Measurement uncertainty, as a new concept, is gradually being introduced into measurement results, and it is easy to confuse it with the measurement error of the original measurement results. This article introduces the concept and evaluation methods of measurement uncertainty, and explores the methods for evaluating the uncertainty of geometric measurements of parts. Basic Concepts of Measurement Uncertainty Measurement uncertainty is a parameter that reasonably characterizes the dispersion of the measurand, representing the degree of unreliability of the measurement result or the degree of doubt about the validity of the measurement result. Measurement uncertainty can be quantitatively evaluated; it is an operational definition, and it can be expressed in absolute and relative forms. When expressed in absolute form, its dimensions are the same as the measurand; when expressed in relative form, it is dimensionless. In actual measurement, the measurement uncertainty must be given along with the measurement result; it is sometimes also called the uncertainty of the measurement result. It is used to characterize the inherent dispersion of the measurand's value within a certain region with a certain probability. Measurement uncertainty is expressed using the standard deviation, and when necessary, it can also be characterized by multiples of the standard deviation or the half-width of the confidence interval. Methods for Evaluating Measurement Uncertainty : ■ Clearly define the measurand and its measurement conditions. ■ Clearly define the measurement principle, measurement method, and measurement equipment used. ■ Analyze the sources of uncertainty and list the standard uncertainty components. ■ When sharing sources of uncertainty, the influence of uncertainty components should be fully considered, and omissions and repetitions should be avoided. ■ When correcting measurement results, the measurement uncertainty introduced by incomplete corrections should be considered. ■ When describing the nature of the source of measurement uncertainty, it should be stated as "uncertainty caused by random influences" or "uncertainty caused by systematic influences." It should not be called "random uncertainty" or "systematic uncertainty." ■ In the final report of measurement results, the last digit of the measurement uncertainty reading should be consistent with the last digit of the measurement result. Analysis and Evaluation of Measurement Uncertainty of Geometric Quantities of Parts Sources of Measurement Uncertainty of the Test Component ■ Measurement uncertainty caused by measurement repeatability (Type A uncertainty). ■ Measurement error caused by the non-perpendicularity of the part's measurement reference plane to the axis and measurement error caused by the angle between the V-shaped iron surface and the flat plate. ■ Measurement uncertainty of the measuring instrument itself. ■ Measurement uncertainty caused by temperature changes. Evaluation of Measurement Uncertainty of the Test Component Type A Measurement Uncertainty Caused by Measurement Repeatability This uncertainty can be obtained through statistical calculation. Table 1 shows the data from 10 measurements of equal precision performed on the test piece, along with the calculated average, residual, and sum of squared residuals. The average of the ten measurements is = 98.026 mm. The degrees of freedom are n = n-1 = 9, where n is the number of measurements. Q = ∑Vi² = 1 × 10⁻⁵, S(qk) = [Q/(n-1)]¹/². The standard deviation of qk can be calculated as S(qk) = (1 × 10⁻⁵/9)¹/² = 0.001 mm. The standard deviation of the arithmetic mean is S(q) = S(qk)/n¹/² = 0.0003 mm. For measurement results, the standard deviation of the arithmetic mean is usually called the Type A standard uncertainty. Therefore, the Type A standard uncertainty of the measured quantity is u(q) = 0.0003 mm . The measurement uncertainty caused by the non-perpendicularity of the part's measurement reference surface to the axis is as follows: The non-perpendicularity of the part's measurement reference surface to the axis causes a small angle between the measured section and the ideal section, forming a measurement error, which is one component of the measurement uncertainty. From Figure 2, we can derive the measurement error Δd = d' - d ≈ d(1/cosα - 1). Therefore, this error is related to the section containing the measured diameter and the tilt angle of the workpiece. The perpendicularity requirement between the part's measurement reference surface and the axis is 0.01 mm, and the height of the measuring surface from the reference surface is approximately 20 mm. Measurements show that the angle between the V-shaped iron surface and the flat plate is a1 = 0.03°. The inclination angle between the axis and the reference plane caused by the perpendicularity of the part's measuring reference plane and the axis is a2 = arctg0.01/20 ≈ 0.03°. a = a1 + a2 = 0.03° + 0.03° = 0.06°. △d = d'－d ≈ a（1/cosα－1） = 5 × 10⁻⁵ mm. Assuming a uniform distribution, the confidence factor K = 1.732, and its standard uncertainty is u2 = △d/K = 5 × 10⁻⁵ mm/1.732 = 3 × 10⁻⁵ mm. Uncertainty of the measuring instrument: The uncertainty of the coordinate measuring machine is U = 2.5 + 3.3L/1000 mm. When the measuring length L = 98 mm, U = 0.0028 mm. If the distribution is uniform, the confidence factor K = 1.732, and its standard uncertainty is u3 = U/K = 0.0028/1.732 = 0.0016 mm. Measurement uncertainty caused by temperature : Through measurement comparison, near 20℃, an increase of one degree results in an increase of 0.001 mm in the measurement result. Under relatively constant temperature conditions of 20°±1℃, the measurement uncertainty caused by temperature is 0.002 mm. If this uncertainty is uniformly distributed, the standard uncertainty is: u4 = 0.002/1.732 = 0.0012 mm. Combined standard uncertainty: The combined standard uncertainty is obtained by combining the above standard uncertainty components. uc = [u (q)² + u²² + u³² + u⁴²]¹/² = (0.0003² + 0.00005² + 0.0016² + 0.0012²)¹/² = 0.002 mm Expanded Uncertainty Assuming the expanded uncertainty distribution is normal, with a confidence probability of 99.73% and a coverage factor K = 3, the expanded uncertainty is: U = 3uc = 3 × 0.002 = 0.006 mm. The measurement result can be expressed as: 98.026 ± 0.006 mm. The following figure shows the tolerance, measurement result, and uncertainty distribution (see Figure 3). The measured value falls within the 98.026 ± 0.006 mm range with a very high probability.