Chapter VII – Experimental Uncertainty Analysis
7.2. Sources of elemental errors
All measurements include errors. There are several sources of errors: A/D converter
errors, sensitivity errors, …. These errors are usually referred to: elemental error
sources. As a consequence, the measurement of a certain variable x will include the
uncertainties due to these sources.
For the purpose of evaluating and comparing these sources of errors, they are divided
into 5 categories:
1. Data acquisition uncertainties
2. Data reduction uncertainties
3. Calibration uncertainties
4. Uncertainties due to methods
5. Other uncertainties
Random variation of the measurand, A/D
conversion uncertainties, …
Errors in interpolation, fitting,…
Errors that originate in the calibration
process.
Assumptions in the methods, instability, …
All these uncertainties are combined using the square root of the sum of squares (RSS):
k

Bx   Bi2 
 i 1 
1
2
Instrumentation and Measurements \ LK\ 2009
m

S x   S i2 
 i 1 
1
2
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Chapter VII – Experimental Uncertainty Analysis
NOTE: For the determination of random uncertainties, we need t-distribution if n < 30.
However, sometimes the input elemental random uncertainties are based on different
sample sizes. In this case a combined value for  is computed using Welch-Satterhwaite
formula:
 m 2
  Si
v x   m i 1

Si4 / vi
 
i 1







Example
One of the methods for measuring the power (P) of rotating machinery is to measure the
rotational speed and the rotational shaft torque and then calculate the power transmitted
through the shaft. The formula for the power is: P     , where  is the torque,
 (= 2 N ) is the rotational speed in radians per second, and N is the number of
revolutions per second. From the measurement of the power of a small engine, the
following information is available:
As per the manufacturer’s information, the accuracy of the torque meter is 0.7 N-m and
the accuracy of the rotational speed-meter is 5 rpm. The mean values of the measured
torque and rotational speed are 165 N-m and 3000 rpm, respectively. In repeating the
speed and torque measurements, it is found that the standard deviations of these
measurements are 4 N-m and 5 rpm, respectively.
a) Calculate the mean value of the power of the engine.
b) If the number of samples used for calculating the standard deviations of the torque
and speed are 10 and 20, respectively, calculate the standard deviation of the
power.
c) Calculate the random and systematic uncertainty of the power.
d) Calculate the total uncertainty of the power at a 95% confidence level.
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