Measurement Uncertainty - Error & Uncertainty Measurement - - PowerPoint PPT Presentation

Instrumentation (and

Process Control)

Fall 1393 Bonab University

Error & Uncertainty

Measurement Uncertainty - Error

• Measurement errors are impossible to avoid
• We can minimize their magnitude by
• Good measurement system design
• Appropriate analysis and processing of measurement data
• All error sources: How to eliminate or reduce their magnitude
• Errors:
• Arise during the measurement process *
• Arise due to later corruption of the measurement signal (by induced noise during transfer of the signal)
• In any measurement system it’s important to:
• Reduce errors to the minimum possible level
• quantify the maximum remaining error that may exist in output reading
• What if system final output is calculated by combining together two or more measurements?
• How each separate measurement error be combined  best estimate of the final output error
• Error main categories:
• Systematic
• Random

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Error & Uncertainty

Measurement Uncertainty - Error

• Systematic:
• Describe errors in the output readings that are consistently on one side of the correct

reading, that is, either all errors are positive or are all negative

• System disturbance during measurement
• The effect of environmental changes
• Bent needles, use of uncalibrated instruments, drift, poor cabling, …
• The remaining is quantified by the quoted accuracy
• Random:
• (precision errors) are perturbations of the measurement in either side of the true value

caused by random and unpredictable effects, such that positive errors and negative errors

• ccur in approximately equal numbers
• mainly small, but large perturbations occur from time to time
• Human observation of analog device + interpolation
• Electrical noise
• Can be largely removed by: many measurements  averaging or other statistical techniques
• The best way is to express them in probabilistic terms (say, 95% CI)

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Error & Uncertainty

Sources of Systematic Error

• Disturbance of the measured system by the act of measurement
• Mercury-in-glass thermometer
• Orifice plate
• General rule: the process of measurement always disturbs

system being measured

• Accurate understanding of the mechanisms of

system disturbance is needed to minimize it

• Case: Electric circuits:
• The´venin’s theorem *
• Rm acts as a shunt
• Rm increase  the ratio = 1
• Practical issues (increasing moving-coil instrument’s Rm)
• Solve: changing the spring constant
• Ruggedness changes, and needs better friction
• So, usually improving one aspect  introduce another problem
• Using active devices improves this limit
• Case: measuring instrument in a bridge circuit

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Error & Uncertainty

Sources of Systematic Error

• Example:

R1 = 400 O; R2 = 600 O; R3 = 1000 O; R4 = 500 O; R5 = 1000 O The voltage across AB is measured by a voltmeter whose internal resistance is 9500 O. What is the measurement error caused by the resistance of the measuring instrument?

• Solution:

RAB=500  measurement error = EO – Em = EO(1-9500/10000) = 0.05 EO  5%

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Error & Uncertainty

Sources of Systematic Error

• Environmental disturbances (modifying inputs)
• static and dynamic characteristics specified for measuring instruments are only valid for

particular environmental conditions

• These specified conditions must be reproduced during calibration
• Its magnitude quantified by:
• sensitivity drift
• zero drift (both included in the specifications)
• Env. Disturbance is difficult to determine
• Example: A small closed box (0.1 kg)  scale says 1kg

(a) a 0.9 kg rat in the box (real input) (b) an empty box with a 0.9 kg bias on the scale due to a temperature change (environmental input) (c) a 0.4 kg mouse in the box together with a 0.5 kg bias (real þ environmental inputs)

• Thus, the magnitude of any environmental input must be measured before the value of the

measured quantity (the real input) can be determined from the output reading of an instrument

• Designers’ choice:
• Reduce the susceptibility of measuring instruments to environmental inputs
• Quantify the effects of environmental inputs and correct for them in the instrument output reading

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Error & Uncertainty

Sources of Systematic Error

• Changes in characteristics due to wear in instrument components (with time)
• Systematic errors can frequently develop over a period of time because of wear in

instrument components

• Recalibration often provides a full solution
• Resistance of connecting leads
• Example: a resistance thermometer
• Often thermometer is separated by 100 meters (20-gauge copper wire is 7 Ω)
• Also: a temperature coefficient of 1 mΩ/oC
• Care:
• Cross section (resistance)
• Route (not to pick up noise)

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Error & Uncertainty

Reduction of Systematic Errors

• Prerequisite : a complete analysis of the measurement system that identifies all

sources of error

• Simple faults: bent meter needles, poor cabling practices…
• other error sources require more detailed analysis and treatment
• Careful Instrument Design
• Reducing the sensitivity (strain gauge to temperature)  cost
• Calibration
• All instruments suffer from drift in their characteristics  it depends on:
• environmental conditions
• Frequency of use
• More frequent calibration = lower drift-related error
• Method of Opposing Inputs
• compensates : effect of an environmental input by introducing an equal and opposite

environmental input that cancels it out

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Error & Uncertainty

Reduction of Systematic Errors

• Method of Opposing Inputs
• Example:
• If the coil resistance Rcoil is sensitive to temperature,

environmental input ( temperature change) will alter the value of the coil current for a given applied voltage  alter the pointer output reading

• Compensation: introducing a compensating resistance Rcomp

where Rcomp has a temperature coefficient equal in magnitude but opposite in sign to that of the coil

• High-Gain Feedback
• Unknown voltage Ei is applied to
• A motor of torque constant Km
• Resistance spring constant Ks
• Effect of environment on motor/spring = Dm/DS

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Error & Uncertainty

Reduction of Systematic Errors

• High-Gain Feedback
• No environment input: displacement Xo = KmKsEi

, but changes with environment

• But if we close the loop:
• Adding amplifier: Ka
• Feedback device: Kf

 high Ka 

• Only Kf !  we have to be concerned only with Df
• Signal Filtering
• corruption of reading by periodic noise
• often at a frequency of 50 Hz caused by pickup through the close proximity to apparatus or

current-carrying cables

• High frequency noise (mechanical oscillation/vibration)
• Appropriate filter (LP

, BP , BS) reduces noise amplitude

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Error & Uncertainty

Reduction of Systematic Errors

• Signal Filtering
• Example: passive RC LP filter
• Manual Correction of Output Reading
• Errors due to
• system disturbance during the act of measurement
• Environmental changes
• a good measurement technician reduce errors
• by calculating the effect of such systematic errors
• making appropriate correction to readings
• Not easy (needs all disturbances quantified)
• Intelligent Instruments
• Contain extra sensors that measure the value of environmental inputs
• Automatically compensate the value of the output reading
• ability to deal very effectively with systematic errors (explained later)

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Error & Uncertainty

Quantification of Systematic Errors

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Error & Uncertainty

• Do all practical steps have been taken to eliminate or reduce the magnitude of

systematic errors?  quantify the maximum likely systematic error

• Quantification of Individual Systematic Error Components
• first complication: exact value for a component = ?  use best estimate:
• Environmental condition errors
• Environment effect?
• Assume midpoint environmental conditions
• specify maximum measurement error as ±x% of the output reading
• If fluctuations occur over a short period of time (random draughts of hot or cold air) this is a

rather a random error

• Calibration errors
• The maximum error just before the instrument is due for recalibration becomes the

basis for estimating the maximum likely error

Quantification of Systematic Errors

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Error & Uncertainty

• Calibration errors
• Example (a pressure transducer ):
• Recalibration frequency: once the measurement error has grown to +1% of the full-scale
• Range: 0 to 10 bar
• How can its inaccuracy be expressed in the form of a ±x% error in the output reading?
• Solution:
• Just before recalibration: error grown to +0.1 bar (1% of 10 bar)
• Half this maximum error, 0.05 bar, should be subtracted from all measurements
• Error:
• just after calibration: -0.05 bar ( -0.5% of FSR)
• just before the next recalibration: +0.05 bar (+0.5% of FSR)
• Inaccuracy due to calibration error: ±0.05% of FSR
• System disturbance (as well as loading) errors
• Maximum likely error = 2x (worst-case system loading)  Likely error: ±x  ±y% FSD

Quantification of Systematic Errors

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Error & Uncertainty

• Calculation of Overall Systematic Error
• Total systemic error: often composed of several separate components
• measurement system loading
• Environmental factors
• Calibration errors
• worst-case prediction of maximum error: simply add up each separate systematic error
• Example: 3 components of systematic error with a magnitude of ±1% each, a worst-case prediction

error: sum of the separate errors = ±3%

• However, it is very unlikely that all components be at their max/min simultaneously
• Usual course of action: combine separate sources root-sum-squares method
• n components:
• Warning: manufacturers data sheets  measurement uncertainty/inaccuracy = best

estimate (manufacturer gives) about performance

• When it’s new, used under specified conditions, and recalibrated at the recommended

frequency

Quantification of Systematic Errors

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Error & Uncertainty

• This can only be a starting point in estimating the measurement accuracy

(achievable in use)

• Many sources (systematic error) may apply in a particular situation (not

included in the accuracy calculation in the manufacturer’s data sheet)

• Example:
• 3 separate sources of systematic error are identified in a measurement system
• After reducing the magnitude of these errors as much as possible, the magnitudes of the three errors

are estimated:

• System loading:

+1.2% (Xm-Xt)

• Environmental changes: 0.8%
• Calibration error:

0.5%

• Calculate the maximum possible total systematic error and the likely system error by the root-sum-

square method.

• Solution:
• The maximum possible system error = ±(1.2 + 0.8 + 0.5)% = ±2.5%
• Applying the root-sum-square: likely error = ± √1.22 + 0.82 + 0.52 = ±1.53%

Sources and Treatment of Random Errors

• Caused by unpredictable variations (precision errors)
• Human observation
• electric noise
• random environmental changes (draught), etc.
• Small perturbations either side of the correct value (positive & negative errors
• ccur in approximately equal numbers)  largely eliminated by averaging
• (but ≠ 0, reason: finite number of measurements)
• The degree of confidence (how close mean value is to the correct value)?
• can be indicated by standard deviation or variance
• parameters describing distribution about the mean value

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Error & Uncertainty

Statistical Analysis of Measurements Subject to Random Errors

• Mean and Median Values
• average value of a set of measurements of a constant quantity:
• Median (was easier for a computer to find)  even number  midway
• Mean (slightly closer to the correct value)
• Example:
• length of a steel bar (mm) is measured by a number of different observers

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Error & Uncertainty Set Mean Median

A

398 420 394 416 404 408 400 420 396 413 430 409 408

B

409 406 402 407 405 404 407 404 407 407 408 406 407

C

409 406 402 407 405 404 407 404 407 407 408 406.5 406 406 410 406 405 408 406 409 406 405 409 406 407

• Which one more

confidence?

• Low-Spread: say range

430-394=36 vs 409-402=7

• Median closer

to mean

Statistical Analysis of Measurements Subject to Random Errors

• Standard Deviation and Variance
• Spread = range between the largest and the smallest value  not a very good way of

examining distribution

• Much better: variance or standard deviation  start with deviation (error)
• di = xi – xmean
• Variance:
• Standard deviation: 
• Definitions: infinite number of data  not in practice
• Finite measurements: xmean ≠ true mean (µ)
• A better prediction: Bessel correction
• Finite number:
• Example:
• Previous sets of measurement

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Error & Uncertainty Set Mean Median spread

V σ

A

409 408 36 137 11.7

B

406 407 7 4.2 2.05

C

406.5 406 8 3.53 1.88

_ _

Statistical Analysis of Measurements Subject to Random Errors

• Graphical Data Analysis Techniques—Frequency Distributions
• Simplest: Histogram
• Bands/bins of equal width across the range of measurement
• # measurements within each band
• Finding the # of bands/bins (Sturgis Rule):
• Example: 23 measurements in set-C
• Bins:  5
• Span: 402-410mm
• Width?  2mm works
• Care in choice of boundaries:
• No measurements on the boundary
• Say, put the middle bin on the Mean (406.5)
• Large enough # of measurement & truly

Random error  symmetry

• Usually, error is of most concern  deviation

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Error & Uncertainty

# Measurements

mm

Gaussian (Normal) Distribution

• Measurement sets that only contain random errors  a distribution with a

particular shape that is called Gaussian

• Frequency of small deviations from the mean >>

the frequency of large deviations

• Measurements in a data set subject to random

errors lie inside deviation boundaries of ±σ  68%

• Lie inside deviation boundaries of ±2σ

 95.4%

• Lie inside deviation boundaries of ±3σ

 99.7%

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±1.96σ  95% (Very common)

Error & Uncertainty

Standard Error of the Mean

• We examined: how measurements with random errors are distributed about the

mean

• However, we know: error exists (mean value of a finite set - true value)
• The standard deviation of mean values of a series of finite

sets of measurements relative to the true mean = standard error of the mean  α

• Question:
• if we use the mean value of a finite set of measurements to

predict the true value  what is the likely error?

• S.d. of error = α  68% of deviations around true value within ±α

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Error & Uncertainty

Mean of 10

Standard Error of the Mean

• Question:
• if we use the mean value of a finite set of measurements to

predict the true value  what is the likely error?

• S.d. of error = α  68% of deviations around true value within ±α
• Means: with 68% certainty that the magnitude of the error

does not exceed |α|

• For data set C , n = 23, σ = 1.88  α = 0.39
• The length (average): 406.5 ± 0.4 (68% confidence limit)
• ±2α  length : 406.5 ± 0.8 (95.4% confidence limit)
• ±3α  length : 406.5 ± 1.2 (99.7% confidence limit)

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Error & Uncertainty

Mean of 10

Not so good

Estimation of Random Error in a Single Measurement (n=1)

• Usually, not practical (repeated measurements  find average)
• Or measured quantity is not constant
• What: likely magnitude of error?
• Often: calculate the error within 95%confidence limits  ± 1.96σ
• However, it was only maximum likely deviation from calculated mean
• Not the true value
• Add: standard error of the mean to the likely maximum deviation value (95%)
• Example:
• A standard mass is measured 30 times (same instrument), σ=0.46  α=0.08
• Now, measure an unknown mass 105.6 kg, how should the mass value be expressed?
• Solution:
• ± 1.96(σ+α) = ±1.06  mass: 105.6±1.06 kg

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Error & Uncertainty

Rogue Data Points (Data Outliers)

• Very large error measurements sometimes occur (random & unpredictable)
• Error magnitude: much larger than the expected random variations
• Sources:
• Sudden transient voltage surges
• Incorrect data recording
• Accepted practice:
• Discard these data points
• Threshold: ±3σ
• Practical problem:
• When a new dataset is measured (S.D. is not known)
• It’s possible to have outlier in measurements
• Simple solution:
• Any new set of measurement  Histogram
• Examine to spot outliers
• Exclude if any  calculate ±3σ to test future measurements

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* * *

Boxplot

Error & Uncertainty

Aggregation of Measurement System Errors

• Usually, 2 or more sources of measurement error
• Total likely error in output reading?
• Forms of aggregation:
• A measurement have both: systematic (±x) & random errors (±y)
• Often likely maximum error:
• A measurement system have several measurement each with separate errors
• Say, different instruments/transducers  add, subtract, multiply, divide
• Example: as S = y + z
• Problem: error term is not expressed as percentage of calculated value for S

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Error & Uncertainty

Aggregation of Measurement System Errors

• Problem: error term is not expressed as percentage of calculated value for S
• Statistical analysis:
• Example: A circuit requirement for a resistance of 550 (2 resistors of nominal values 220

and 330 in series)

• If each resistor has a tolerance of ±2%, the error in the sum?
• It can be shown that the error (e) for subtraction is the same

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Error & Uncertainty