Introduction to uncertainty analysis¶
From the great lecture on Fluke website fluke lecture on uncertainty
- True value is uknown and unpredictable - nature
- error is a measured quantity value minus a reference quantity value
- error is a physical reality, a difference between measured and true values
- error is not the same as uncertainty, error causes uncertainty
- not to be confused with production error or mistake
Uncertainty is a number that characterizes the dispersion of values attributed to a measurand, based on the information in hand (including theory, our assumptions and our measurements)
this is an interval about the measured value that we are reasonably confindent that the true value lies within. How confident? typically we would like to be confident 95% Our main goal is to estimate the uncertainty as correctly as possible. Only after that we might want to reduce it.
quality of measurement is expressed as a comparison of the measurement uncertainty to the measured value (e.g. 1 mm uncertainty in respect 1 meter or to 1 micrometer)
if uncertainty is unknown how can we judget the quality of anything? or our manufacturing? of our product? of our theory?
the objective of a measurement is ALWAYS: to determine the value of the measurand, so it begins with : define a measurand, - the definition has to be completed by defining the accuracy required
the result of the measurement is only an estimate of the value of the measurand and is only complete when it is accompanied by the statement of uncertainty of that estimate.
The first step in determining the uncertainty is to understand the theory, i.e. the relationship between the measurand and all the input quantities:
Then we need to evaluate the standard uncertainty of each input quantity
Standard uncertainty is defined as an uncertainty of the result of measurement expressed as standard deviation
Type A - evaluated by the statistical analysis of a series of observations. - can contain operator variance, repeatibility, reproducibility, environmental conditons
Type B - evaluated by other means that statistical analysis: uncertainty due to standards, resolution, data acquisition, bias corrections (thermal expansion has its own uncertainty)
For \(n\) measurements: \((x_1, x_2, x_3, ..., x_n)\) calculate the mean: $\(\bar{x} = 1/n \sum_{i=1}^{n} x_i\)$ calculate the experimental standard deviation: $\(S(x_i) = \sqrt{\sum_{i=1}{n} \frac{(x_i - x_n)^2}{n-1}}\)$ calculate experimental standard deviation of the mean: $$S(\bar{x}) = \frac{S(x_i)}{\sqrt{n}} = u(x) $$
- If your measurement is the average of a series of measurements, use standard deviation of the mean
- If it's a repeatibility study that will serve as an estimate for many tests, use a standard deviation of the separate tests
- There are also other ways to estimate type A evaluation, e.g. linear regression
68% is k = 1 under the Gaussian curve, extend it with k = 2
Type B: from theory, handbooks, etc.
- in normal distributions we used k = 1, 2, 3 to explain coverage factor
- in other distributions, we need to find out one sigma - standard deviation of it to make it similar to type A
- for each distribution we need to know how to divide from 100% confidence to 1 std: uniform distribution (digital resolution, or we have no information or little information, most conservative) the probabiliy under a rectangle is 100%, outside is zero, divide by \(\sqrt{3}\), if manufacturer does not specify information about coverage factors, use rectanglar for worst case
- U-shaped (derivative of a sine function), e.g. temperature, microwave measurements, by \(\sqrt{2}\)
sensitivity coefficients: 1. start with defining the relation of measurand to the input variables. e.g. direct relation is when we calibrate a meter with voltage source: $$ V_{disp} = V_{source}$, i.e. sensitivity coefficient is 1. 2. measure power: $\(P = i^2 R\)$, $\(\partial P/\partial i = 2 i R\)$, so the sensitivity coefficient is \(2iR\) 3. how do we use sensitivity coefficients? $\(u(P_i) = \partial P/\partial i \, u(i)\)$
We made few steps out of 8 steps: 1. defined measurand in terms of the equation, identified input variables and defined what uncertainty we want to achieve 2. identified error components or uncertainty components due to those errors 3. evaluated components by type A or B 4. identified their distributions 5. converted them to a standard deviation of the mean value or of the component 6. Now we need to combine those to a single value of our measurand uncertainty
Correlations: if quantities are unrelated they are presumably uncorrelated, if we use 10 resistors of the same value, calibrated from the same source, they might be correlated
- if uncertainties are uncorrelated use root of squared sums $\(u_c = \sqrt{u_x^2 + u_y^2 + u_z^2}\)$
- if they are 100% correlated, then use the worst case: $\(u_c = u_x + u_y + u_z\)$
- or use combinations: $\(u_c = \sqrt{u_x^2 + (u_y + u_z)^2}\)$
- now we have combined standard uncertainty - this represents one standard deviation of a normal distribution.
- note we use here a central limit theorem that different distributions all convolve to a normal distribution
-
increase the confidence interval by coverage factor of k = 2 to get approximately 95%
-
Often one soure of uncertainty, e.g. number of repeated measurements has low degrees of freedom and dominates the uncertainty estimate (not enough freedom to play random as expected)
- In this case we need to use Welch-Satterhwaite formula to determine effective degrees of freedom and use Student's t-table
GUM 8-step process paraphrased 1. Describe measured value of your measurement process 2. List the input quantities 3. Determine the uncertainty for each input quantity 4. Evaluate any covariances/correlations in input quantities 5. Calculate the measured value to report 6. Correctly combine the uncertainty components 7. Multiply the combined uncertainty by the coverage factor 8. Report the result in the proper format
Typical sources of uncertainty (not an inclusive list) 1. Definition of consider: if the component is not in your uncertainty budget there should be documentation as to why it was not considered to be signficant - no way to forget 2. calibration uncertainty - always there, either you do calibration or use manufacturer's calibration 3. instrument long stability - no instrument works the same for a long time, things get rusty 4. Misc. random elements - look around, Murphy is watching 5. Resolution - always there 6. Environment - always there, in multiple forms: temperature, pressure, humidity, dust, table water level, etc. 7. Instrument bias - no instrument is perfect 8. Add more - that is a chance you missed something important that will make the measurement useless at the end
The course and the 8-step provides the framework for assessing uncertainty, but ** it cannot substitute for critical thinking, intelectual honesty, and professional skill ** the evaluation of uncertainty is neither a routine task not a purely mathematical one: it depends on detailed knowledge of the nature of the measurand (pressure, temperature, wind velocity, surface roughness, stresses) and the measurement process. The quality and utility of the uncertainty quoated for a result of measurement therefore ultimately depend on the undertsanding, critical analysis, and integrity of those who contribute to the assignment of its value.
from IPython.display import display
# JUPYTER NOTEBOOK: Surface Roughness Measurement with GUM-Based Uncertainty Analysis
# Author: Alex Liberzon and Perplexity Pro
# Course: Metrology for Mechanical Engineering
# Purpose: Single notebook implementing surface roughness measurement and full GUM-based uncertainty analysis
import numpy as np
import pandas as pd
from scipy import stats
import matplotlib.pyplot as plt
plt.style.use('seaborn-v0_8')
np.random.seed(42)
print('=' * 100)
print('SURFACE ROUGHNESS MEASUREMENT WITH GUM-BASED UNCERTAINTY ANALYSIS') # For reproducibility
print('A Comprehensive Example for Mechanical Engineering Students')
print('=' * 100)
scenario = {'Objective': 'Measure average surface roughness (Ra) of a machined steel surface', 'Instrument': 'Stylus profilometer (contact method)', 'Evaluation length (L)': '4.0 mm', 'Cutoff wavelength (λc)': '0.8 mm', 'Number of measurements (n)': 10, 'Environment': '20±2°C, 45±10% RH'}
print('\nMEASUREMENT SCENARIO:')
for k, _v in scenario.items():
# Measurement scenario metadata
print(f'- {k}: {_v}')
==================================================================================================== SURFACE ROUGHNESS MEASUREMENT WITH GUM-BASED UNCERTAINTY ANALYSIS A Comprehensive Example for Mechanical Engineering Students ==================================================================================================== MEASUREMENT SCENARIO: - Objective: Measure average surface roughness (Ra) of a machined steel surface - Instrument: Stylus profilometer (contact method) - Evaluation length (L): 4.0 mm - Cutoff wavelength (λc): 0.8 mm - Number of measurements (n): 10 - Environment: 20±2°C, 45±10% RH
# Measurement data: 10 repeated Ra measurements (in micrometers, μm)
measurements = np.array([1.58, 1.62, 1.59, 1.61, 1.64, 1.57, 1.60, 1.63, 1.58, 1.61])
n = len(measurements)
xbar = np.mean(measurements)
s = np.std(measurements, ddof=1) # sample standard deviation
sem = s / np.sqrt(n)
print("1) MEASUREMENT DATA AND BASIC STATISTICS")
print("-" * 60)
print(f"n = {n}")
print(f"Measurements (μm): {measurements}")
print(f"Mean (x̄): {xbar:.4f} μm")
print(f"Standard deviation (s): {s:.4f} μm")
print(f"Standard error of mean (SEM): {sem:.4f} μm")
print(f"Range: {np.min(measurements):.2f} – {np.max(measurements):.2f} μm")
1) MEASUREMENT DATA AND BASIC STATISTICS ------------------------------------------------------------ n = 10 Measurements (μm): [1.58 1.62 1.59 1.61 1.64 1.57 1.6 1.63 1.58 1.61] Mean (x̄): 1.6030 μm Standard deviation (s): 0.0231 μm Standard error of mean (SEM): 0.0073 μm Range: 1.57 – 1.64 μm
# Type A uncertainty: u_A = s / sqrt(n)
u_A = s / np.sqrt(n)
nu_A = n - 1 # degrees of freedom for Type A
print("2) TYPE A UNCERTAINTY (REPEATABILITY)")
print("-" * 60)
print("Formula: u_A = s / √n")
print(f"Calculation: u_A = {s:.4f} / √{n} = {u_A:.5f} μm")
print(f"Degrees of freedom (ν_A) = {nu_A}")
2) TYPE A UNCERTAINTY (REPEATABILITY) ------------------------------------------------------------ Formula: u_A = s / √n Calculation: u_A = 0.0231 / √10 = 0.00731 μm Degrees of freedom (ν_A) = 9
# Type B uncertainties: define sources and convert to standard uncertainties
# For each source: stated ±value (μm), distribution (normal or rectangular), divisor
# Normal distribution with k=2 → standard uncertainty = stated / 2
# Rectangular distribution → standard uncertainty = stated / √3
uncertainty_sources = {'Instrument calibration': {'value': 0.025, 'distribution': 'normal', 'divisor': 2.0, 'description': 'Calibration certificate (k=2)'}, 'Stylus tip radius': {'value': 0.02, 'distribution': 'rectangular', 'divisor': np.sqrt(3), 'description': 'Stylus geometry filtering effect'}, 'Measurement force': {'value': 0.015, 'distribution': 'rectangular', 'divisor': np.sqrt(3), 'description': 'Contact force variation'}, 'Environmental vibration': {'value': 0.012, 'distribution': 'rectangular', 'divisor': np.sqrt(3), 'description': 'Floor/acoustic disturbances'}, 'Temperature variation': {'value': 0.008, 'distribution': 'rectangular', 'divisor': np.sqrt(3), 'description': 'Thermal expansion effects'}, 'Digital filtering': {'value': 0.01, 'distribution': 'rectangular', 'divisor': np.sqrt(3), 'description': 'Algorithmic filtering uncertainty'}}
type_b_std = {}
print('3) TYPE B UNCERTAINTY SOURCES')
print('-' * 60)
for _src, _p in uncertainty_sources.items():
u_std = _p['value'] / _p['divisor']
type_b_std[_src] = u_std
print(f"{_src}: ±{_p['value']:.3f} μm | {_p['distribution']} | Divisor = {_p['divisor']:.3f} → u = {u_std:.5f} μm ({_p['description']})")
u_B_sq = sum((u ** 2 for u in type_b_std.values()))
u_B = np.sqrt(u_B_sq)
# Combine Type B components (assumed independent)
print(f'\nCombined Type B uncertainty: u_B = √(Σ u_i²) = {u_B:.5f} μm')
3) TYPE B UNCERTAINTY SOURCES ------------------------------------------------------------ Instrument calibration: ±0.025 μm | normal | Divisor = 2.000 → u = 0.01250 μm (Calibration certificate (k=2)) Stylus tip radius: ±0.020 μm | rectangular | Divisor = 1.732 → u = 0.01155 μm (Stylus geometry filtering effect) Measurement force: ±0.015 μm | rectangular | Divisor = 1.732 → u = 0.00866 μm (Contact force variation) Environmental vibration: ±0.012 μm | rectangular | Divisor = 1.732 → u = 0.00693 μm (Floor/acoustic disturbances) Temperature variation: ±0.008 μm | rectangular | Divisor = 1.732 → u = 0.00462 μm (Thermal expansion effects) Digital filtering: ±0.010 μm | rectangular | Divisor = 1.732 → u = 0.00577 μm (Algorithmic filtering uncertainty) Combined Type B uncertainty: u_B = √(Σ u_i²) = 0.02162 μm
# Combined standard uncertainty: u_c = √(u_A² + u_B²)
u_c = np.sqrt(u_A ** 2 + u_B ** 2)
nu_eff = u_c ** 4 / (u_A ** 4 / nu_A)
# Welch-Satterthwaite degrees of freedom (approximation):
# Type B often treated as large ν (>> 30), so ν_eff ≈ dominated by Type A
confidence = 0.95 # since u_B treated as large ν
alpha = 1 - confidence
k_1 = 2.0 if nu_eff >= 30 else stats.t.ppf(1 - alpha / 2, df=nu_eff)
U = k_1 * u_c
# Coverage factor k: for ν_eff >= ~30, k ≈ 2; else use t-distribution
print('4) COMBINED STANDARD UNCERTAINTY AND EXPANDED UNCERTAINTY')
print('-' * 60)
print(f'Combined standard uncertainty: u_c = √(u_A² + u_B²) = {u_c:.5f} μm')
print(f'Effective degrees of freedom (ν_eff) ≈ {nu_eff:.2f}')
print(f'Coverage factor (k) for {int(confidence * 100)}% confidence: {k_1:.2f}')
print(f'Expanded uncertainty: U = k × u_c = {U:.4f} μm')
print(f'\nFINAL RESULT: Ra = {xbar:.3f} ± {U:.3f} μm (k = {k_1:.1f}, {int(confidence * 100)}% confidence)')
print(f'Relative expanded uncertainty: {100 * U / xbar:.1f}%')
4) COMBINED STANDARD UNCERTAINTY AND EXPANDED UNCERTAINTY ------------------------------------------------------------ Combined standard uncertainty: u_c = √(u_A² + u_B²) = 0.02282 μm Effective degrees of freedom (ν_eff) ≈ 854.29 Coverage factor (k) for 95% confidence: 2.00 Expanded uncertainty: U = k × u_c = 0.0456 μm FINAL RESULT: Ra = 1.603 ± 0.046 μm (k = 2.0, 95% confidence) Relative expanded uncertainty: 2.8%
variance_total = u_c ** 2
rows = []
rows.append({'Uncertainty Source': 'Repeatability (Type A)', 'Type': 'A', 'Distribution': 'Normal', 'Standard Uncertainty (μm)': u_A, 'Contribution (%)': u_A ** 2 / variance_total * 100})
for _src, u in type_b_std.items():
rows.append({'Uncertainty Source': _src, 'Type': 'B', 'Distribution': uncertainty_sources[_src]['distribution'].title(), 'Standard Uncertainty (μm)': u, 'Contribution (%)': u ** 2 / variance_total * 100})
budget_df = pd.DataFrame(rows)
budget_df['Standard Uncertainty (μm)'] = budget_df['Standard Uncertainty (μm)'].round(6)
budget_df['Contribution (%)'] = budget_df['Contribution (%)'].round(1)
print('5) UNCERTAINTY BUDGET TABLE')
print('-' * 60)
display(budget_df)
budget_df.to_csv('uncertainty_budget.csv', index=False)
summary_df = pd.DataFrame({'Parameter': ['Mean Ra', 'Std deviation', 'Type A', 'Type B', 'Combined (u_c)', 'Coverage k', 'Expanded (U)', 'Relative U %'], 'Value': [xbar, s, u_A, u_B, u_c, k_1, U, 100 * U / xbar], 'Unit': ['μm', 'μm', 'μm', 'μm', 'μm', '-', 'μm', '%']})
summary_df.to_csv('surface_roughness_uncertainty_analysis.csv', index=False)
print('\nSaved:')
print('- uncertainty_budget.csv')
print('- surface_roughness_uncertainty_analysis.csv')
5) UNCERTAINTY BUDGET TABLE ------------------------------------------------------------ Saved: - uncertainty_budget.csv - surface_roughness_uncertainty_analysis.csv
| Uncertainty Source | Type | Distribution | Standard Uncertainty (μm) | Contribution (%) | |
|---|---|---|---|---|---|
| 0 | Repeatability (Type A) | A | Normal | 0.007311 | 10.3 |
| 1 | Instrument calibration | B | Normal | 0.012500 | 30.0 |
| 2 | Stylus tip radius | B | Rectangular | 0.011547 | 25.6 |
| 3 | Measurement force | B | Rectangular | 0.008660 | 14.4 |
| 4 | Environmental vibration | B | Rectangular | 0.006928 | 9.2 |
| 5 | Temperature variation | B | Rectangular | 0.004619 | 4.1 |
| 6 | Digital filtering | B | Rectangular | 0.005774 | 6.4 |
# Horizontal bar chart of contributions
_fig, ax = plt.subplots(figsize=(8, 5))
colors = ['C0' if t == 'A' else 'C1' for t in budget_df['Type']]
ax.barh(budget_df['Uncertainty Source'], budget_df['Contribution (%)'], color=colors)
ax.set_xlabel('Contribution to total variance (%)')
ax.set_title('Uncertainty Budget for Surface Roughness Measurement')
for i, _v in enumerate(budget_df['Contribution (%)']):
ax.text(_v + 0.5, i, f'{_v:.1f}%', va='center')
ax.grid(True, axis='x', alpha=0.3)
plt.tight_layout()
plt.show()
print('6) SURFACE PROFILE SIMULATION AND Ra CALCULATION')
print('-' * 60)
L = 4.0
dx = 0.001
x = np.arange(0, L + dx, dx)
N = len(x)
form = 0.5 * np.sin(2 * np.pi * x / L)
waviness = 0.3 * np.sin(2 * np.pi * x / 0.8) + 0.2 * np.cos(2 * np.pi * x / 0.6)
rough = 0.8 * np.random.normal(0, 0.4, N)
rough = rough + 0.6 * np.sin(2 * np.pi * x / 0.05) * np.exp(-x / 2.0)
profile = form + waviness + rough
lc = 0.8
window_pts = int(max(3, round(lc / dx)))
kernel = np.ones(window_pts) / window_pts
profile_lp = np.convolve(profile, kernel, mode='same')
roughness_profile = profile - profile_lp
roughness_profile = roughness_profile - np.mean(roughness_profile)
Ra_sim = np.mean(np.abs(roughness_profile))
Rq = np.sqrt(np.mean(roughness_profile ** 2))
Rt = np.max(roughness_profile) - np.min(roughness_profile)
skew = stats.skew(roughness_profile)
kurt = stats.kurtosis(roughness_profile)
print(f'Sampling interval: {dx * 1000:.0f} μm | Points: {N} | λc: {lc} mm')
print(f'Simulated Ra: {Ra_sim:.3f} μm')
print(f'Rq: {Rq:.3f} μm | Rt: {Rt:.3f} μm | Skewness: {skew:.3f} | Kurtosis: {kurt:.3f}')
_fig, _axs = plt.subplots(3, 1, figsize=(10, 10), constrained_layout=True)
_axs[0].plot(x, profile, lw=1, label='Raw profile')
_axs[0].plot(x, profile_lp, lw=1, alpha=0.8, label='Low-pass (form+waviness)')
_axs[0].set_title('Raw Surface Profile and Low-Pass Component')
_axs[0].set_xlabel('Position x (mm)')
_axs[0].set_ylabel('Height (μm)')
_axs[0].grid(True, alpha=0.3)
_axs[0].legend()
_axs[1].plot(x, roughness_profile, color='C3', lw=1)
_axs[1].axhline(0, color='k', lw=0.8, alpha=0.6)
_axs[1].set_title(f'Filtered Roughness Profile (Ra ≈ {Ra_sim:.3f} μm)')
_axs[1].set_xlabel('Position x (mm)')
_axs[1].set_ylabel('Height (μm)')
_axs[1].grid(True, alpha=0.3)
_axs[2].hist(roughness_profile, bins=60, color='C2', alpha=0.9)
_axs[2].set_title('Height Distribution of Roughness Profile')
_axs[2].set_xlabel('Height (μm)')
_axs[2].set_ylabel('Count')
_axs[2].grid(True, axis='y', alpha=0.3)
plt.show()
6) SURFACE PROFILE SIMULATION AND Ra CALCULATION ------------------------------------------------------------ Sampling interval: 1 μm | Points: 4002 | λc: 0.8 mm Simulated Ra: 0.367 μm Rq: 0.458 μm | Rt: 3.395 μm | Skewness: 0.091 | Kurtosis: -0.108
print('7) SENSITIVITY ANALYSIS')
print('-' * 60)
n_list = np.array([3, 5, 10, 15, 20, 30])
uA_list = s / np.sqrt(n_list)
for n_i, uA_i in zip(n_list, uA_list):
print(f'n = {n_i:2d} → u_A = {uA_i:.5f} μm')
cal_list = np.array([0.01, 0.015, 0.02, 0.025, 0.03, 0.035])
U_list = []
for cal in cal_list:
u_cal = cal / 2
u_B_temp_sq = u_cal ** 2 + sum((u ** 2 for src, u in type_b_std.items() if src != 'Instrument calibration'))
u_B_temp = np.sqrt(u_B_temp_sq)
u_c_temp = np.sqrt(u_A ** 2 + u_B_temp ** 2)
U_temp = k_1 * u_c_temp
U_list.append(U_temp)
print('\nCalibration uncertainty vs. expanded uncertainty (all else constant):')
for cal, Uv in zip(cal_list, U_list):
print(f'±{cal:.3f} μm → U = {Uv:.4f} μm')
_fig, _axs = plt.subplots(1, 2, figsize=(12, 4))
_axs[0].plot(n_list, uA_list, marker='o')
_axs[0].set_title('Effect of Number of Measurements on u_A')
_axs[0].set_xlabel('Number of measurements (n)')
_axs[0].set_ylabel('u_A (μm)')
_axs[0].grid(True, alpha=0.3)
_axs[1].plot(cal_list, U_list, marker='s', color='C3')
_axs[1].set_title('Effect of Calibration Uncertainty on U')
_axs[1].set_xlabel('Calibration uncertainty (± μm)')
_axs[1].set_ylabel('Expanded uncertainty U (μm)')
_axs[1].grid(True, alpha=0.3)
plt.tight_layout()
plt.show()
7) SENSITIVITY ANALYSIS ------------------------------------------------------------ n = 3 → u_A = 0.01335 μm n = 5 → u_A = 0.01034 μm n = 10 → u_A = 0.00731 μm n = 15 → u_A = 0.00597 μm n = 20 → u_A = 0.00517 μm n = 30 → u_A = 0.00422 μm Calibration uncertainty vs. expanded uncertainty (all else constant): ±0.010 μm → U = 0.0395 μm ±0.015 μm → U = 0.0410 μm ±0.020 μm → U = 0.0431 μm ±0.025 μm → U = 0.0456 μm ±0.030 μm → U = 0.0486 μm ±0.035 μm → U = 0.0518 μm
print('8) SUMMARY AND KEY LEARNING POINTS')
print('-' * 60)
print(f'Final result: Ra = {xbar:.3f} ± {U:.3f} μm (k = {k_1:.1f}, 95% confidence)')
print(f'Relative expanded uncertainty: {100 * U / xbar:.1f}%')
top = budget_df.sort_values('Contribution (%)', ascending=False)
print('\nTop contributors to uncertainty:')
print(top[['Uncertainty Source', 'Contribution (%)']].head(5).to_string(index=False))
print('\nKey learning points:')
points = ['GUM provides a systematic framework for uncertainty evaluation.', 'Type A uncertainty captures random repeatability (u_A = s/√n).', 'Type B uncertainty aggregates systematic effects from known sources.', 'Combine independent components using root-sum-of-squares: u_c = √(u_A² + u_B²).', 'Use coverage factor k (e.g., ~2 for 95%) to report expanded uncertainty U = k × u_c.', 'Focus improvement efforts on the largest contributors in the uncertainty budget.', 'Surface roughness measurement is sensitive to calibration, stylus geometry, and force control.', 'Always report the coverage factor and confidence level along with the result.']
for _p in points:
print(f'- {_p}')
8) SUMMARY AND KEY LEARNING POINTS
------------------------------------------------------------
Final result: Ra = 1.603 ± 0.046 μm (k = 2.0, 95% confidence)
Relative expanded uncertainty: 2.8%
Top contributors to uncertainty:
Uncertainty Source Contribution (%)
Instrument calibration 30.0
Stylus tip radius 25.6
Measurement force 14.4
Repeatability (Type A) 10.3
Environmental vibration 9.2
Key learning points:
- GUM provides a systematic framework for uncertainty evaluation.
- Type A uncertainty captures random repeatability (u_A = s/√n).
- Type B uncertainty aggregates systematic effects from known sources.
- Combine independent components using root-sum-of-squares: u_c = √(u_A² + u_B²).
- Use coverage factor k (e.g., ~2 for 95%) to report expanded uncertainty U = k × u_c.
- Focus improvement efforts on the largest contributors in the uncertainty budget.
- Surface roughness measurement is sensitive to calibration, stylus geometry, and force control.
- Always report the coverage factor and confidence level along with the result.