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. V1 V2 V3 A2 A1 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 = A1*A2 = 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 => caused by small errors in measurement process • Error => 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 you’re 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 measurement’s consistency of getting the same values of measurand on different trials • Repeatibility = getting the same value when exposed to the same stimulus

9. Xi Xi Xi Xi Xo Xo Xo Xo 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 << Xo or Xi >> Xo) Bad Precision (Large. Std) Good Accuracy (Xi ~ Xo) Bad Precision (Large. Std) Bad Accuracy (Xi << Xo or Xi >> Xo) Xi = Where the measurement is supposed to be Xo = Mean of Data

10. Example of Precision and Accuracy Good Precision (Sm. Std) Bad Accuracy (Xi << Xo or Xi >> Xo) Good Precision (Sm. Std) Good Accuracy (Xi ~ Xo) Bad Precision (Large. Std) Bad Accuracy (Xi << Xo or Xi >> 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

13. 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

14. 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

16. 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

17. 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

19. 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