3.1: Component Uncertainty

1. Errors are properties of all measurements
   1. \(\hat{x}_{\mathrm{gage}} = x_{\mathrm{actual}} + \epsilon\), where \(\epsilon\) arises from contributing error sources
   2. Results should be presented as value ± uncertainty range
2. Categories of measurement error
   1. Systematic error – constant offset to the true value
   2. Random errors – variations about the measurand value
   3. Comparison of types:

      Systematic		Random
      Not determined statistically		Scatter of data measurements allows statistics
      Estimated via comparison methods		Affected by repetition and resolution
      1. Calibration
      2. Comparison to other references
      3. Engineering experience

3. Measurand uncertainty analysis 
   1. Identifies the ± range for individual sensor and resulting measurand
   2. Like magnitude of a rank \( \mathbb{R} \) vector, magnitude determined by root-sum-square: \[ u_{r} = \pm \sqrt{u_{0}^{2} + u_{1}^{2} + u_{2}^{2} + \ldots + u_{J}^{2}} \]
      1. Since it's a summation → all units must match (e.g., cannot add \( \pm 0.03 \) cm and 3% of measurement)
      2. Assumes unit sensitivity where none are more important than others
      3. Built from two types: zero-order and instrument uncertainty
   3. Zero-order uncertainty
      1. Based on resolution (physical divisions of transducer stage)
      2. Variations create random scatter above/below scale value
      3. Equation: \( u_{0} = \pm 0.5 \times \text{resolution scale} \)
      4. Explains importance of significant figures in science
   4. Instrument errors
      1. Determined through calibration methods or manufacturer specs
      2. Types:
         1. Hysteresis: approaching value from increasing or decreasing rate
         2. Linearity: consistency of slope
         3. Sensitivity: impact of small and large disturbances
         4. Repeatability: reach same value over and over again?
         5. Zero-shift: positive or negative bias near the lowest numerical values
         6. Stability: consistency when independent variable is constant
         7. Thermal drift: temperature dependence of sensor
      3. Equation: \[ u_{I} = \sqrt{u_{H}^{2} + u_{S}^{2} + \ldots} \]
   5. Total resulting uncertainty: \[ u_{r} = \pm\sqrt{u_{0}^{2} + u_{I}^{2}} \]

   6. Exercise \(\PageIndex{1}\)

      A vehicle speedometer is graduated in 5-mph increments. The OEM indicates instrument accuracy of ±4% of the reading.

      If the driver claims operating the vehicle at 60 mph, what is the associated uncertainty?

      Answer

      The resolution uncertainty is half the increment size: \(u_{0} = \pm \frac{1}{2} 5 = \pm 2.5\) mph

      The instrument uncertainty is a calculation: \(u_{I} = \pm 0.04 (60) = \pm 2.4\) mph; since no other instrument uncertainties present, there is no need to find a magnitude via the root-sum-square. 

      The resulting uncertainty is the combination of both: \( u_{r} = \pm \sqrt{2.5^{2} + 2.4^{2}} = \pm 3.5 \).

                                     ... I swear officer, I could not be going anything more than 63.5 miles per hour

   7. Important distinctions (when reading words):
      1. % of the measurement → multiply result by percentage
      2. % FSO (full scale output) → use full sensor range