The Influence of Magnification on Measurement Accuracy

Abstract

This article presents an experimental and statistical investigation of how optical magnification influences calibration constants, measurement results, and uncertainty in a digital optical microscope. Measurements were performed on reference gauge blocks with nominal lengths from 1.0 mm to 1.5 mm at five magnification levels (1×–5×) to quantify the effect of magnification on dimensional accuracy. A combined statistical methodology integrating non-parametric hypothesis testing and bootstrap-based uncertainty analysis was developed to evaluate data distributions and validate the use of a normal coverage factor (k = 2) for expanded uncertainty. The results showed that magnification has a statistically significant effect on the measured lengths for most standards, with the smallest combined standard uncertainty achieved at approximately 4× magnification. The uncertainty budget analysis revealed that the dominant component arises from the microscope’s declared Maximum Permissible Error (MPE), while type A and reference-standard components contribute only marginally. All expanded uncertainties remained within the declared MPE limits, confirming the reliability and traceability of the measurement process. Practical recommendations were proposed for selecting optimal magnification and for implementing calibration verification procedures at each zoom level. The presented methodology provides a validated framework for minimizing uncertainty in image-based dimensional measurements using digital optical microscopes.

1. Introduction

Optical microscopes are widely used in dimensional metrology for evaluating small geometrical features, surface structures, and fine mechanical parts. With ongoing miniaturization in modern manufacturing and materials engineering, the ability to obtain accurate and traceable dimensional data at the microscale has become increasingly important. Digital measuring microscopes—based on image acquisition and pixel calibration—provide a non-contact, fast, and flexible solution for such measurements [1,2,3,4]. The dimensional evaluation in these systems relies on converting image pixel distances into physical units using a calibrated scale factor, which depends on magnification, optical setup, and detector parameters.

In a digital optical microscope, the real measured length L_real is calculated as

$$L_{real}=N_{pix}·S_{pix},$$

(1)

where N_pix is the number of pixels between two measured points, and S_pix is the pixel size in micrometers per pixel, determined during calibration with a traceable standard. The pixel size depends on the magnification $M$, the optical system geometry, and the physical dimensions of the image sensor:

$$S_{pix}={\displaystyle \frac{p_{sensor}}{M}},$$

(2)

where p_sensor is the pixel pitch of the sensor (e.g., in µm). Thus, any change in magnification alters the calibration constant S_pix, which directly affects the computed length L_real and the corresponding uncertainty [5,6,7].

The total measurement uncertainty is influenced by several parameters, including calibration accuracy, repeatability of edge detection, optical aberrations, focusing, and illumination stability [8,9,10,11]. Each of these contributors can be represented by a standard uncertainty u_i, and the combined standard uncertainty u_c is determined according to the law of propagation of uncertainty:

$$u_c=\sqrt{{\sum }_i{\left(c_i{u}_i\right)}^2},$$

(3)

where $c_i$ are the sensitivity coefficients describing how each component affects the result [12,13,14,15]. The expanded uncertainty $U$ represents the interval expected to contain the true value with a defined level of confidence and is expressed as

$$U=k·u_c,$$

(4)

where $k$ is the coverage factor; typically $k=2$ for a confidence level of approximately 95%. In practice, this approach assumes normal distribution of measured data; however, image-based measurements sometimes exhibit non-Gaussian behavior, especially when the number of samples is small or when edge-detection algorithms introduce bias [15,16,17]. In such cases, alternative methods such as bootstrap analysis or non-parametric statistics are required to properly estimate $k$.

The measurement accuracy in digital microscopy is therefore highly dependent on the selected magnification. Higher magnifications generally improve spatial resolution and edge localization but can amplify random and systematic errors related to optical imperfections or calibration drift [17,18,19,20,21]. Conversely, lower magnifications provide a larger field of view and better stability but at the expense of resolution and measurement precision. Selecting an optimal magnification thus represents a trade-off between image quality, measurement repeatability, and uncertainty [22,23].

Several recent studies have addressed aspects of optical measurement uncertainty and calibration. Gao et al. [1] presented a metrological calibration model for optical areal surface measuring instruments, emphasizing traceability and systematic uncertainty. Bellantone et al. [2] experimentally analyzed uncertainty propagation in optical measurements of micro-injected products. Hooshmand et al. [3] proposed a comprehensive framework for uncertainty evaluation in optical surface topography measurement using virtual instrumentation. Zhou et al. [4] developed a high-precision visual dimension measurement method with multi-prism projection, demonstrating how system geometry affects calibration linearity. Earlier works, such as those by Bao et al. [6] and Gadelmawla et al. [7], established the foundation for microscopic calibration and pixel-based uncertainty evaluation. More recently, Batista et al. [8] investigated uncertainty propagation in optical micro-flow measurements, while Daher et al. [9] examined the validity limits of digital optical microscopy for dimensional analysis.

Despite these efforts, the direct quantitative relationship between optical magnification and measurement uncertainty in pixel-based systems has not been sufficiently characterized. Zhang et al. [10] and Li et al. [11] confirmed that magnification can influence calibration constants and the effective scale factor of optical instruments. Leach [17] and Brown et al. [24] noted that magnification-related effects, such as distortion and aberration, can introduce additional uncertainty components that are often neglected in standard uncertainty budgets. Furthermore, the Maximum Permissible Error (MPE) declared by microscope manufacturers may depend on magnification [25,26,27,28,29,30], yet practical verification across magnifications is rarely performed [27,31,32,33,34,35,36].

Surface metrology research further emphasizes the importance of proper calibration and magnification control. Pawlus et al. [18] and Podulka et al. [19] discussed how optical parameters and measurement area selection influence roughness parameters. Vorburger and Blateyron [20,22] reviewed the comparison of optical and stylus methods for surface measurement and proposed classification frameworks. Poon and Bhushan [21] compared optical, AFM, and contact profilometry, highlighting differences in uncertainty sources. Blateyron and Leach [17,22], as well as De Chiffre et al. [37], presented detailed methodologies for characterizing surface texture and the role of optical magnification in uncertainty budgets.

Standardization efforts (ISO 25178 [32], ISO 21920 [38]) and guidelines from NIST [39,40] provide the basis for traceability and uncertainty estimation in optical and surface topography measurements, but they still lack explicit recommendations regarding magnification-dependent effects. Hence, the metrological traceability of optical microscopes must include a detailed analysis of magnification-related contributions to total uncertainty.

Given these gaps, the present study investigates the influence of magnification on the accuracy and uncertainty of measurements performed using a digital optical microscope operating on a pixel-calibration principle. The objectives are to determine whether variations in magnification lead to statistically significant changes in the measured values, deviations from nominal dimensions, and expanded uncertainties, and to evaluate whether these deviations remain within the declared MPE.

Based on the literature review and experimental motivation, the following research criteria are defined:

Criterion 1: Identification of the influence of optical magnification on measurement uncertainty and repeatability.

Criterion 2: Verification of whether measurement deviations at different magnifications remain within the declared MPE.

Criterion 3: Determination of the optimal magnification level that minimizes combined uncertainty while maintaining adequate field of view.

Criterion 4: Statistical verification of the normality assumption in the measurement data for reliable uncertainty estimation.

The study focuses on determining whether variations in magnification lead to statistically significant changes in the measured values, deviations from nominal dimensions, and expanded uncertainties.

This work provides three main contributions:

(1)

Quantitative evaluation of how magnification affects calibration constants, measured values, and uncertainty in a digital optical microscope.

(2)

Development of a statistical methodology combining non-parametric hypothesis testing and bootstrap-based uncertainty analysis for image-based dimensional measurements.

(3)

Proposal of practical guidelines for selecting optimal magnification to achieve minimal measurement uncertainty and compliance with MPE limits.

In many industrial and research applications, the relationship between optical magnification and measurement uncertainty has direct practical consequences. For example, in the dimensional inspection of micro-mechanical components, small deviations in magnification may alter pixel size and thereby bias edge-based feature extraction [41,42]. In microelectronics and high-density interconnect manufacturing, inappropriate magnification selection can reduce edge contrast or increase the influence of diffraction, resulting in inconsistent metrology outcomes [43]. Similar challenges have been reported in precision tooling and surface-quality evaluation, where magnification-dependent optical aberrations and calibration drift limit achievable measurement accuracy [44]. Previous studies emphasize that pixel-scale calibration, magnification error, and optical design characteristics are often dominant contributors to uncertainty when measuring features in the sub-millimeter to millimeter range [44,45], underscoring the need for systematic investigation of magnification selection in digital optical microscopes. These practical considerations further justify the relevance of the present study.

The rest of this paper is organized as follows. Section 2 describes the experimental setup and calibration procedure of the optical microscope. Section 3 presents the statistical analysis of the measurement data and the uncertainty evaluation. Section 4 discusses the findings regarding the relationship between magnification, uncertainty, and instrument performance. Finally, Section 5 concludes the paper with recommendations for further research and industrial applications.

2. Materials and Methods

Optical digital microscopes differ in their declared Maximum Permissible Error (MPE), which characterizes the largest deviation between the measured and the true value under specified conditions. Depending on the optical design and calibration strategy, manufacturers specify two general types of microscopes:

• Microscopes with magnification-dependent MPE, where the declared measurement accuracy changes with the selected objective or zoom level;
• Microscopes with constant MPE, where the measurement accuracy is assumed to remain uniform across the entire magnification range.

The difference between these two approaches arises from the optical configuration, calibration model, and image scaling method.

In microscopes with magnification-dependent MPE, the system magnification directly affects the calibration factor Spix (µm/pixel). Since the optical path length, numerical aperture, and field curvature change with zoom level, the image scale is non-linear, leading to variation in pixel pitch and therefore in measurement accuracy [1,2,3,4]. This behavior is particularly evident in microscopes with variable zoom optics or interchangeable lenses that are not individually calibrated [5,6].

Conversely, microscopes with constant MPE utilize fixed optical magnifications with factory calibration curves, telecentric lens designs, or internal electronic correction algorithms to maintain stable scaling across the zoom range [7,8,9]. Telecentric objectives minimize perspective distortion and maintain constant magnification regardless of object distance, significantly improving dimensional stability [10,11,12].

As noted by Copeland et al. [5] and Leach [17], optical aberrations such as field curvature, chromatic shift, and distortion contribute to scale variation, especially in systems where the calibration is performed only at a reference magnification. Li et al. [11] and Zhou et al. [4] confirmed that calibration constants vary systematically with magnification and can introduce non-linear scaling errors if not corrected.

Microscopes maintaining constant MPE often include internal reference grids or autocalibration routines that compensate for these changes automatically [17,18,19]. However, even in such cases, environmental conditions, focusing distance, and image interpolation may introduce additional uncertainty components [20,21,22].

The influence of magnification on measurement uncertainty thus depends on both optical and algorithmic design factors. In the context of dimensional metrology, it is therefore necessary to experimentally verify whether the uncertainty and deviation from nominal values remain within the manufacturer’s declared MPE for each magnification setting [23,46,47,48]. This verification provides a direct assessment of the microscope’s metrological capability and its suitability for traceable measurements [31,49].

This study employs a microscope with magnification-dependent MPE intentionally, since it allows a systematic investigation of how optical zoom levels affect measurement accuracy, calibration constants, and uncertainty.

By comparing measurement results across magnifications, it is possible to determine whether:

• The declared MPE correctly represents the true measurement capability;
• The uncertainty decreases proportionally with magnification;
• Calibration corrections can compensate for scale variation.

The analysis thus provides insight into the metrological reliability of optical microscopes across their magnification range, contributing to more precise uncertainty budgeting and standardization in digital optical metrology.

All measurements were carried out using a digital microscope Insize ISM-DL520 (INSIZE Co., Ltd., Suzhou, China) (Figure 1), whose Maximum Permissible Error (MPE) depends on the applied magnification. The corresponding MPE values specified by the manufacturer are summarized in

Table 1. Dependence of MPE on magnification [50].

Objective Magnification	MPE [µm]
1.0×	±6
2.0×	±6
3.0×	±5
4.0×	±4
5.0×	±4

.

The magnifications 1×–5× were deliberately selected because they cover all three manufacturer-specified MPE regimes of the INSIZE ISM-DL520 microscope (Table 1) and therefore provide a representative spread of accuracy conditions for evaluating the influence of magnification on measurement uncertainty. Although the instrument supports additional intermediate magnification values, these were not included in the present study because the 1×–5× range offers a balanced compromise between field of view, depth of field and spatial resolution for the 1.0–1.5 mm gauge blocks investigated here. Furthermore, using discrete magnification steps with distinct MPE values enables a clearer interpretation of differences in uncertainty contribution.

At substantially higher magnifications, several additional effects typically become significant—reduced field of view, reduced depth of field, higher susceptibility to focus drift and environmental vibrations, increased impact of optical aberrations, and diffraction-induced loss of edge contrast—all of which can increase measurement uncertainty [44,51,52,53,54,55]. In future studies, additional magnification levels with different MPE specifications may be incorporated to further examine the generality of the observed trends.

Before the measurement campaign, the microscope was calibrated using a glass scale (Figure 2) with a maximum permissible error of z_max = ±0.2 µm and a calibrated length of 10 mm.

The MPE values used in the uncertainty budget correspond to the manufacturer-specified maximum permissible error for each magnification setting of the Insize ISM-DL520 microscope. Such manufacturer-provided limits are commonly used as conservative uncertainty contributors in optical and dimensional metrology [51]. To confirm the practical validity of these values, repeated measurements of the calibrated gauge blocks were analyzed, and all observed deviations remained within the limits declared by the manufacturer. Therefore, the MPE specification was adopted as an appropriate and traceable contribution to the uncertainty budget, although a full independent calibration of the microscope would provide additional verification and is recommended for future work.

The calibration procedure was performed for each magnification level, resulting in the determination of correction coefficients (mm/pixel). These coefficients are listed in

Table 2. Calibration coefficients for each magnification.

Objective Magnification	Calibration Coefficient (mm/Pixel) [µm]
1.0×	0.00602
2.0×	0.00302
3.0×	0.00199
4.0×	0.00149
5.0×	0.00120

and show that with increasing magnification, fewer pixels correspond to a unit of measured length.

The measurements were performed on ceramic parallel gauge blocks, Grade 1 (Figure 3), with nominal lengths of 1.0 mm, 1.1 mm, 1.2 mm, 1.3 mm, 1.4 mm, and 1.5 mm. Each gauge block was accompanied by a calibration certificate specifying its systematic error, summarized in

Table 3. Nominal length and systematic error of the gauge blocks.

Nominal Length [mm]	Systematic Error [µm]
1.0	0.0
1.1	+0.03
1.2	+0.08
1.3	+0.01
1.4	+0.06
1.5	−0.07

.

Each gauge block was measured at five magnifications (1×, 2×, 3×, 4×, and 5×). For all experiments, each gauge block at each magnification level was measured 50 times (n = 50 independent repetitions per group). All descriptive statistics, hypothesis tests and bootstrap procedures reported in this work are therefore based on samples of this size, unless explicitly stated otherwise. The measurements were conducted in a climate-controlled laboratory maintained at 20 ± 1 °C, so the influence of temperature on the measurement results was considered negligible.

The influence of temperature on the gauge blocks was considered negligible based on a quantitative estimation. The length change due to thermal expansion can be expressed as

$$∆L=\alpha ·L·∆T$$

(5)

where $\alpha$ is the coefficient of thermal expansion, $L$ is the nominal gauge block length, and $\mathsf{\Delta }T$ is the temperature variation. The ceramic gauge blocks used in this study have a coefficient of thermal expansion of $\alpha =9.2\times {10}^{-6}\hspace{0.17em}{\mathrm{K}}^{-1}$. For the maximum investigated gauge block length of $L=1.5 \mathrm{m}$m and a conservative temperature variation of $\mathsf{\Delta }T=1 \mathrm{K}$ (equivalent to 1 °C), the resulting length change is approximately $\mathsf{\Delta }L\approx 0.014$ µm. This value is at least one order of magnitude smaller than the smallest uncertainty component included in the uncertainty budget and therefore does not contribute significantly to the overall measurement uncertainty.

In addition to temperature control, relative humidity and atmospheric pressure were continuously monitored during all measurements to ensure stable environmental conditions. No active vibration-isolation table was employed; however, the laboratory environment exhibits low levels of background mechanical disturbance, and no vibration-induced artifacts were observed in the acquired microscope images. Previous studies have shown that, under typical laboratory conditions, vibrations contribute negligibly to optical dimensional measurement uncertainty compared with dominant contributors such as optical aberrations, pixel-scale calibration, and focus stability [17,51]. Therefore, the influence of vibrations was considered negligible for the purposes of this study.

Prior to each measurement, the microscope calibration was verified to ensure measurement accuracy. The gauge block was placed at the center of the field of view, with its surface aligned horizontally using two angular standards to ensure proper orientation relative to the optical axis. Measurements were taken in the central area of each gauge block, where the surface flatness and optical conditions were optimal. A schematic representation of the measurement setup and sample positioning is shown in Figure 4.

3. Statistical Analysis of Measurement Data

To assess whether the measurements of individual gauge blocks obtained at different magnifications differ significantly, a statistical analysis was performed. The first step consisted of evaluating whether the datasets corresponding to each magnification follow a normal distribution. The normality assessment was performed using the Lilliefors test, which is a modification of the Kolmogorov–Smirnov test suitable when the population mean and variance are unknown [1]. The null hypothesis assumes that the data follow a normal distribution; if the resulting p-value exceeds 0.05, the null hypothesis is not rejected.

Subsequently, the homogeneity of variances among groups (magnifications) was examined using Levene’s test [56]. This test evaluates whether the variances are statistically equal across the compared groups. If the p-value is greater than 0.05, the assumption of equal variances is accepted.

Depending on the outcomes of both preliminary tests, one of the following was selected for further analysis:

• One-way ANOVA, if all datasets were normally distributed and variances were homogeneous, or
• Kruskal–Wallis test, if at least one dataset violated the assumption of normality or variance homogeneity.

The results of the normality and homogeneity tests are summarized in

Table 4. Results of normality and homogeneity testing for gauge blocks.

Gauge Block [mm]	p(1.0×)	p(2.0×)	p(3.0×)	p(4.0×)	p(5.0×)	Levene p	Selected Test
1.0	p < 0.001	p < 0.001	p < 0.001	p < 0.001	p < 0.001	p < 0.001	Kruskal–Wallis
1.1	p < 0.001	p < 0.001	0.0016	p < 0.001	p < 0.001	0.1005	Kruskal–Wallis
1.2	p < 0.001	p < 0.001	p < 0.001	p < 0.001	0.0029	p < 0.001	Kruskal–Wallis
1.3	p < 0.001	0.3230	p < 0.001	p < 0.001	0.0362	p < 0.001	Kruskal–Wallis
1.4	p < 0.001	p < 0.001	p < 0.001	p < 0.001	0.0041	0.4826	Kruskal–Wallis
1.5	p < 0.001	0.1825	p < 0.001	p < 0.001	p < 0.001	0.0148	Kruskal–Wallis

. For each gauge block, the p-values of the Lilliefors test are presented for all magnifications, together with the p-value of Levene’s test and the final decision on the appropriate statistical test. For p-values below the reporting resolution of the statistical software, results are indicated using threshold notation p < 0.001.

The p-values in Table 4 belong to the Lilliefors normality test and the Levene test for homogeneity of variances. Although each group contains the same number of repeated measurements (n = 50), the p-values differ because the test statistics are sensitive to the underlying distributional characteristics of each magnification setting. For the Lilliefors test, variation in p-values arises from differences in distributional shape, particularly skewness, kurtosis, and the discrete nature of pixel-based measurements, which often deviate slightly from Gaussian behavior in optical microscopy. For the Levene test, p-values reflect differences in within-group dispersion; magnification-dependent variance changes are expected due to focus stability, edge-detection sensitivity and optical resolution effects, leading naturally to non-uniform variance structure across magnifications. Consequently, even with identical sample sizes, the distributional and dispersion characteristics produce varying p-values for the Lilliefors and Levene tests. These effects are well documented in meteorological data, where measurement distributions may deviate from strict normality and display heteroscedasticity across measurement conditions [52,53,54,55,56,57,58,59,60].

Although data transformation techniques such as logarithmic or Box–Cox transformations were considered, they did not consistently restore normality nor reduce heteroscedasticity among the magnification groups. Similar observations have been reported in optical and dimensional metrology studies, where transformation often fails to address distributional deviations caused by small-scale edge-detection variability and magnification-dependent contrast behavior [61,62]. Furthermore, applying a transformed scale would reduce the interpretability of the results in a practical measurement context. For these reasons, and given the robustness of rank-based non-parametric methods, the Kruskal–Wallis test was selected as the most appropriate approach.

Since the preliminary analysis indicated that none of the datasets satisfied the assumptions of normality and homogeneity of variances, all subsequent evaluations were performed using the non-parametric Kruskal–Wallis test. This test is a rank-based equivalent of one-way ANOVA and is suitable for comparing more than two independent groups without assuming normal data distribution [63].

The Kruskal–Wallis test was applied separately to the measurements of each gauge block. The null hypothesis states that all magnification groups originate from the same population, i.e., their median values are statistically equal. If the resulting p-value is lower than 0.05, the null hypothesis is rejected, indicating that at least one magnification level differs significantly from the others. Representative results of the Kruskal–Wallis analysis are summarized in

Table 5. Summary of Kruskal–Wallis test results for gauge blocks.

Gauge Block [mm]	Mean of All Values [mm]	Kruskal–Wallis p-Value	Interpretation
1.0	0.9973	<0.0001	Differences are statistically significant
1.1	1.0998	<0.0001	Differences are statistically significant
1.2	1.1988	<0.0001	Differences are statistically significant
1.3	1.3002	0.1221	No statistically significant differences
1.4	1.3995	<0.0001	Differences are statistically significant
1.5	1.4994	<0.0001	Differences are statistically significant

.

For each gauge block, the MATLAB R2024b (The MathWorks, Inc., Natick, MA, USA) script generated:

• the mean and median of the measured values at each magnification,
• a boxplot illustrating the data distribution,
• and the overall p-value of the Kruskal–Wallis test.

When statistically significant differences were found (p < 0.05), a post hoc comparison using MATLAB’s multcompare() function was automatically performed to identify which magnifications differed from each other.

For all measured gauge blocks, the p-values were below 0.05, indicating that the results obtained at different magnifications differ significantly. This suggests that the optical magnification has a measurable influence on the detected values, likely due to calibration precision, pixel resolution, and microscope optics.

In addition to numerical results, graphical outputs illustrate the data distribution across magnifications using boxplots. These visualizations highlight the variation trends and confirm the statistical findings.

The boxplot is a graphical representation of data distribution based on five key statistical descriptors: minimum, first quartile (Q1), median (Q2), third quartile (Q3), and maximum. The central box represents the interquartile range (IQR), which contains the middle 50% of all measured values. The horizontal line inside the box indicates the median (Q2), showing the central tendency of the dataset. The “whiskers” extend to the most extreme data points that are not considered outliers, typically up to 1.5 × IQR from the box boundaries.

Red crosses represent outliers, i.e., data points lying outside the 1.5 × IQR. These points may result from random measurement variability or systematic effects. The height and position of each box show both the dispersion and the central value of the data for a given magnification.

Comparing boxes across magnifications provides insight into how the measurement results shift or spread with changing optical magnification. Differences in median positions indicate systematic shifts, while differences in box heights (IQR) or whisker lengths suggest changes in measurement repeatability or precision.

Figure 5 illustrates the distribution of 50 measurements of the 1.0 mm gauge block obtained at magnifications of 1×, 2×, 3×, 4×, and 5×. The Kruskal–Wallis test yielded p < 0.0001, indicating statistically significant differences between the magnifications.

The median measured values for each magnification are: 1× = 0.9957 mm, 2× = 0.9993 mm, 3× = 0.9972 mm, 4× = 0.9999 mm, and 5× = 0.9959 mm (see the table in the figure). The data show that the measured value slightly depends on the applied magnification. The smallest magnifications (1× and 3×) tend to underestimate the nominal length, whereas higher magnifications (especially 4×) yield results closer to or slightly above the nominal value.

The spread of the boxes also differs among magnifications, suggesting that measurement repeatability varies with magnification. The smallest spread (narrow box) occurs at 4× magnification, indicating the most consistent results, while larger spreads at 1× and 3× magnifications imply higher variability, possibly due to lower effective resolution or calibration uncertainty.

Outliers (marked with red crosses) were observed for magnifications 2×, 3×, and 4×, but their occurrence is infrequent and does not significantly affect the median values.

Overall, the boxplot confirms that magnification has a statistically significant effect on the measured length. The optimal precision and consistency for the 1.0 mm gauge block were achieved at 4× magnification.

Figure 6 presents the distribution of 50 measurements of the 1.1 mm gauge block obtained at magnifications of 1×, 2×, 3×, 4×, and 5×. The Kruskal–Wallis test yielded $p=0.0001$, confirming statistically significant differences between the measurement groups corresponding to different magnifications.

The median measured values for each magnification are: 1× = 1.1023 mm, 2× = 1.1019 mm, 3× = 1.0989 mm, 4× = 1.0992 mm, and 5× = 1.1001 mm (see the table in the figure). The results show that the measured values slightly depend on the applied magnification, with the smallest differences observed between 3×, 4×, and 5×. The 1× magnification produces the highest median value, indicating a minor overestimation of the nominal length, while higher magnifications yield values closer to the nominal size.

The variation in box height across the magnifications suggests that the repeatability of measurements improves with increasing magnification. The narrowest boxes are observed at 4× and 5× magnification, reflecting better measurement stability and consistency. In contrast, a slightly larger spread at 1× magnification indicates higher measurement variability, likely due to reduced resolution and alignment precision.

Outliers (marked by red crosses) appear across all magnifications, most notably at 1× and 5×, but their influence on the median values remains negligible.

Overall, the boxplot confirms a statistically significant influence of magnification on the measured length of the 1.1 mm gauge block. The most stable and precise measurement results were achieved at 4× and 5× magnification, where the distribution is narrowest and closest to the nominal value.

Figure 7 shows the distribution of 50 measurements of the 1.2 mm gauge block obtained at magnifications of 1×, 2×, 3×, 4×, and 5×. The Kruskal–Wallis test yielded $p<0.0001$, confirming statistically significant differences among the measurement groups.

The median measured values for each magnification are: 1× = 1.1987 mm, 2× = 1.2015 mm, 3× = 1.2006 mm, 4× = 1.1999 mm, and 5× = 1.1987 mm (see the table in the figure). The results reveal that the measured values fluctuate slightly with magnification. The measurements at 2× and 3× magnification exhibit slightly higher medians than the nominal value, while those at 1× and 5× are closer to or marginally below it.

The box heights (interquartile ranges) vary noticeably between magnifications, indicating differences in repeatability. The smallest spread is observed at 3× and 4× magnification, suggesting the best consistency and measurement stability at these settings. Conversely, the 1× and 5× magnifications show a wider spread, implying greater variability likely associated with lower resolution or focus precision.

Outliers (marked with red crosses) appear at all magnifications except 1×, most frequently at 2× and 4×, but they do not significantly affect the median values.

Overall, the boxplot demonstrates that magnification has a statistically significant effect on the measured length of the 1.2 mm gauge block. The most precise and stable results were obtained at 3× and 4× magnification, where both the variability and deviation from the nominal length are minimal.

Figure 8 shows the distribution of 50 measurements of the 1.3 mm gauge block obtained at magnifications of 1×, 2×, 3×, 4×, and 5×. The Kruskal–Wallis test yielded p = 0.1221, indicating no statistically significant differences among the measurement groups.

The median measured values for each magnification are: 1× = 1.3011 mm, 2× = 1.2999 mm, 3× = 1.3003 mm, 4× = 1.3007 mm, and 5× = 1.3002 mm (see the table in the figure). The results show that all median values are very close to the nominal length of 1.3 mm, with deviations within ±0.002 mm.

The interquartile ranges are comparable across magnifications, suggesting similar repeatability and measurement stability. Slightly larger spread is observed at 1× and 2× magnifications, while 3×–5× exhibit more compact boxes, indicating marginally better consistency at higher magnifications.

Outliers (marked with red crosses) appear at all magnifications, most frequently at 1× and 2×, but they do not substantially affect the overall distribution or median values.

Overall, the boxplot demonstrates that magnification does not have a statistically significant effect on the measured length of the 1.3 mm gauge block. The measurements remain highly consistent across all magnifications, confirming stable measurement performance and accuracy for this scale.

Figure 9 presents the distribution of 50 measurements of the 1.4 mm gauge block obtained at magnifications of 1×, 2×, 3×, 4×, and 5×. The Kruskal–Wallis test yielded p < 0.0001, confirming statistically significant differences among the measurement groups.

The median measured values for each magnification are: 1× = 1.3975 mm, 2× = 1.4008 mm, 3× = 1.4000 mm, 4× = 1.3999 mm, and 5× = 1.3997 mm (see the table in the figure). The medians show a slight upward shift at higher magnifications, with 2×–4× producing values slightly above the nominal length, while 1× is marginally below it.

The interquartile ranges are smallest at 3× and 4× magnification, indicating the best repeatability and measurement stability in these settings. The 1× group shows the widest spread, suggesting lower precision, likely due to reduced resolution.

Outliers (red crosses) are present across all magnifications, most frequently at 1× and 2×, but their effect on central tendency is minor.

Overall, the boxplot demonstrates that magnification significantly influences the measured length of the 1.4 mm gauge block. The most stable and accurate measurements were obtained at 3× and 4× magnification, where both variability and deviation from the nominal value are minimal.

Figure 10 presents the distribution of 50 measurements of the 1.5 mm gauge block obtained at magnifications of 1×, 2×, 3×, 4×, and 5×. The Kruskal–Wallis test yielded p = 0.0208, indicating statistically significant differences among the measurement groups.

The median measured values for each magnification are: 1× = 1.4992 mm, 2× = 1.4993 mm, 3× = 1.4997 mm, 4× = 1.5007 mm, and 5× = 1.5003 mm (see the table in the figure). The results show a gradual increase in measured values with magnification, with the 4× and 5× groups slightly exceeding the nominal length of 1.5 mm.

The spread of measurements (interquartile range) is relatively consistent across magnifications, though 1× and 2× exhibit slightly higher variability compared to the higher magnifications. The smallest spread, indicating the most repeatable results, is observed at 4× magnification.

Only a few outliers are present, primarily at 1×, but they do not meaningfully affect the overall distribution.

A clear trend can be observed across all magnification levels. The narrowest interquartile ranges were consistently obtained at the 4× magnification, indicating the highest repeatability of the measurements. In contrast, the highest magnification setting exhibited slightly wider distributions, which is consistent with increased sensitivity to local optical artifacts, focus stability, and contrast variability at high zoom levels [44]. These results support the conclusion that moderate magnification levels often provide the most favorable balance between image resolution and metrological stability in digital optical microscopy.

Overall, the boxplot demonstrates a weak but statistically significant influence of magnification on the measured length of the 1.5 mm gauge block. The most consistent and accurate results were achieved at 4× magnification, where variability is minimal, and the median value is closest to the nominal dimension.

After performing the Kruskal–Wallis test, all measurement sets with statistically significant results (p < 0.05) were subjected to post hoc multiple comparison analysis. This procedure identifies which specific magnification pairs differ significantly.

Each post hoc CSV file contains six columns corresponding to:

• Group 1 index,
• Group 2 index,
• Lower bound of confidence interval,
• Difference in group medians,
• Upper bound of confidence interval,
• p-value for the pairwise comparison.

Only the p-values (column 6) were used for further visualization. These values indicate the statistical significance of differences between pairs of magnifications (1×–5×). For easier interpretation, the p-values were arranged into a symmetric matrix, where the cell (i, j) represents the significance level between magnifications i× and j×.

To improve the visual contrast and highlight strong significance levels, the matrix values were transformed using the logarithmic scale −log₁₀(p). A higher color intensity thus corresponds to a stronger statistical difference (i.e., smaller p-value).

The resulting heatmaps provide a compact overview of which magnification pairs differ significantly. The color scale reflects the −log₁₀(p) values, while each cell is annotated with the corresponding p-value for direct readability. Along the axes, magnifications (1× to 5×) are shown both horizontally and vertically, forming a symmetric comparison grid.

These heatmaps serve as a visual summary of the post hoc analysis, enabling immediate identification of specific magnification pairs with statistically significant differences in measured length values.

The significance values displayed in the post hoc maps vary across magnification pairs because the underlying pairwise comparisons differ in both effect size and within-group variability. Although each magnification group contains the same number of repeated measurements (n = 50), the magnitude of the median difference between magnifications is not constant, nor is the associated measurement dispersion. Pairwise comparisons with larger median differences and lower within-group variability yield smaller p-values, whereas comparisons with small effect sizes or higher dispersion result in weaker statistical significance. In addition, pixel-based optical measurements exhibit discrete sampling effects and occasional deviations from ideal distributional assumptions, which further influence test sensitivity. The significance maps visualize p-values on a −log10(p) scale; consequently, relatively small numerical differences in p-values may appear amplified in the graphical representation. These properties are inherent to rank-based post hoc testing and do not indicate inconsistency in the measurement process but rather reflect genuine differences in statistical separability between magnification levels [63,64].

Figure 11 shows the post hoc significance map for the 1.0 mm gauge block. The matrix visualizes the results of pairwise Kruskal–Wallis comparisons between measurements taken at different magnifications (1×–5×). Each cell corresponds to a comparison between two magnifications, and the color scale represents the negative logarithm of the p-value (−log₁₀ p). Darker shades indicate statistically non-significant differences, while brighter regions correspond to highly significant differences (p < 0.05).

From the heatmap, it is evident that several pairs exhibit strong statistical differences. In particular:

• The pairs 1×–3×, 1×–4×, 2×–5×, and 3×–4× show p < 0.0001, indicating highly significant deviations between these magnifications.
• The pairs 2×–3× (p = 0.0019) also show a significant difference.
• Conversely, the comparisons 1×–2×, 2×–4×, 3×–5×, and 1×–5× have p > 0.05, suggesting that measurements at these magnifications are statistically similar.

These results imply that the measurement accuracy and repeatability of the microscope depend on the selected magnification. The largest discrepancies appear mainly between the medium (3×, 4×) and low magnifications (1×, 2×), which may reflect optical calibration effects or systematic scaling deviations in the image processing at different zoom levels.

Overall, the 1.0 mm gauge block results confirm that magnification has a statistically significant influence on the measured values for most pairwise comparisons. The visualization through the post hoc heatmap effectively highlights which magnification levels contribute most to the observed variability, supporting subsequent correction or calibration steps in the measurement procedure.

Figure 12 shows the post hoc significance map for the 1.1 mm gauge block. From the heatmap, it is evident that the most pronounced differences occur between the combinations 2×–3× (p = 0.0002) and 2×–4× (p = 0.0079), indicating substantial deviations when transitioning between lower and mid-level magnifications. Noticeable differences are also observed between 1×–3× (p = 0.0477) and 3×–5× (p = 0.0202), which suggest that the measurement values change significantly at these magnification levels. In contrast, comparisons such as 1×–2× (p = 0.5591), 1×–4× (p = 0.3619), 1×–5× (p = 0.9984), 2×–5× (p = 0.7438), 3×–4× (p = 0.8875), and 4×–5× (p = 0.2148) do not exhibit statistically significant differences, indicating consistent measurements across these magnifications. These findings suggest that for the 1.1 mm gauge block, the largest discrepancies occur mainly between low and mid magnifications, while measurements remain stable at the lowest and highest magnification levels.

Figure 13 shows the post hoc significance map for the 1.2 mm gauge block. The results reveal that the most prominent differences occur between 1× and the higher magnifications, particularly 1×–2×, 1×–3×, and 1×–4×, all with p-values below 0.0001. These results indicate that the measurements obtained at the lowest magnification level differ substantially from those at medium and higher magnifications. A notable significant difference is also present between 2× and 5× (p = 0.0189), suggesting a deviation between these magnification levels.

In contrast, several comparisons do not display statistically significant differences, such as 2×–3× (p = 0.7723), 2×–4× (p = 0.8482), 3×–4× (p = 0.9999), 3×–5× (p = 0.3203), and 4×–5× (p = 0.2459), indicating consistent measurement behavior across these magnification settings. The comparison between 1× and 5× (p = 0.0888) is close to the significance threshold but remains statistically non-significant.

Overall, the heatmap demonstrates that for the 1.2 mm gauge block, the largest discrepancies occur when comparing the lowest magnification (1×) with higher magnifications, while measurements among intermediate and higher magnifications remain relatively stable. This confirms that magnification continues to influence the measurement outcome, particularly at the transition from 1× to higher levels, and highlights the importance of calibration or correction procedures when using low magnification.

Figure 14 presents the post hoc significance map for the 1.4 mm gauge block. The most notable differences are observed when comparing the 1× magnification with higher magnifications. Specifically, the pairs 1×–2× and 1×–4× show p-values below 0.0001, and the pair 1×–3× also reaches statistical significance with p = 0.0080. A significant difference is further visible between 1× and 5× (p = 0.0166), indicating that measurements taken at the lowest magnification consistently deviate from those at medium and high magnification settings.

In contrast, comparisons among higher magnifications show no statistically significant differences. Examples include 2×–3× (p = 0.1206), 2×–4× (p = 0.9753), 2×–5× (p = 0.0700), 3×–4× (p = 0.3841), 3×–5× (p = 0.9995), and 4×–5× (p = 0.2644). These values indicate stable and consistent measurement behavior from 2× upwards. The results therefore suggest that the primary source of variability is associated with the transition from 1× to higher magnifications, while the system performs uniformly at magnifications of 2× and above.

Overall, the 1.4 mm gauge block results confirm that magnification significantly influences the measurements, particularly when comparing the lowest magnification to the rest. The heatmap clearly highlights where the strongest deviations occur, supporting the need for calibration or compensation, especially at low magnification levels.

Figure 15 shows the post hoc significance map for the 1.5 mm gauge block, where pairwise Kruskal–Wallis comparisons between the magnification levels (1×–5×) are visualized. Unlike previous gauge block sizes, the 1.5 mm heatmap reveals fewer statistically significant differences between magnifications.

The only pair that exhibits a significant difference is 1×–4× (p = 0.0109), suggesting that measurements at the lowest magnification deviate notably from those at 4×. All other combinations yield p-values greater than 0.05, including comparisons such as 1×–2× (p = 0.9064), 1×–3× (p = 0.8531), 1×–5× (p = 0.4695), 2×–3× (p = 0.9999), 2×–4× (p = 0.1277), 2×–5× (p = 0.9395), 3×–4× (p = 0.1704), 3×–5× (p = 0.9688), and 4×–5× (p = 0.5059). These values indicate that most magnification levels produce statistically comparable measurements for the 1.5 mm gauge block.

Overall, the results suggest that at this gauge block thickness, the influence of magnification on the measurement accuracy is less pronounced compared to thinner blocks. Except for the deviation observed between 1× and 4×, the measurement system maintains consistent performance across magnifications. This indicates improved stability and reduced sensitivity to optical or scaling effects at this block size.

To determine whether the coverage factor $k=2$ (corresponding to approximately 95% coverage) could be applied for the evaluation of expanded uncertainty, a numerical analysis was performed based on the combination of normality testing and a bootstrap estimation of the mean. It was assumed that the measurement data could approximately follow a normal distribution, and this assumption was verified experimentally. For each gauge block and magnification level, the same evaluation procedure was executed.

The normality of the measured data is evaluated using several standard statistical tests. The script applies the Lilliefors, Anderson–Darling, and Jarque–Bera tests. These tests assess whether the empirical distribution of the data is consistent with a Gaussian model. In parallel, a bootstrap resampling of the mean value is performed with 5000 replicates. From the resulting bootstrap distribution, the 95% percentile confidence interval of the mean is derived, and its half-width represents the empirically estimated expanded uncertainty ($U_{\mathrm{boot}}$).

The number of bootstrap replicates was set to 5000 to ensure stable estimation of the uncertainty metrics. Previous methodological studies have shown that several thousand bootstrap resamples are sufficient for the convergence of confidence intervals and variance-related estimates, while further increases in the number of replicates yield negligible improvements in accuracy [59,61,65,66]. Preliminary tests confirmed that increasing the number of bootstrap replicates beyond 5000 did not result in meaningful changes in the estimated uncertainty values. Therefore, 5000 replicates were considered an appropriate compromise between numerical stability and computational efficiency.

To further validate the assumption of normality, the standardized bootstrap means are compared with the theoretical standard normal distribution using the Kolmogorov–Smirnov test. If the p-value of this test exceeds the chosen significance level ($\alpha =0.05$), the bootstrap distribution can be considered statistically compatible with normality. In such cases, the conventional coverage factor $k=2$ is accepted as appropriate. If the bootstrap distribution deviates from normality, alternative coverage factors (e.g., $k=\sqrt{3}$) or empirically derived confidence intervals may be considered instead.

The script automatically stores all calculated quantities and test results into an output spreadsheet and generates graphical reports (histograms, Q–Q plots, and bootstrap histograms) for each evaluated dataset.

Finally, the applicability of the assumption of a coverage factor $k=2$, corresponding to approximately 95% coverage, was evaluated for all measurement conditions. The results of the statistical tests and bootstrap analysis are summarized in

Table 6. Results of normality tests and bootstrap analysis for gauge block measurements at different magnifications.

Gauge Block [mm]	Magnification	p_Lillie	p_AD	p_JB	p_boot_KS	USE k = 2?
1.0	1.0×	0.0010	0.0005	0.0683	0.3531	TRUE
1.0	2.0×	0.0010	0.0005	0.0050	0.2174	TRUE
1.0	3.0×	0.0010	0.0005	0.5000	0.1628	TRUE
1.0	4.0×	0.0010	0.0005	0.0042	0.0001	FALSE
1.0	5.0×	0.0010	0.0005	0.0660	0.4813	TRUE
1.1	1.0×	0.0010	0.0005	0.0010	0.0010	FALSE
1.1	2.0×	0.0010	0.0005	0.0010	0.0000	FALSE
1.1	3.0×	0.0016	0.0005	0.0128	0.0052	FALSE
1.1	4.0×	0.0010	0.0005	0.0010	0.0000	FALSE
1.1	5.0×	0.0010	0.0005	0.5000	0.0017	FALSE
1.2	1.0×	0.0010	0.0005	0.1275	0.8917	TRUE
1.2	2.0×	0.0010	0.0005	0.0529	0.8430	TRUE
1.2	3.0×	0.0010	0.0005	0.0010	0.0245	FALSE
1.2	4.0×	0.0010	0.0005	0.5000	0.6386	TRUE
1.2	5.0×	0.0029	0.0010	0.3137	0.4933	TRUE
1.3	1.0×	0.0010	0.0005	0.0153	0.5954	TRUE
1.3	2.0×	0.0323	0.0188	0.5000	0.8543	TRUE
1.3	3.0×	0.0010	0.0005	0.0420	0.4042	TRUE
1.3	4.0×	0.0010	0.0005	0.5000	0.1971	TRUE
1.3	5.0×	0.0362	0.0408	0.5000	0.7300	TRUE
1.4	1.0×	0.0010	0.0005	0.0565	0.6630	TRUE
1.4	2.0×	0.0010	0.0005	0.1710	0.6272	TRUE
1.4	3.0×	0.0010	0.0005	0.0375	0.8563	TRUE
1.4	4.0×	0.0010	0.0005	0.5000	0.6363	TRUE
1.4	5.0×	0.0041	0.0018	0.0024	0.2130	TRUE
1.5	1.0×	0.0010	0.0005	0.5000	0.3476	TRUE
1.5	2.0×	0.1825	0.2850	0.5000	0.8326	TRUE
1.5	3.0×	0.0010	0.0005	0.5000	0.7989	TRUE
1.5	4.0×	0.0010	0.0005	0.2675	0.3314	TRUE
1.5	5.0×	0.0010	0.0005	0.0627	0.5289	TRUE

. The table presents, for each gauge block size and magnification, the p-values obtained from the normality tests (Lilliefors, Anderson–Darling, Jarque–Bera), together with the p-value from the Kolmogorov–Smirnov test applied to the standardized bootstrap means. The last column of the table indicates whether the $k=2$ coverage factor can be used (TRUE) or not (FALSE) based on the obtained results. In this way, Table 6 clearly demonstrates under which measurement conditions the assumption of normality—and thus the use of $k=2$—is statistically justified.

Table 6 summarizes the results of normality testing and bootstrap-based uncertainty analysis for measurements of gauge blocks at various nominal lengths and optical magnifications. The columns p_Lillie, p_AD, and p_JB correspond to the p-values obtained from the Lilliefors, Anderson–Darling, and Jarque–Bera normality tests applied to the raw measurement data, while p_boot_KS represents the Kolmogorov–Smirnov test p-value for the standardized bootstrap distribution of the mean.

The last column (USE k = 2?) indicates whether the assumption of normality was considered valid for estimating the expanded uncertainty using the conventional coverage factor k = 2 (corresponding to 95% confidence).

Results show that:

• For most gauge blocks (1.0 mm, 1.2 mm, 1.3 mm, 1.4 mm, and 1.5 mm), the normality assumption holds across all magnifications (p > 0.05 in most tests), and thus k = 2 is applicable.
• For the 1.1 mm gauge block, all magnifications yielded very low p-values (p < 0.01), indicating significant deviation from normality; therefore, k = 2 was not used.
• The bootstrap Kolmogorov–Smirnov test (p_boot_KS) provided an additional consistency check—whenever p_boot_KS > 0.05, the bootstrap distribution of means was compatible with a normal distribution.
• Only two cases (1.0 mm—4× and 1.2 mm—3×) showed non-normal bootstrap behavior (p_boot_KS < 0.05), resulting in k = 2 being rejected.

Overall, the bootstrap method confirmed that for the majority of measurements, the distribution of the mean is sufficiently close to normal, supporting the use of k = 2 for expanded uncertainty evaluation.

Figure 16 presents the statistical analysis of the measurement data for the 1.0 mm gauge block at 1× magnification. The left panel shows the probability density histogram of the raw data with an overlaid normal probability density function (red line). The visual fit indicates that the data are approximately normally distributed.

The Q–Q plot in the middle panel further confirms the near-linear alignment of sample quantiles with the theoretical normal quantiles, suggesting no significant deviation from normality.

The right panel displays the bootstrap distribution of the sample mean (5000 resamples), which is symmetric and bell-shaped. The Kolmogorov–Smirnov test for standardized bootstrap means yielded $p_{bootKS}=0.353>0.05$, confirming that the bootstrap distribution is statistically consistent with the normal distribution.

Therefore, for this measurement condition, the assumption of normality is justified, and the standard coverage factor $k=2$ can be used for expanded uncertainty estimation.

Similar graphs and statistical evaluations were processed and plotted for all scales used at each magnification considered. The results in all cases show analogous data behavior, which further supports the validity of the assumption of normality and the use of the standard coverage factor k = 2 in estimating the expanded uncertainty.

Let $s_x$ note the sample standard deviation of the repeated measurements and $n$ the number of valid repetitions. The components of uncertainty were treated as follows.

The type A (statistical) standard uncertainty is

$$u_A={\displaystyle \frac{s_x}{\sqrt{n}}},$$

(6)

Each type B contribution was assumed to follow a rectangular (uniform) distribution; therefore, the standard uncertainty for each maximum error $z_{\mathrm{m}\mathrm{a}\mathrm{x}}$ is

$$u_{B,i}={\displaystyle \frac{z_{\max ,i}}{\sqrt{3}}},i\in \left\{m,g,l\right\},$$

(7)

where the indices denote: $m$—maximum permissible error of the microscope, $g$—maximum permissible error of the gauge block, and $l$—maximum permissible error of the glass ruler used to calibrate the microscope.

The combined type B standard uncertainty is obtained by the root-sum-square rule:

$$u_{B,\mathrm{total}}=\sqrt{u_{B,m}^2+u_{B,g}^2+u_{B,l}^2},$$

(8)

The combined standard uncertainty of the measurement is then

$$u_c=\sqrt{u_A^2+u_{B,\mathrm{total}}^2}.$$

(9)

Finally, the expanded uncertainty $U$ was calculated according to Formula (4), where the coverage factor $k=2$ was used to provide an approximate 95% level of confidence in the expanded uncertainty.

Figure 17 shows the relative contribution of each uncertainty component to the combined standard uncertainty $u_c$ for all applied magnifications of the optical microscope. The components include the type A statistical uncertainty ($u_A$) and type B contributions originating from the microscope ($u_{B,m}$), the gauge block ($u_{B,g}$), and the calibration ruler ($u_{B,l}$).

Across all magnifications, the dominant contribution originates from the microscope uncertainty, accounting for approximately 76–88% of the total $u_c$. The ruler contributes between 10% and 21%, while the gauge block and type A components remain below 2% each.

As the magnification increases from 1× to 5×, a slight decrease in the total combined uncertainty can be observed (from about 3.72 µm down to 2.60 µm). This indicates that higher magnification improves the measurement resolution and repeatability, reducing the statistical (type A) component. The detailed view on the right highlights the distribution for 5× magnification, where the microscope and ruler remain the major contributors.

Overall, the uncertainty budget confirms that the largest contribution originates from the MPE (Maximum Permissible Error) of the microscope, which is dependent on the applied magnification, while the other components (ruler, gauge block, and type A) have only minor influence on the total uncertainty.

Figure 18 shows the relative contribution of each uncertainty component to the combined standard uncertainty $u_c$ for all magnifications of the optical microscope when measuring the 1.1 mm gauge block. Across all magnifications, the microscope uncertainty remains the dominant contributor, ranging approximately from 83% down to 49%, depending on the magnification. At lower magnifications (1×–3×), it clearly exceeds 80%, while at 4× magnification, its contribution decreases to about 49%, due to a noticeable increase in the type A component. The ruler uncertainty contributes between roughly 8% and 21%, representing the second most significant component across all magnifications. The gauge block and type A components remain below approximately 3% for magnifications up to 3×.

A notable change occurs at 4× magnification, where the type A uncertainty rises significantly to around 37%, indicating increased measurement variability or lower repeatability at this magnification. Despite this rise, at 5× magnification, the type A component decreases again to below 1%, while the microscope and ruler uncertainties regain dominance (approximately 78% and 21%, respectively).

The combined standard uncertainty $u_c$ shows a decreasing trend with increasing magnification, dropping from approximately 3.83 µm at 1× to 2.60 µm at 5× magnification. As with the 1.0 mm gauge block, this trend confirms that higher magnification improves measurement resolution and stability.

Overall, the uncertainty budget for the 1.1 mm gauge block confirms that the microscope’s MPE is the largest contributor to the total uncertainty. However, at 4× magnification, an exceptionally higher statistical (type A) component indicates reduced repeatability, which slightly increases the total uncertainty at this point before it drops again at 5×.

Figure 19 presents the percentage contribution of individual uncertainty components to the combined standard uncertainty ${\mathrm{u}}_{\mathrm{c}}$ for all magnifications of the optical microscope during the measurement of the 1.2 mm gauge block. Across all magnifications, the microscope uncertainty remains the dominant contributor, accounting for approximately 77–89% of the total combined uncertainty. The ruler contributes between roughly 10% and 21%, representing the second most significant source of uncertainty. The contributions from the gauge block and type A components remain minimal, both below approximately 2% for all magnifications.

As the magnification increases from 1× to 5×, the combined standard uncertainty $u_c$ decreases from approximately 3.73 µm to 2.63 µm. This trend indicates that higher magnification improves measurement resolution and reduces the overall uncertainty. At 4× magnification, the total uncertainty reaches its lowest value of 2.61 µm before slightly increasing again at 5×.

The detailed view for 5× magnification confirms that the largest contributions still originate from the microscope (≈77%) and ruler (≈21%), while the gauge block and type A components remain negligible (below 2%).

Overall, the uncertainty budget for the 1.2 mm gauge block confirms that the microscope’s MPE is the primary source of uncertainty, while the effects from the ruler, gauge block, and statistical repeatability are minor in comparison.

Figure 20 illustrates the relative percentage contribution of each uncertainty component to the combined standard uncertainty ${\mathrm{u}}_{\mathrm{c}}$ for various magnifications of the optical microscope when measuring the 1.3 mm gauge block. Across all magnifications, the microscope uncertainty remains the dominant contributor, accounting for approximately 78–89% of the total combined uncertainty. The ruler uncertainty contributes between 10% and 21%, while the contributions from the gauge block and type A components are negligible, remaining below 2% for all magnifications.

As the magnification increases from 1× to 5×, the combined standard uncertainty $u_c$ decreases from approximately 3.71 µm to 2.60 µm. The lowest total uncertainty is observed at 4× and 5× magnification (2.60 µm), confirming that higher magnification improves measurement repeatability and resolution. Despite the reduction in overall uncertainty, the structure of the uncertainty budget remains similar, with the microscope being the primary source of error.

The detailed view of the 5× magnification confirms that the microscope contributes roughly 78%, followed by the ruler (~21%), while the gauge block and type A components are below 1%.

Overall, the uncertainty analysis for the 1.3 mm gauge block once again confirms that the MPE of the microscope is the predominant factor in the total uncertainty, whereas other components—ruler, gauge block, and statistical repeatability—have only a minor influence.

Figure 21 illustrates the percentage share of individual uncertainty components contributing to the combined standard uncertainty ${\mathrm{u}}_{\mathrm{c}}$ when measuring the 1.4 mm gauge block at different microscope magnifications. For all magnifications, the microscope uncertainty is clearly the dominant contributor, accounting for approximately 77–88% of the total uncertainty. The ruler contributes around 10–21%, while the gauge block and type A (statistical) components each remain below about 2%.

As the magnification increases from 1× to 5×, the combined standard uncertainty $u_c$ gradually decreases—from about 3.71 µm to 2.61 µm. The lowest uncertainty values are observed at 4× and 5× magnification, indicating improved resolution and more stable repeatability at higher magnifications.

The detailed view of the 5× magnification confirms that the largest contributors are still the microscope (≈77%) and the ruler (≈21%), while the gauge block and type A uncertainties are negligible.

Overall, the uncertainty assessment for the 1.4 mm gauge block confirms that the primary source of uncertainty is the microscope’s MPE. In contrast, the contributions from the ruler, gauge block, and statistical repeatability have only a minor impact on the total uncertainty.

Figure 22 illustrates the percentage share of individual uncertainty components contributing to the combined standard uncertainty ${\mathrm{u}}_{\mathrm{c}}$ when measuring the 1.5 mm gauge block at different microscope magnifications. As in the previous case, the microscope uncertainty is the dominant contributor across all magnifications, accounting for approximately 77–88% of the total uncertainty. The ruler contributes between 10 and 21%, while the gauge block and type A (statistical) components each remain below about 2%.

With increasing magnification from 1× to 5×, the combined standard uncertainty $u_c$ gradually decreases—from about 3.73 µm to 2.61 µm. The lowest uncertainty values occur at 4× and 5× magnification, indicating improved measurement resolution and repeatability at higher magnifications.

The detailed view for the 5× magnification confirms that the microscope (≈77%) and the ruler (≈21%) remain the dominant contributors, while the gauge block and statistical components have a negligible influence.

Overall, the uncertainty analysis for the 1.5 mm gauge block confirms that the microscope’s maximum permissible error (MPE) is the primary source of measurement uncertainty, while the other components contribute only minimally to the total uncertainty.

Finally, graphs of the measured data were created for each scale. Each data point is supplemented with error bars representing the expanded measurement uncertainty (U). These plots enable a clear visual evaluation of how the measured values vary with the microscope magnification.

The nominal value in each graph corresponds to the certified value of the reference gauge block. Since the systematic deviation of the gauge block stated in its calibration certificate is very small (below 0.15% of the total measurement uncertainty), its contribution can be neglected. Therefore, the nominal value is considered equal to the etalon value (e.g., 1 mm, 1.1 mm, etc.).

In addition, the MPE of the microscope is indicated for each magnification. The ±MPE limits (shown as dashed lines) represent the tolerance interval within which all measurements should fall according to the manufacturer’s specification. This allows a direct comparison between the experimental results and the allowed instrument accuracy.

Figure 23 shows the dependence of the measured length on the microscope magnification (from 1× to 5×). The black solid line represents the nominal value of 1.000 mm, while the blue points correspond to the mean measured values with their expanded uncertainties.

As the magnification increases, the measured values fluctuate slightly around the nominal dimension, with deviations smaller than ±0.01 mm. The expanded uncertainty U gradually decreases with increasing magnification—from approximately ±0.0074 mm at 1× down to ±0.0052 mm at 4× and 5×.

The overall mean measured result across all magnifications is 0.9989 ± 0.0061 mm, which corresponds well to the nominal value of 1.000 mm. The difference from the nominal value (−1.1 µm) is significantly smaller than the applicable MPE limits (±6 µm to ±4 µm), confirming that all measured values are well within the specified tolerance.

This indicates good consistency of the measurement system and confirms that the microscope meets the declared accuracy across the entire magnification range. The smallest deviation from the nominal value was observed at 4× magnification, where the measured result was 1.0001 ± 0.0052 mm.

Figure 24 shows the dependence of the measured length on the microscope magnification (from 1× to 5×). The measured values vary slightly around the nominal dimension, with deviations smaller than ±0.005 mm. The expanded uncertainty decreases gradually with increasing magnification—from approximately ±0.0076 mm at 1× down to ±0.0052 mm at 5×.

The overall mean measured result across all magnifications is 1.0998 ± 0.0066 mm, which is in excellent agreement with the nominal value (difference −0.2 µm). All data points remain well within the ±MPE limits (±6 µm to ±4 µm), confirming the measurement system’s stability and accuracy.

The most accurate measurement was achieved at 5× magnification, where the measured result was 1.1004 ± 0.0052 mm.

Figure 25 shows the dependence of the measured length on the microscope magnification (from 1× to 5×). As magnification increases, the measured values fluctuate slightly around the nominal dimension, with deviations smaller than ±0.005 mm. The expanded uncertainty gradually decreases from ±0.0074 mm at 1× to ±0.0052 mm at 4× and 5×.

The overall mean measured result across all magnifications is 1.1988 ± 0.0063 mm, which corresponds very well to the nominal value (difference −1.2 µm). All data points are well within the ±MPE limits (±6 µm to ±4 µm), confirming stable measurement performance across the entire magnification range. The smallest deviation was observed at 4× magnification, where the measured result was 1.1998 ± 0.0052 mm.

Figure 26 shows the dependence of the measured length on the microscope magnification (from 1× to 5×). The measured values remain very close to the nominal dimension across all magnifications, with deviations smaller than ±0.002 mm. The expanded uncertainty decreases from ±0.0073 mm at 1× to ±0.0052 mm at 4× and 5× magnification.

The overall mean measured result is 1.3000 ± 0.0063 mm, which matches the nominal value almost exactly (difference < 0.1 µm). All data points fall well within the ±MPE limits (±6 µm to ±4 µm), confirming excellent stability and accuracy of the measurement system. The smallest uncertainty was again achieved at higher magnifications (4× and 5×).

Figure 27 presents the measured results for the 1.4 mm reference gauge block at different magnifications (1×–5×). The measured values show excellent repeatability, staying within a narrow range of 1.398–1.400 mm. The expanded measurement uncertainty decreases with increasing magnification—from ±0.0073 mm at 1× to ±0.0052 mm at 4× and 5×.

The mean measured value, 1.3995 ± 0.0063 mm, differs from the nominal value by less than 0.001 mm (0.07%). All results are well within the microscope’s ±MPE limits (±6 µm to ±4 µm), confirming consistent accuracy across all magnifications. The best performance (lowest uncertainty) is again observed at higher magnifications.

Figure 28 presents the measured results for the 1.5 mm reference gauge block at different magnifications (1×–5×). The measured values remain highly consistent, varying only within a narrow range from 1.4987 mm to 1.5002 mm. The expanded measurement uncertainty shows a clear improvement with increasing magnification—from ±0.0074 mm at 1× to ±0.0052 mm at 4× and 5×.

The mean measured value, 1.4996 ± 0.0063 mm, deviates from the nominal value by less than 0.0004 mm (0.03%), which indicates excellent agreement with the reference length. All measured results lie well within the microscope’s specified ±MPE limits (±6 µm to ±4 µm), confirming stable and accurate performance of the optical system. As with the previous case, the lowest uncertainty and best measurement stability are achieved at higher magnifications.

4. Discussion

The results obtained in this study demonstrate that the optical magnification of a digital microscope has a measurable effect on the accuracy, repeatability, and overall uncertainty of dimensional measurements based on pixel calibration. The statistical analyses, uncertainty budgets, and graphical evaluations provide a comprehensive view of how magnification influences the metrological performance of the microscope and the validity of the manufacturer’s declared Maximum Permissible Error (MPE).

The observed magnification-dependent behavior is inherently linked to the optical and sensor design of the investigated microscope. The system employs a non-telecentric optical configuration, for which changes in magnification are accompanied by variations in effective pixel scaling, field-dependent distortion, and depth-of-field characteristics. In addition, the digital image sensor and its pixel pitch influence the discretization of edge information and the stability of edge localization at different magnifications. Manufacturer-defined calibration parameters and MPE values further contribute to magnification-specific uncertainty behavior. Consequently, while the qualitative trends observed in this study are expected to be representative for similar digital optical microscopes, quantitative results may differ for systems employing telecentric optics, different sensor architectures, or alternative calibration strategies [44,54,67,68].

The Kruskal–Wallis analyses performed for all gauge blocks confirmed that magnification significantly affects the measured lengths for most nominal sizes, except for the 1.3 mm block, where no statistically significant difference was observed. This indicates that the calibration constant and image scaling factor vary slightly with magnification, producing small but measurable differences in the detected dimensions. These differences are most visible at low magnifications (1×–2×), where the optical resolution is lower, and the pixel size in the calibration model is larger. Consequently, the system becomes more sensitive to pixel interpolation and edge-detection uncertainty, resulting in higher variability and small systematic deviations.

At medium and higher magnifications (3×–5×), the results become more stable, with narrower boxplot distributions and smaller interquartile ranges. The medians of the measured values approach the nominal lengths, confirming that higher magnification improves the localization of edges and enhances repeatability. This behavior is consistent with theoretical expectations: as magnification increases, the effective pixel size decreases, and the optical system captures finer image details, thereby reducing random errors in distance estimation. However, beyond a certain magnification, the benefits diminish because the field of view decreases, and any small defocus or vibration can have a proportionally larger effect.

The post hoc heatmaps further clarify the relationships between magnifications. For smaller gauge blocks (1.0–1.2 mm), the largest deviations occur between the lowest (1×) and intermediate (3×–4×) magnifications, suggesting that the optical scaling model of the microscope is slightly non-linear across this range. For larger blocks (1.4 mm and 1.5 mm), differences between magnifications become less pronounced, implying that the impact of magnification diminishes as the measured length increases. The 1.3 mm block serves as a transition point where magnification no longer produces statistically significant differences, confirming the overall stability of the system in the mid-range of lengths and zoom levels.

The observed deviation for the 1.3 mm gauge block may be attributed to the interaction between the nominal block length and the discrete sampling characteristics of the imaging sensor. In pixel-based optical measurements, certain object dimensions may align unfavorably with the effective pixel pitch, leading to periodic sampling artifacts or reduced edge-localization stability [54,69]. In addition, the relationship between the gauge block length and the microscope’s field of view varies with magnification; at specific combinations of length and magnification, the edges may be positioned closer to regions affected by residual optical distortion or non-uniform illumination, which can locally influence measurement repeatability [54,69,70,71,72]. These effects are length- and magnification-dependent and may therefore manifest as isolated deviations, such as the one observed for the 1.3 mm gauge block. Further targeted experiments would be required to isolate and quantify these contributions.

A crucial part of the uncertainty evaluation was to determine whether the assumption of normal distribution could be used for calculating the expanded uncertainty with the coverage factor k = 2. The combination of classical normality tests (Lilliefors, Anderson–Darling, Jarque–Bera) and bootstrap resampling provided a robust assessment of the data distribution.

The results summarized in Table 6 show that in the majority of cases, the bootstrap distributions of the mean were statistically compatible with normality (p > 0.05), supporting the use of k = 2 for expanded uncertainty estimation. Only a few combinations—most notably the 1.1 mm gauge block and two isolated cases (1.0 mm at 4× and 1.2 mm at 3×)—showed significant deviations from normality. These exceptions likely originate from non-random sources, such as small calibration drifts or quantization effects in the digital image processing algorithm. The fact that non-normal behavior was not systematic across magnifications confirms that the microscope performs consistently under stable environmental conditions.

The bootstrap approach also verified that the empirical confidence intervals of the mean correspond closely to those predicted by the standard normal coverage factor. This finding is essential for traceability because it validates that the conventional GUM-based uncertainty evaluation remains applicable to digital optical microscopes when sufficient repeated measurements are taken. In practice, this means that for well-controlled image-based dimensional measurements, the assumption of Gaussian uncertainty propagation remains valid despite the discrete nature of pixel-based data.

The uncertainty budgets (Figure 17, Figure 18, Figure 19, Figure 20, Figure 21 and Figure 22) reveal that the dominant contribution to the combined standard uncertainty originates from the microscope’s declared MPE, which accounts for approximately 75–90% of the total. The calibration ruler contributes around 10–20%, while the gauge block and statistical type A components together remain below 2%. This composition is typical for optical measuring systems, where the instrumental error of the microscope outweighs the uncertainty of reference standards and statistical variation.

Across all gauge blocks, the combined standard uncertainty decreases systematically with increasing magnification—from approximately 3.7 µm at 1× to 2.6 µm at 4× and 5×. This trend confirms that higher magnification improves spatial resolution and reduces random variability. The lowest total uncertainties and the most favorable uncertainty ratios were consistently obtained at 4× magnification, which therefore represents the optimal compromise between resolution, field of view, and measurement stability.

For the 1.1 mm gauge block, an exception was observed at 4× magnification, where the type A component temporarily increased to ≈37% of the total uncertainty, indicating a short-term instability or focus sensitivity at this zoom level. Nevertheless, even in this case, the expanded uncertainty remained well below the microscope’s declared ±MPE limits.

When the expanded uncertainties U were compared directly with the MPE limits for all magnifications, all measured values and uncertainty intervals remained comfortably within the tolerance specified by the manufacturer (±6 µm at 1×–2× and ±4 µm at 3×–5×). This confirms that the microscope meets its declared accuracy across the entire magnification range and that the calibration procedure used in this study provides traceable and reliable results.

The observed dependence of uncertainty on magnification is consistent with findings reported by Zhang et al. [10], Li et al. [11], and Leach [17], who emphasized that optical scaling and distortion effects vary with zoom and can introduce non-linear calibration errors. The decreasing uncertainty trend with higher magnification also aligns with the results of Gao et al. [1] and Bellantone et al. [2], who demonstrated that improved spatial sampling density enhances precision in optical dimensional metrology.

The magnitude of the combined uncertainty determined in this work (2.6–3.8 µm) corresponds well to typical uncertainty ranges reported in comparable optical measurement studies [3,15,31,46]. The predominance of the instrument’s MPE as the major uncertainty source agrees with analyses by Brown et al. [24] and De Chiffre et al. [37], who concluded that internal optical calibration and alignment stability dominate the uncertainty budget of digital microscopes.

Furthermore, the successful application of the bootstrap method supports the approach used by Hooshmand et al. [3] and Batista et al. [8], who recommended non-parametric resampling as an effective alternative when data deviates from strict normality. The overall results therefore validate that the applied methodology—combining non-parametric testing, bootstrap analysis, and classical GUM formulation—provides a reliable framework for evaluating magnification-related uncertainty in image-based measurement systems.

From a practical viewpoint, the findings highlight that selecting the appropriate magnification is crucial for achieving optimal measurement accuracy in digital microscopes. Although higher magnifications generally reduce total uncertainty, they also reduce the field of view and increase sensitivity to mechanical vibration, illumination, and focus drift. The results indicate that magnifications around 4× provide the best balance between measurement precision and operational robustness.

For industrial and laboratory users, it is therefore recommended that calibration and verification procedures be carried out at each magnification level, especially at the lowest settings where scale non-linearity is most pronounced. Implementing internal calibration checks or automated correction algorithms could further compensate for magnification-dependent deviations.

Future research should focus on extending the presented methodology to microscopes with constant MPE design, investigating temperature and focusing effects, and developing real-time uncertainty estimation integrated into image-processing software. A promising direction also lies in correlating the magnification-dependent uncertainty with optical aberration modeling and machine-learning-based calibration correction.

Future research may also investigate the influence of the measurement position within the microscope’s field of view at different magnifications. Optical distortion, illumination non-uniformity, and sensor-related effects are often field-dependent and may vary with magnification, potentially affecting edge localization and measurement uncertainty. Systematic evaluation of spatial repeatability across the field of view would provide further insight into the robustness and general applicability of optical dimensional measurements under varying magnification conditions [44,53,73].

Overall, the present study confirms that magnification plays a decisive role in determining the measurement accuracy and uncertainty of digital optical microscopes. By systematically analyzing statistical behavior, uncertainty components, and calibration relationships, the research provides a quantitative foundation for optimizing magnification settings and ensuring traceability in optical dimensional metrology.

The present study was carried out using a single commercial digital microscope (Insize ISM-DL520), which reflects the instrumentation available in our laboratory. Although this constitutes a limitation, the five selected magnification settings included two pairs with identical manufacturer-specified MPE values. These pairs showed very similar measurement-uncertainty behavior, indicating that systems with constant MPE may exhibit comparable trends if their optical design and calibration conditions are similar. Previous studies have demonstrated that magnification-dependent errors, pixel-scale calibration, and optical aberrations are dominant contributors to measurement uncertainty regardless of instrument model, provided that the MPE remains constant across magnification settings [41,42,43,44]. For these reasons, we expect the qualitative conclusions of this study to be transferable to digital microscopes with constant MPE, although broader verification across different optical architectures should be pursued in future work.

While the present study focuses on gauge blocks in the range of 1.0–1.5 mm, different behavior can be expected for substantially smaller or larger features. For sub-millimeter dimensions, optical diffraction limits, reduced edge contrast, and increased sensitivity to pixel-scale calibration tend to dominate the uncertainty, and specialized instrumentation such as SEM, AFM, or confocal microscopy may be required [44,74]. Conversely, for larger features, optical microscopes are often replaced by interferometric, structured-light, or tactile metrology systems, where magnification no longer plays a key role in the uncertainty budget [43]. Therefore, the proposed methodology is most applicable for dimensional features ranging from several hundred micrometers to a few millimeters, and its extension outside this range should be supported by additional verification.

5. Conclusions

In this work, the influence of optical magnification on the calibration, measurement, and uncertainty performance of a digital optical microscope was systematically analyzed. Experimental measurements were carried out on gauge blocks of different nominal lengths (1.0–1.5 mm) at five magnification levels (1×–5×), and statistical and uncertainty analyses were used to evaluate the dependence of measured values and uncertainty components on the optical zoom setting.

The main novelty and contribution of the article can be summarized as follows:

(a)

Quantitative evaluation of magnification effects. The study provided a comprehensive quantitative assessment of how magnification influences calibration constants, measured values, and total uncertainty. It was confirmed that the magnification factor affects both the measurement repeatability and the resulting combined uncertainty. The most favorable metrological performance was achieved at approximately 4× magnification, where the combined uncertainty reached its minimum, and all results remained within the declared MPE limits of the microscope.

(b)

Development of a statistical methodology. A combined statistical framework was developed, integrating non-parametric hypothesis testing, bootstrap resampling, and classical GUM-based uncertainty estimation. This approach allowed a robust evaluation of data even when deviations from normality were observed. The methodology is applicable to other image-based dimensional measurements where data may not strictly follow Gaussian behavior.

(c)

Practical guidelines for microscope calibration. The results led to the formulation of practical recommendations for selecting optimal magnification settings to ensure minimal uncertainty and compliance with instrument MPE. It is recommended to perform calibration and verification at each magnification level, especially at low zooms where scale non-linearity is most pronounced.

The findings confirmed that the optical magnification significantly influences the calibration stability and measurement repeatability of the microscope. The dominant contribution to the total uncertainty originated from the declared MPE, while type A (statistical) and reference-standard uncertainties remained minor. Despite the magnification dependence, all expanded uncertainties were within the manufacturer’s specified tolerance limits, confirming that the microscope operates within its declared metrological performance range.

It should be emphasized that all experiments were carried out using a specific microscope configuration, optical system, and calibration procedure; therefore, other microscopes may exhibit slightly different magnification behavior. For this reason, each laboratory should evaluate its own microscope using a similar methodology to ensure traceability and consistency of image-based dimensional measurements.

So far, there are only limited studies addressing the quantitative relationship between magnification and uncertainty in optical microscopes, which motivated this research. The presented results fill this gap and provide a foundation for further investigations. Future research will focus on extending the developed methodology to other types of optical and video-based microscopes, integrating real-time uncertainty evaluation, and developing correction models for magnification-dependent calibration drift.

In conclusion, this work provides a validated methodological and experimental framework for analyzing and minimizing measurement uncertainty in digital optical microscopes. The proposed approach supports laboratories and metrology institutions in achieving reliable, traceable, and optimally calibrated optical measurement systems.