1. MECH 373 Instrumentation and Measurements
Lecture 17
Experimental Uncertainty Analysis (Chapter 7)
• Introduction
• Types of Errors
• Propagation of Uncertainties
• Consideration of Systematic and Random
Components of Uncertainty
• Sources of Elemental Error
• Step-by-Step Procedure for Analysis

2. Consideration of Systematic and Random Components of Uncertainty
• In detailed uncertainty analyses, it is usually desirable to keep separate track of the systematic or bias uncertainty (B) and the random uncertainty (P).
• A systematic error does not vary during experiments and is independent of the sample size, whereas, the random uncertainty depends on the sample size.
• Due to this reason it is desirable to handle them separately in the uncertainty analysis.
Random Uncertainty
• The random uncertainty is estimated using the t-distribution.
• If the variable x is measured n times, then the standard deviation of the sample can be determined from
• The mean can be determined from

3. Consideration of Systematic and Random Components of Uncertainty
• For a given confidence level, a value of t can be obtained from Table 6.6 for the degree of freedom v (v = n – 1).
• The random uncertainty for a single measurement of the variable x can be obtained from
• The random uncertainty in the mean of x is determined from
where is the estimate of the standard deviation of the mean.
• Usually Sx is determined in tests with a large sample size (n > 30).
• In addition, a confidence level of 95% is commonly used to determine the random uncertainty interval.
• For these two conditions, the value of t is 2.0.

4. Consideration of Systematic and Random Components of Uncertainty
Systematic or Bias Uncertainty
• The systematic uncertainty (Bx) in the measured variable x, remains constant if the test is repeated under the same conditions.
• The systematic uncertainties include those errors which are known but have not been eliminated from the measurement process.
• Unlike random uncertainty, which has a well-established statistical basis, if the measurement conditions remain the same, the systematic uncertainty does not change and does not lend itself to rigorous statistical analysis.
• The systematic uncertainty is the 95% confidence level estimate of the limits of a true systematic error.
• That is, for systematic uncertainty, a confidence level of 95% implies that the actual error will be less than the estimate 95% of the time an estimate is made.
• It should be noted that the systematic and random uncertainties need to be evaluated with the same confidence level to be combined (usually 95%).
• The effect of systematic and random uncertainties on the measured result is shown in the figures in the next slide.

5. Consideration of Systematic and Random Components of Uncertainty
• The sample mean of these measurements x also differs from the population mean and has an uncertainty interval of ±Px.
• The systematic and random uncertainties are combined to obtain the total uncertainty.
• For mean of x
• For single measurement of x
• It is not necessary that the systematic uncertainty interval be symmetric.
• See Examples 7.3 and 7.4

6. Example 7.3 (P187)
Load cell to measure the mass of chemical mixture during batch process. From 10 measurements (M), average mass = 750Kg. From a large number of previous measurements, standard deviation = 15Kg (t = 2 for 95% Confidence level due to large sample size. Assumed no random uncertainty.)
Assuming 95% confidence level:
(a) The standard deviation and random uncertainty of each measurement.
(b) The standard deviation and random uncertainty of the mean value of the ten measurements.

7. For (a) - For each measurement (M=1):
Example 7.3 (P187) Solution
For (a) - For each measurement (M=1):
95% Confidence level, t =2.
Sx = 15kg.
Px = tSx = 2Sx = 30Kg.
For (b) – Average value of measurement:
Mean value x = 750Kg. M = 10
Sxmean = Sx/(M)1/2 = 4.7Kg.
Pxmean = 2Sxmean = 9.4Kg

8. Example 7.4 (P187)
Ten samples (M=10) of heating value (KJ/Kg) measured by calorimeter. These are:
48530, 48980, 50210, 49860, 48560, 49540, 49270, 48850, 49320, 48680
(a) The random uncertainty of each measurement (M=1).
(b) the random uncertainty of the mean value of the measurements (M = 10).
(c) the random uncertainty of the mean value of the measurements, assuming that S was calculated on the basis of a large sample (n>30); but the same value as computed in parts (a) and (b).

9. Example 7.4 (P187) Solution xmean = ∑xi / n = 49180 KJ/Kg.
Sx = [∑(xi – x mean)2/(n – 1)]1/2 = 566.3KJ/Kg
(a) - Random uncertainty of each sample (M=1):
95% Confidence level, v = n-1 = 9, t= 2.26 from Table 6.6.
Pi = tSx = 2.26*566.3 = 1280KJ/Kg.
(b) - Random uncertainty of the mean (M=10):
Sxmean = Sx/(M)1/2
Pxmean = tSx/(M)1/2 = 2.26*566.3/(10)1/2 = 404.7KJ/Kg
(C) – Random uncertainty of the mean; assuming large sample size (M=10)
Pxmean = 2.0*566.3/(10)1/2 = 358.2KJ/Kg.

10. Example 7.5 (P188) Solution (Max possible combined value)
Mean value = xmean = ∑xi / n = KJ/Kg.
Random uncertainty of mean (95% CL) Pxmean = Kj/Kg
Random uncertainty of a single value Px = 1280 KJ/Kg.
Systematic uncertainty 1.5% of 100,000KJ/Kg (95% CL)= 1500KJ/Kg.
A, Total uncertainty on the mean value of the measurement:
Wxmean = (Bx^2 + Pxmean^2)^0.5 = (1500^ ^2)^0.5
= 1554 KJ/Kg. (3.1% of the mean value).
B, Total uncertainty of a single measurement of 49,500 KJ/Kg having 95% CL.
Px = 1280 KJ/Kg.
Wxmean = (Bx^2 + Pxmean^2)^0.5 = (1500^ ^2)^0.5
= 1972 KJ/Kg ( 4.0% of the measured value of 49,500 KJ/Kg)

11. Sources of Elemental Error
• In a typical measurement system, there are a large number of error sources.
• Each component in a chain can also contribute a number of different types of error.
• These error sources are known as elemental error sources, and each can generate either a systematic or a random error.
• Normally, there will be several elemental error sources in the measurement of each variable x, and the uncertainty in x will be a combination of the uncertainties due to these sources.
• To identify and compare measurement uncertainties, the elemental errors are grouped into five categories:
- Calibration uncertainties
Data-acquisition uncertainties – Measuring equipment
- Data-reduction uncertainties – Sampling, FFT lines and resolution.
Uncertainties due to methods – Interpolation.
- Other uncertainties – Environment, material and manufacturing etc.

12. Sources of Elemental Error
Calibration Uncertainties
• Calibration uncertainties are uncertainties that originate in the calibration process.
• Calibration minimizes resulting systematic errors, some residual errors remain, resulting in calibration uncertainty..
• Calibration errors tend to enter through these principal sources:
Uncertainty in calibration process.
Randomness in the calibration process.
Uncertainty in standards.

13. Sources of Elemental Error
Data-Acquisition Uncertainties
• All uncertainties, that arise when the actual measurements are made, are referred to as data acquisition uncertainties.
• Among errors produced are random variation of the measurand, installation effects, A/D conversion uncertainties, and uncertainties in recording or display devices.
• Some common data acquisition errors are listed below.

14. Sources of Elemental Error
Data-Reduction Uncertainties
• Data-reduction uncertainties are caused by a variety of errors and approximations that are used in the data-reduction process. For example, interpolation, curve fitting, and correlation.
• The rounding-off of the resulted values also introduces the error called truncation error.
Uncertainties due to Methods
• These are uncertainties originate from techniques or methods inherent in the measurement process
• Example of these uncertainties are assumptions or constants in the calculation routines, disturbance effects caused by instrumentation, spatial effects, and uncertainties due to instability, non-repeatability, and hysteresis of test processes.
• Since elemental uncertainties are combined by using the square root of the sum of the squares (RSS), it does not matter into which category an elemental error is placed.
• Rather, the categories are primarily a bookkeeping and diagnostic tool.

15. Sources of Elemental Error
• If one category dominates the errors, then it is the category to which corrective action should be applied first.
• To obtain the systematic and random uncertainties of a measurement, elemental systematic and random uncertainties must be combined as illustrated below:

16. Sources of Elemental Error
• The systematic uncertainties and the standard deviations are combined as
where, k is the number of elemental systematic uncertainties, and m is the number of random uncertainties in the measurement of the variable x.
• The random uncertainty for a single measurement of the variable x can then be obtained from
• The random uncertainty in the mean can be obtained from
where t is the value of the Student’s t for the degrees of freedom v and the appropriate level of confidence and M is the number of times the final value of x was measured.

17. Sources of Elemental Error
• In order to apply the t-distribution to the resulting value of S, it is necessary to have a value for v. If the sample sizes are greater than 30, then the t is only a function of the confidence level.
• In some cases, the input elemental random uncertainties are based on different samples with different sample sizes.
• If any of the samples has a size less than 30, it is necessary to find a combined value of v, as
where, vi is the degree of freedom of the individual elemental error (number of data values in the sample, minus 1), m is the number of elemental random uncertainties.
• The process of combining elemental errors is demonstrated in Examples 7.7 and 7.8.

18. Step-by-Step Procedure for Uncertainty Analysis
Following is a step-by-step procedure for estimating the overall uncertainty of measurements.
1. Define the measurement process
• This step involves reviewing test objectives, identifying all independent parameters and their nominal values, and defining the functional relationship between the independent parameters and the test results.
2. List all elemental error sources
• This step includes making a complete and exhaustive list of all possible error sources for each measured parameter.
• To identify all uncertainties and maintain clear bookkeeping, group the uncertainties into categories based on their source: calibration, data acquisition, data reduction, methods, and others.
3. Estimate the elemental errors
• In this step, the systematic uncertainties and standard deviations must be estimated.

19. Step-by-Step Procedure for Uncertainty Analysis
• If data are available to estimate the standard deviation of a parameter, or if the error is known to be random in nature, then it should be classified as random uncertainty; otherwise, classify it as systematic uncertainty.
• It is important to estimate all uncertainties to the same confidence level and for small samples, to identify the number of degrees of freedom associated with each standard deviation.
• The Table 7.1 can be used as a guideline. Examples 7.7. and 7.8 illustrate this step.

20. Step-by-Step Procedure for Uncertainty Analysis
4. Calculate the systematic and random uncertainties for each measured variable
• Systematic uncertainties and standard deviations for the variable identified in Step 1 are calculated in this step.
• Data obtained in Step 3 are used in the calculations.
5. Propagate the systematic uncertainties and standard deviations all the way to the result(s)
• In this step, propagate the systematic and random uncertainties of the measured variables to the final test result(s).
• Care should be taken so that the same confidence level is observed in all calculations.
6. Calculate the total uncertainty of the results
• In this step, combine the systematic and random uncertainties to obtain the total uncertainty of the result(s).

21. Example 7.7 (P193) Transducer uncertainty
• Measuring pressure in chemical reaction tank using high quality pressure transducer. Manufacturing specifications of transducer:
Range: ±3000KPa (3MPa).
Sensitivity: ± 0.25% of FS.
Linearity: ± 0.15% of FS.
Hysteresis: ± 0.10% of FS.
A large number of auxiliary tests been performed to check the repeatability of the pressure transducer at 1500KPa (working pressure). Sx from these test = 10KPa.
A large number of tests at the same pressure performed in data transmission system, Sx = 5KPa. (Separate test)
A/D converter – random uncertainty for 95% confidence level Px = 3KPa.
Assumed large sample size, t = 2 for 95%CL, Px =2.Sx, Sx = 3/2 = 1.5KPa.

22. Example 7.7 (P193)
• (a) - Find random uncertainty of the pressure measurement (single value): Calculating max. value.
Sx (total) = ( )1/2 = 11.3KPa.
Px = tSx = 2.0*11.3 = 22.6KPa. (t=2 for large sample size with 95% confidence level, M=1 when one measurement is taken).
(b) – Systematic uncertainty of the pressure transducer:
Sensitivity = (0.25*3000)/100 = 7.5KPa.
Linearity = (0.15*3000)/100 = 4.5KPa.
Hysteresis = (0.1*3000)/100 = 3.0KPa
Bx = ( )1/2 = 9.2KPa.
(C) – Total uncertainty at 95% confidence level:
Wx = (Bx2 + Px2)1/2 = ( )1/2 = 22.8KPa
(Maximum deviation from the true value)

23. Example 7.8 – Uncertainty in temp. sensor
Chemical process – Find temperature sensor accuracy:
Systematic errors:
Calibration uncertainty - ± 0.5C
Spatial variation - ± 2C.
Installation effect - ± 1C.
Linear relationship assumption pressure/voltage - ± 1C.
Total systematic uncertainty:
Bx = (0.5^2 + 2^2 + 1^2 + 1^2)^0.5 = 2.5C.
Random errors:
Sensor repeatability – 20 Measurements at the average value of 150C, Sx1 = 1.5C.
Data transmission system test – 10 Measurements , Sx2 = 0.5C.

24. Example 7.8 – Uncertainty in temp. sensor – contd.
Random errors:
Combined Sx = (Sx1^2 + Sx2^2)^0.5 = (1.5^ ^2)^0.5 = 1.58C.
V1 = = 19, V2 = = 9
Combined v = (1.5^2 +0.5^2) / (1.5^4/ ^4/9) = 22.9, say 23.
V = 23, 95% confidence level, t = 2.07 from Table 6.6
Random uncertainty of each temperature measurement:
M = 1,
Px = tSx = 2.07x1.58 = 3.3C.
Total uncertainty of measurement with 95%CL:
Wx = (Bx^2 + Px^2)^0.5 = (2.5^ ^2)^0.5 = 4.14C.