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Steps in Uncertainty Calculations
 Determining the uncertainty of pressure and temperature measurement devices involves a series of steps to assess and quantify the possible errors in measurements.
 Determining the uncertainty in measurements involves several systematic steps to ensure accuracy and reliability. Here’s a detailed outline of the steps involved in uncertainty calculations:
Steps in Uncertainty Calculations
1. Define the Measurement Process
 Clearly define what is being measured and the conditions under which the measurements are made.
 Specify the instruments and techniques used for the measurement.

Example of Temperature measurement process: How to Measure Temperature in Experiments with Best Methods
 Example of Pressure measurement process: Measurement of Fluid Pressure: Principles and Devices
2. Identify Sources of Uncertainty
 List all potential sources of uncertainty that might affect the measurement. These can include:
 Instrument resolution
 Calibration uncertainty
 Environmental conditions
 Measurement procedure
 Operator influence
 Drift over time
3. Quantify Individual Uncertainties
 Manufacturer Specifications: Refer to the instrument’s manual or data sheet to find the specified uncertainties.
 Calibration Data: Use uncertainties provided in the calibration certificate of the instrument.
 Experimental Data: Conduct repeated measurements and statistical analysis to determine the standard deviation (repeatability).
 Environmental Effects: Evaluate the influence of environmental conditions on the measurement.
4. Convert Uncertainties to Standard Uncertainties
 If uncertainties are not already given as standard deviations, convert them. For example:
 For a rectangular distribution (e.g., resolution), the standard uncertainty is the halfwidth divided by the square root of 3:
 For a triangular distribution, the standard uncertainty is the halfwidth divided by the square root of 6:
5. Combine Standard Uncertainties
 Combine the individual standard uncertainties using the root sum of squares (RSS) method to find the combined standard uncertainty (ucu):
6. Determine the Expanded Uncertainty
 Multiply the combined standard uncertainty by a coverage factor (k) to obtain the expanded uncertainty (U).
 The coverage factor depends on the desired confidence level (e.g., k=2k for approximately 95% confidence):
 U=k⋅uc
7. Report the Uncertainty
 Report the measurement result along with the expanded uncertainty and the confidence level.
 For example: Measured value at 95% confidence level
Example Calculation for Pressure Measurement
 Measurement Process:
 Measuring pressure using a digital pressure gauge.
 Conditions: room temperature, stable environment.
 Identify Sources of Uncertainty:
 Instrument resolution: ±0.1 kPa
 Calibration uncertainty: ±0.2 kPa
 Environmental factors: ±0.05 kPa
 Repeatability (standard deviation from repeated measurements): ±0.1 kPa
 Quantify Individual Uncertainties:
 Instrument resolution (U1)
 Calibration uncertainty (U2)
 Environmental factors (U4)
 Repeatability (U5)
 Combine Standard Uncertainties, Uc= U1 + U2,+ U2+……+Un
 Determine the Expanded Uncertainty:
Identify the Sources of Uncertainty
For both pressure and temperature measurement devices, uncertainties can arise from several sources, including:
 Instrument Resolution: The smallest change that can be detected by the instrument.
 Calibration: Uncertainties associated with the calibration process.
 Environmental Factors: Temperature, humidity, and other environmental conditions affecting the measurement.
 Drift: Changes in the instrument’s performance over time.
 Measurement Procedure: Variations introduced by the method of measurement.
 Operator Influence: Differences in how different operators use the instrument.
Quantify Each Source of Uncertainty
Each identified source of uncertainty needs to be quantified, usually in terms of a standard deviation. This can be done through:
 Manufacturer Specifications: Use the instrument’s data sheet to find specified uncertainties.
 Calibration Certificates: Refer to the calibration certificate for uncertainties associated with the calibration process.
 Repeatability Studies: Perform repeated measurements to assess the repeatability of the instrument.
3. Combine the Uncertainties
 Combine the individual uncertainties to obtain the overall uncertainty using the root sum of squares (RSS) method.
 The combined standard uncertainty ($u_{c}$) is given by:
Expanded Uncertainty
 To express the uncertainty with a higher level of confidence, calculate the expanded uncertainty (U) by multiplying the combined standard uncertainty by a coverage factor ($k$), typically 2 for a 95% confidence level.
 U=k⋅uc
Example of Pressure Measurement Device
 Instrument Resolution: ±0.1 kPa
 Calibration Uncertainty: ±0.2 kPa
 Environmental Factors: ±0.05 kPa
 Repeatability: ±0.1 kPa
 Combined Standard Uncertainty: U=2*0.25=0.5 kPa
Example of Temperature Measurement Device
 Instrument Resolution: ±0.1°C
 Calibration Uncertainty: ±0.15°C
 Environmental Factors: ±0.05°C
 Repeatability: ±0.1°C
 Combined Standard Uncertainty:U=2*0.212=0.424°C
Conclusion
 By following these steps, you can determine the uncertainty of pressure and temperature measurement devices, providing a quantifiable measure of the confidence in your measurements
Reference
 Michalski et al, Estimating uncertainty of temperature measurements