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Analyzing Uncertainty and Errors. ISAT 253 Spring 2005. Objectives. Understand the concept of uncertainty Define measurement uncertainties and errors For a set of measurements, learn to calculate specific Sensitivities Accuracies Precisions. Objectives. Define measurement errors

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Analyzing uncertainty and errors

Analyzing Uncertainty and Errors

ISAT 253

Spring 2005


  • Understand the concept of uncertainty

  • Define measurement uncertainties and errors

  • For a set of measurements, learn to calculate specific

    • Sensitivities

    • Accuracies

    • Precisions

Dr. Ken Lewis


  • Define measurement errors

  • Differentiate between systematic and random errors

Dr. Ken Lewis

Key concepts



Precision error

Bias error



Calibration standards

Measurement standards

Key Concepts

Dr. Ken Lewis


  • Resolution

    • The smallest increment of a unit of measure that an instrument can detect or measure.

  • Accuracy

    • How close the measurement is to the “true value”

  • Precision

    • The consistent repeatability of a measurement.

Dr. Ken Lewis

Recall types of error
Recall -- Types of Error

  • Bias error ( average of the measurements – true)

    • Non random

    • Systematic

    • Destroys accuracy

  • Precision error (measurement readings – average)

    • Random

    • Hard to control without changing the measurement system

Dr. Ken Lewis

Error summary
Error summary

Precision Error

Random Error

Bias Error

Systematic Error






Dr. Ken Lewis

Which type of error is it
Which type of error is it?

  • A carpenter bought a piece of lumber at Lowe’s and measured it in the store as 8’-2”. At home, when she measured it again to cut it, she measured it as 8’-1” using the same tape measure.

  • We measure the pH of a solution of 0.1 mol acetic acid and 0.2 mol ammonium acetate as 4.8 at 25°C, but standards show it should be 4.78.

  • A statistical process control (SPC) gauge is 6 microns high every time it is used.

Dr. Ken Lewis

Quantifying uncertainty
Quantifying uncertainty

  • Ten measurements were made on a battery

  • The true voltage is known to be 6.11 volts.

  • The average of the measurements is

    • 6.03 volts


The resolution error

The systematic error or accuracy

The precision

Dr. Ken Lewis

Quantifying uncertainty1
Quantifying uncertainty

The resolution uncertainty or resolution error.

±0.01 V

Dr. Ken Lewis

Quantifying uncertainty2
Quantifying uncertainty

The systematic error or accuracy

Accuracy = average value - true value

Accuracy = 6.03 V - 6.11 V = -0.08 V

-0.08 V

Dr. Ken Lewis

Quantifying uncertainty3
Quantifying uncertainty

The precision

Precision = Maximum deviation from the average

Precision = ±|5.96 V - 6.03 V| = ±0.07 V

±0.07 V

Dr. Ken Lewis

Quantifying uncertainty4
Quantifying uncertainty

  • Caution

    • Be sure that the precision statement isn’t based on one bad measurement

Dr. Ken Lewis

Reporting uncertainties
Reporting Uncertainties

  • Uncertainties can be reported as:

    • a number in the measurement units

    • a percentage of the instrument’s full scale

    • a percentage of the measurement itself

  • Resolution error can be reported as:

    • ± 1 of the least significant digit

    • ± ½ of the least significant digit

Dr. Ken Lewis

Numerical example
Numerical Example

  • Consider our battery/voltmeter problem:

    • Average measured value = 6.03 V

    • Assume meter range = 0-10 V

  • Precision can be reported in three ways:

    • Measurement units: 6.03 V ± 0.07 V

    • Percent of full scale: 6.03 V ± 0.7% FS

    • Percent of the measure: 6.03 V ± 1%

Dr. Ken Lewis


  • A measuring system is designed to operate over a finite range

    • 0 – 500°C

    • 20 – 200 psig

    • 0 – 300 lbs

  • The range given describes the limits of proper response

  • What happens outside the range is no gauranteed.

Dr. Ken Lewis

Analyzing uncertainty and errors

  • The span is the difference between the high and low of the range

Dr. Ken Lewis

Accuracy example 5 full scale
Accuracy – example ±5% full scale

  • Problem: below the full scale reading the error will be greater than ±5%.

  • 0 – 200°C ± 10°C

  • So at 30°C, the reading will be somewhere between 20°C and 40°C

Dr. Ken Lewis


  • The slope of the line relating input to output

Dr. Ken Lewis

Sensitivity calculation
Sensitivity Calculation

For example




Dr. Ken Lewis

Other measurement problems1
Other Measurement Problems

  • Hysteresis

    • Friction

    • Mechanical flexure of internal parts

    • Electrical capacitance

  • Will usually appear random

Dr. Ken Lewis

Example a tachometer
Example – A tachometer

  • Tachometer measures shaft rotation speeds in the range of 0 – 5000rpm

    • Accuracy: ±5% FS

    • Hysteresis: 30 rpm

    • Zero offset: 200 rpm

  • What is the maximum error you expect in a shaft speed reading of 3500 rpm?

Accuracy uncertainty


±30 rpm

Zero offset 200 rpm

U = 250rpm + 30rpm + 200 rpm

U = 580rpm


  • A process wherein a set of measurements are made of measurand values that can determined independently

  • Readings are compared to the known ‘true’ values and errors determined

  • Implies that the measurements are referenced against a measurementstandard.

Dr. Ken Lewis

What is a meter measurement standards
What is a meter? Measurement Standards

  • 1793 Govt. of France decrees the unit of length to be 10-7 of the earth’s quadrant passing through Paris and called the meter.

  • 1889 Treaty of the Meter (Conférence Général des Poids et Mésures, CGPM) established a platinum-iridium bar.

Dr. Ken Lewis

What is a meter measurement standards1
What is a meter? Measurement Standards

  • 1960 definition based on the krypton86 radiation from an electrical discharge lamp.

  • 1983: The meter is the SI unit of length and is defined as the length of the path traveled by light in a vacuum during the time interval of 1/299,792,458 of a second.

Dr. Ken Lewis

What is a second measurement standards
What is a second?Measurement standards

  • The second is the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom.

  • Other base units – great site


Dr. Ken Lewis

What is the point measurement standards
What is the POINT?Measurement Standards

  • There are many basic standards.

  • Before the French Revolution every different duchy had their own version of standards of weight, length, etc.

    • Need to standardize to allow transfers of knowledge

    • Need to have standards to allow calibration of instruments to make results reliable and interchangeable.

Dr. Ken Lewis

What s the point here 200 000 000 mars polar lander
What’s the Point? Here $200,000,000Mars Polar Lander

  • December 3, 1999

    • EDWARD EULER, Lockheed Martin: The mistake was that we had to give the Jet Propulsion Lab some data that is used to compute very, very small little thrust pulses onboard the spacecraft. And we did give them the data in the wrong units... and in English units, and it should have been in metric. And they used the data as if it were metric, and underestimated the magnitude of these small, little pulses that come out of the jets of the Orbiter by about a factor of five. And that in turn made it very difficult to get the proper navigation, or determine the position and velocity of the spacecraft, which eventually led to the failure.

Dr. Ken Lewis

Measurement standards
Measurement Standards

  • Many standards;

    • 1 foot is 12 inches

    • 2.54 centimeters is exactly 1 inch

    • There are 28 grams in 1 ounce

    • A CD is 12 centimeters in diameter

    • All video players (VHS) can interpret correctly any VHS tape

    • The electric voltage and current in California is the same as it is in New Hampshire

Dr. Ken Lewis

Seven si base units
Seven SI Base Units

Dr. Ken Lewis


  • All primary standards except mass “can” be reproduced in a good well equipped laboratory.

  • The standard for mass ‘International Prototype Kilogram’ is a Pt—Ir cylinder kept in Paris France

  • Standards for all other physical variables are;

    • Derived from the base standards

    • Physical laws

Dr. Ken Lewis

Example force
Example -- Force

  • Recall Newton’s second law

    • Force = mass X acceleration

  • Acceleration = meter/second/second

    • =length/second2

Dr. Ken Lewis

Some si derived units
Some SI Derived Units

Dr. Ken Lewis