Theories of Measurement

1
Theories of Measurement
2
Basics of Measurements

• Measurement assignment of numerals to represent
  physical properties
• Two Types of Measurements for Data
• Qualitative Non-numerical or verbally
  descriptive also have 2 types
• Nominal no order or rank ex list
• Ordinal allows for ranking but differences
  between data is meaningless ex alphabetical list
• Quantitative Numerical Ranking also have 2
  types
• Interval meaningless comparison ex calendar
• Ratio based on fixed or natural zero point ex
  weight, pressure, Kelvin

3
Definition Decibels

• dB 20 log (Gain) where Gain Voutput/ Vinput
  can also be in current or power
• Why bother? Easier math because you can add and
  subtract db instead of multiplying and dividing
• A1 V2/V1 A2 V3/V2
• Total Gain A1A2 V2/V1 V3/V2 now if
  everything was in dB
• Total Gain A1 (dB) A2 (dB)
• Calculation of Gain given dB
• dB 20 Log (output/ input)
• Output input 10dB/20

4
Decibel Example Problem 18

• Question
• An amplifier has 3 amplifier states and a 1 db
  attenuator in cascade. Assuming all impedances
  are matched, what is the overall gain if the
  amplifiers are 5, 10, 6 dB? Express your answer
  in dB and nondB form.
• Solution
• Gain 5 dB 10 dB 6 dB -1 dB 20 dB
• or
• 20 dB 20 log (Gain)
• Gain 1020/20 10

5
Variation and Error

• Variation gt caused by small errors in
  measurement process
• Error gt caused by limitation of machine
• Data will exhibit variation where you will see a
  distribution in data. You can quantify
  distribution by calculating mean, variance, and
  standard deviation
• Mean where Xi
  data point and N Total number of points
• Example data points 2,3,3,4,3 Mean Xbar (2
  3 3 4 3 ) / 5 3
• Variance
• Example Variance (2-3)2 ( 3-3) 2 (3-3)2
  (4 3)2 (3 3)2 /5 2 / 5 0.4
• Standard Deviation
• Example Standard Deviation (0.4)1/2
• Note with small populations use N-1 instead of N

6
Root Mean Square (RMS)

• RMS used in electrical circuits
• VRMS RMS value in voltage
• T time interval from t1 to t2
• V(t) time varying voltage signal
• With a sine wave

7
Three Categories of Measurement

• Direct Measurement holding a measurand up to a
  calibrated standard and comparing two ex meter
  stick
• Indirect Measurement Measuring something other
  than actual measurement this is typically done
  when direct measurement is difficult to obtain or
  is danger ex blood pressure
• Example blood pressure can be obtained using a
  catheter with pressure transducer or can be
  obtained using Korotkoff Sounds
• Neural activity of brain, direct measurement
  would be implanting of electrodes or use of
  indirect measurement of fMRI
• Null Measurement Compared calibrated source to
  an unknown measurand and adjust till one or other
  until difference is zero
• Electrical Potentiometer used in Wheatstone
  Bridge

8
Definitions of Factors that Affect Measurements

• Error normal random variation not a mistake, if
  you have a nonchanging parameter and you measure
  this repeatedly the measurement will not always
  be precisely the same but will cluster around a
  mean Xo. The deviation around Xo error term
  where you can assume your measurement is Xo as
  long is deviation is small.
• Validity Statement of how well instrument
  actually measures what it is supposed to measure
  ex youre developing a blood pressure sensor with
  a diaphram that has a strain gauge. This
  instrument is only valid if the deflection of the
  strain gauge is correlated to blood pressure
• Reliability and Repeatability
• Reliability statement of a measurements
  consistency of getting the same values of
  measurand on different trials
• Repeatibility getting the same value when
  exposed to the same stimulus

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Definitions of Factors that Affect Measurements
continued

• 4. Accuracy and Precision
• Accuracy Freedom from error, how close is a
  measurement to a standard ex. Goldman tonometer
  vs other tonometers or blood pressure cuff with
  catheter mean value of normal distribution is
  close to true value
• Precision exactness of successive measurements,
  has small standard deviations and variance under
  repeated trials

Good Precision (Sm. Std) Good Accuracy (Xi Xo)
Good Precision (Sm. Std) Bad Accuracy (Xi ltlt Xo
or Xi gtgt Xo)
Bad Precision (Large. Std) Good Accuracy (Xi
Xo)
Bad Precision (Large. Std) Bad Accuracy (Xi ltlt
Xo or Xi gtgt Xo)
Xi Where the measurement is supposed to be Xo
Mean of Data
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Example of Precision and Accuracy
Good Precision (Sm. Std) Bad Accuracy (Xi ltlt Xo
or Xi gtgt Xo)
Good Precision (Sm. Std) Good Accuracy (Xi Xo)
Bad Precision (Large. Std) Bad Accuracy (Xi ltlt
Xo or Xi gtgt Xo)
Bad Precision (Large. Std) Good Accuracy (Xi
Xo)
11
Tactics to Decrease Error on Practical
Measurements

• Make Measurements several Times
• Make Measurements on Several Instruments
• Make successive Measurements on different parts
  of instruments (different parts of ruler)

12
Definitions of Factors that Affect Measurements
cont.

• Resolution Degree to which a measurand can be
  broken into identifiable adjacent parts ex
  pictures dpi (dots per square inch)
• Another Example is the number of levels of
  resolution ex multimeter or binary data word

Less Resolution
More Resolution
3
3
2.5
2
2
1.5
1
1
Binary Resolution if you have 8 Bit that will
represent 10 V what is the resolution of the
system? Resolution 10 0 / 255 39 mV per
bit 8 bits gives you 28 256 values or 256 -1
255 segments
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Error

• Measurement Error Deviation between actual value
  of measurand and indicated value produced by
  instrument
• Categories of Error
• Theoretical Error the difference between the
  theoretical equation and the simplified math
  equation Ex Mean arterial blood pressure is
  theoretically Pbar 1/T ? t1t2 P(t) dt where
  clinically people use the first order
  approximation
• MAP Diastolic (Systolic Diastolic)/3
• Theoretical Error Pbar - MAP
• Static Error Errors that are always present even
  in unchanging system and therefore are not a
  function of time or frequency
• Reading Static Error Misreading of Digital
  display output
• Parallax Reading Error error when Not measure
  straight on (water in measuring cup
• Interpolation Error Error in estimating correct
  value
• Last Digit Bobble Error Digital display
  variations when the LSB varies between 2 values
• Environmental Static Error Temperature,
  pressure, electromagnetic fields, and radiation
  can change output ex electrical components are
  rated as industrial temperature itemp 85 to -50
  oC
• Characteristic Static Errors Residual Error that
  is not reading or environment ex zero offset,
  gain error, processing error, linearity error,
  hysteresis, repeatibility or resolution or
  manufacturing deficiences
• Quantization Error Error due to digitization of
  data and is the value between 2 levels

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Error Cont.

• Dynamic Error When a measurand is changing or is
  in motion during measurement process ex inertia
  of mechanical indicating devices during
  measurement of rapidly changing parameters ex
  analog meters or frequency, slew rate limitation
  of instrumentation
• Instrument Insertion Error Measurement process
  should not significantly alter phenomenon being
  measured ex if you are measuring body temp and
  performing laser surgery the laser will heat the
  surrounding area and not give an accurate body
  temperature another example is when you add a
  device such as a flowmeter you might add thereby
  changing length and diameter or you may add
  turbulence thus altering flow

15
Methodology to offset Measurement Error

• Procedure minimize error contributions with a
  voltmeter you want a high input impedance
  compared to rest of circuitry.
• 
  Ideally Vo R2/ (R1 R2) (V 0)

R1
V -
However when you have a ground current Ig going
through ground resistance ,you can have an
increase or decrease in voltage Vo by IgRg
R2
Vo
Ground Plan
Rg

• Solution You can use many instruments to measure
  same parameter and average results to decrease
  measurement error
• 
  

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Error Contribution Analysis

• Error Budget Analysis to determine allowable
  error to each individual component to ensure
  overall error not too high.
• Error Calculation
• Why not take just summation of the average?
  Because noise error can be positive and negative
  thus canceling thus your math calculation will
  show less error that what truly exists.
• Also need to depict standard deviation because
  need to denote spread in your data

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Operation Definitions

• To keep procedure constant person such that if
  different people do a measurement on different or
  the same instruments they will attain the same
  results
• Example of Standards
• ANSI
• ETSI
• ITU
• AAMI
• IEEE
• TIA/ EIA

18
Summary

• Define and understand how to depict system gain
  in dB and non dB format
• Define 2 Types of Measurement
• Calculate Mean, Variance and Standard Deviation
• Define 3 categories of Measurement
• Explain 5 factors that Affect Measurement
• Define Accuracy and Precision
• Define 4 types of Error
• Describe one way to avoid Error
• What is an Error Budget and how do you calculate
  Error
• What are Standards and why are they important

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Homework

• Read Chapters 3, 4, 5
• Hand in Homework Problems
• Chapter 3 Problems 16, 17, 21
• Chapter 4 Questions and Problems 5, 18, 19,
  21, 22