The difference between a measurement result and the mean of an infinite number of measurement results for the same measurand under repeatability conditions.
③ Gross Error: An error that exceeds the expected value under specified conditions.
II. Precision:
Precision is subdivided into:
Accuracy (Accuracy): The effect of systematic error on measurement results.
Precision (Precision): The effect of random error on measurement results.
Accuracy (Accuracy): The combined effect of systematic and random errors on measurement results.
Precision is a term from error theory and is a different concept from measurement uncertainty. In error theory, the quantitative characteristics of precision can be expressed by the current measurement uncertainty (for measurement results) and limiting error (for measuring instruments). For measurements, high precision does not necessarily mean high accuracy, and high accuracy does not necessarily mean high precision; however, high accuracy (Accuracy) means both high accuracy and high precision. Precision is an abbreviation for accuracy (Accuracy). At present, the use of the term "precision" is not encouraged.
III. Measurement Uncertainty
Definition: A parameter, associated with the result of a measurement, that characterizes the dispersion of the values that could reasonably be attributed to the measurand.
① This parameter may be, for example, a standard deviation or its multiple, or the half-width of an interval with a stated confidence level.
② Measurement uncertainty consists of multiple components. Some of these components may be estimated from the statistical distribution of results from a series of measurements and characterized by experimental standard deviations. Other components may be estimated from assumed probability distributions based on experience or other information and may also be characterized by standard deviations.
③ The measurement result should be understood as the best estimate of the value of the measurand, and all uncertainty components contribute to the dispersion, including those arising from systematic effects (e.g., components related to correction values and reference measurement standards).
From this, it can be seen that measurement uncertainty, error, and precision are defined differently. Therefore, the conceptual differences also lead to differences in evaluation methods.
IV. Main Differences Between Measurement Error and Measurement Uncertainty
1. Difference in definition: Error represents a point on the number axis, while uncertainty represents an interval on the number axis;
2. Difference in evaluation method: Error is evaluated as systematic error and random error, while uncertainty is evaluated as Type A and Type B;
3. Difference in concept: Systematic error and random error are idealized concepts, while uncertainty only uses estimated values;
4. Difference in expression: Error cannot be expressed in the form of ±, while uncertainty can only be expressed in the form of ±;
5. Difference in combination method: Errors are combined by algebraic addition, while uncertainties are combined by the root-sum-square method;
6. Difference in measurement results: Errors can directly correct measurement results, while uncertainty cannot correct measurement results; by definition, error is only related to the true value, while uncertainty is related to factors affecting the measurement;
7. Difference in acquisition method: Error is obtained through measurement, while uncertainty is obtained through evaluation;
8. Difference in operational method: Systematic error and random error are difficult to operate, while uncertainty evaluation is easy to operate;
Error and measurement uncertainty are interrelated; that is, measurement error also contains uncertainty, and conversely, the evaluated uncertainty also has error.
Precision is evaluated according to the classification of errors, but its method of error combination differs from that of measurement uncertainty. Systematic errors are combined algebraically, random errors are combined by the root-sum-square method, and the combination of systematic and random errors may be based on standard deviations or limiting errors. Therefore, the combination methods are not uniform.
