Even if the sources of error in a measurement are controlled and corrected for, there will still be uncertainty associated with it. It is important to quantify this uncertainty so that we can estimate a range in which true value of the measurement will lie with a certain level of confidence e.g. 95%.
Estimation of measurement uncertainty has three important steps
- Identify sources of uncertainty in your measurement such as repeatability, temperature, resolution etc.
- Quantify these sources to estimate standard uncertainties using statistical techniques and available data.
- Combine these standard uncertainties using root sum square and expand it by multiplying with a coverage factor (e.g., K=2) to achieve a higher confidence level, such as 95%.
Some common sources of measurement uncertainty in calibration of vernier caliper are
- Repeatability of the vernier
- Resolution of the vernier
- Uncertainty of Reference gauge blocks or ring Gauge
- Temperature differential between the vernier & gauge block or ring gauge
- Uncertainty of Coefficient of Thermal Expansion (CTE)
Repeatability of the vernier
Repeat each measurement five times; calculate mean and standard deviation (s). Calculate the standard uncertainty as below
$$U_{rep} = \frac{s}{\sqrt{n}}$$As this uncertainty is estimated through statistical means, it is type “A”.
Resolution of the Vernier
This is a type “B” uncertainty with rectangular distribution. Standard uncertainty due to resolution will be determined as follows
$$U = \frac{\frac{\text{Resolution}}{2}}{\sqrt{3}}$$Uncertainty of Reference Standard
This is also a type “B” uncertainty with normal distribution. Determine the uncertainty mentioned in the certificate of the gauge blocks or ring gauge. Divide the said value with the coverage factor (K) mentioned in the certificate, which is usually 2.
$$U_{ref} = \frac{U_{cert}}{K}$$Uncertainty due to Temperature differential
During calibration, temperature of the Vernier and the gage block are generally not the same. As a result they will experience uneven thermal expansion or contraction and this will induce an error in the measurement. We can reasonably assume, based on experience, that under laboratory conditions, the temperature differential does not exceed 0.5℃. Uncertainty contribution for this type of difference between block and vernier is calculated by the equation for rectangular distribution as below:
$$U_{temp} = \frac{L \times \Delta T \times \alpha}{\sqrt{3}}$$Where L is the Length of Gauge Block
ΔT is the temperature differential
α is the coefficient of thermal expansion
√3 is the division for rectangular distribution
Since the vernier and gauge block are made of steel, the coefficient of thermal expansion is 11.5 μm/m ℃.
Uncertainty of Coefficient of Thermal Expansion (CTE):
Vernier and gauge block are made of steel and commonly used CTE for steel is 11.5 μm/m ℃. Despite being a satisfactory value for most engineering calculations, the value of CTE is not precise, and uncertainty due to it needs to be estimated. Here we can reasonably assume that the value CTE might vary by a maximum of 10% of length per degree Celsius. The equation for calculation will be same as in last section
$$U_{temp} = \frac{L \times \Delta T \times \alpha}{\sqrt{3}}$$Combined Uncertainty
When the uncertainty from each source has been estimated combine them as follows
$$U_c = \sqrt{U_{rep}^2 + U_{res}^2 + U_{ref}^2 + U_{temp}^2 + U_{cte}^2}$$Expanded Uncertainty
The expanded uncertainty is intended to produce an interval about the result that has a high probability of containing the true value of the measurand.
$$U_E = K \times U_c$$Here K is the coverage factor and we will take the value of K=2 (95% Probability)
Example
Suppose we calibrated a vernier at 25 mm using a gauge block. We repeated the measurement thrice. Recorded measurement were as follow
24.99 mm, 25.00 mm, 24.99 mm, 25.00 mm, 25.00 mm
Mean of these readings is 24.996 mm whereas Standard deviation comes out to be 0.005477
Urep will be calculated as below (Refer Para 4.5.1)
$$U_{rep} = \frac{0.005477}{\sqrt{5}} = 0.002449\ \text{mm}$$Resolution of the Vernier was 0.01. Therefore Ures will be calculated as below (Refer Para 4.5.2)
$$U_{res} = \frac{0.01}{2\sqrt{3}} = 0.002887\ \text{mm}$$√3 is the factor of rectangular distribution.
Uncertainty of the reference gauge block given in the calibration certificate is 0.00014 mm. Coverage factor (K) given on the certificate is K=2 so Uref will be
$$U_{ref} = \frac{0.00014}{2} = 0.00007\ \text{mm}$$We can reasonably expect, based on experience that under laboratory conditions, the temperature differential between the gauge block and vernier does not exceed 0.5℃ during calibration. Since gauge blocks & Vernier are both made of steel so coefficient of thermal expansion (α) is 11.5μm/m℃. Utemp will be calculated as follows
$$U_{temp} = \frac{(0.025 \times 11.5 \times 0.5)}{\sqrt{3}} = 0.082\ \mu\text{m} \text{ or } 0.000082\ \text{mm}$$Here again √3 is the factor for rectangular distribution.
Uncertainty of CTE will be calculated as mentioned below. The maximum change in CTE can be assumed to be within 10% of length per degree celsius i.e 1.15 μm/m℃
$$U_{cte} = \frac{(0.025 \times 1.15 \times 0.5)}{\sqrt{3}} = 0.033\ \mu\text{m} \text{ or } 0.000033\ \text{mm}$$After we have estimated the each uncertainty component, the combined uncertainty will be calculated using root sum square
$$U_{c} = \sqrt{(0.002449)^2 + (0.002887)^2 + (0.00007)^2 + (0.000082)^2 + (0.000033)^2}$$ $$U_{c} = 0.0037869\ \text{mm}$$Assuming a coverage factor K=2 (95%), calculate the expanded uncertainty as follows:
$$U_{E} = 2 \times 0.0037869 = 0.007573\ \text{mm}$$Now this measurement will be expressed as 24.996 ± 0.007573 mm with a confidence level of 95%.
It can be interpreted as that we are 95% confident that true value of measurement is between 24.996 ± 0.007573 mm (or 24.9884mm – 25.0035mm).