1. ME 120 Experimental Methods
Accuracy, Precision, and
Error in Measurement
BJ Furman
17AUG05
BJ Furman SJSU MAE
Error in Measurement
Error = ε ≡ xmeas - xtrue
Error is present in every measurement
Best we can do is estimate a bound on ε with some
level of certainty
-u ≤ ε ≤ u (n:1)
where u is the uncertainty estimated at odds of n:1,
or equivalently,
xmeas- u ≤ xtrue ≤ xmeas+ u (n:1)
Wider interval at higher odds
Narrower interval at lower odds
BJ Furman SJSU MAE
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2. The Goal
Statements of uncertainty associated with
measurement results
“A measurement result is complete only when
accompanied by a quantitative statement of its
uncertainty.”
Guidelines for Evalutating and Expressing the Uncertainty of NIST Measurement
Results, NIST Tech. Note 1297 1994 Edition
Example
• Y= y ± uc(y) 95% confidence or (n:1) odds
BJ Furman SJSU MAE
Probability and Odds
Probability ≡ the likelihood of a particular event taking
place measured with reference to all possible events.
The probability of rolling ‘snake eyes’ [••] with a pair of dice is
1/36
Odds ≡ the ratio of the probability of an event to the
probability that it does not happen:
odds = p/(1-p)
The odds of rolling snake eyes with a pair of dice is
(1/36)÷(35/36) ⇒ 1 in 35
Helpful in magnifying small or large probabilities by expressing
them as a ratio of whole numbers.
What probability does odds of 19:1 correspond to?
BJ Furman SJSU MAE
2

3. Important Definitions
Calibration=the process of applying known values of
a measurand to determine the response of a
measurement system
Static Calibration=calibration using a constant measurand
• Ex. Weights applied to a scale
Dynamic Calibration=calibration using a time-dependent
measurand
• Ex. Sinusoidal, step, impact
Sensitivity=the change of output (measured value)
per unit of input (measurand). Sensitivity is the slope of
the static calibration curve at a given value of the input.
BJ Furman SJSU MAE
Calibration, Sensitivity, and Linearity
Sensor Calibration
40
35
30
Output, y units
25
K=sensitivity
20
15
10
5
0
0 5 10 15 20 25
Measurand input, x units
BJ Furman SJSU MAE
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4. Important Definitions , cont.
Accuracy=the closeness of a measured value
to the actual value being measured
Repeatability (precision)=the ability of an
instrument to repeat an output when
measuring a given quantity under identical
conditions
Resolution=the smallest increment of change
in the measured value that can be determined
from the readout or recording instrument
BJ Furman SJSU MAE
Accuracy, Repeatability, and Systematic Error
Accuracy
Repeatability
BJ Furman SJSU MAE
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5. Calibration and Repeatability
Sensor Calibration Repeated Several Times
40
35
30
Output, y units
meas. 1
25
meas. 2
20 meas. 3
meas. 4
15
meas. 5
10
5
0
0 5 10 15 20 25
Measurand input, x units
BJ Furman SJSU MAE
Components of Uncertainty
Random effects (random or precision errors)
Occur differently for each measurement
Creates a distribution of values
Arise from uncontrolled variables, e.g., _________________
Can use statistical methods to estimate the likely range that
the true value lies in
Systematic effects (systematic or bias errors)
Occur the same way for each measurement
• Ex: 1st down measurement chain – too many links
• Ex: Bubble level - vial misaligned
Arise from:
• Calibration error
• Loading error
• Measurement disturbance
• Aging of components in measurement equipment
• Uncontrolled variables
BJ Furman SJSU MAE
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6. Components of Uncertainty, cont.
Systematic effects, cont.
In general, need more than statistical methods to
determine
• Can estimate using:
Comparison with more accurate standards
Compare with a different method of measuring the same
variable
Self-calibration (reversal) sometimes
Inter-laboratory comparison
Experience
BJ Furman SJSU MAE
Calibration and Hysteresis
Sensor Calibration with Hysteresis
35
30
25 g
sin
Output, y units
rea
20
inc
15
ing
eas
ecr
10
d
5
0
0 5 10 15 20 25
Measurand input, x units
BJ Furman SJSU MAE
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7. References
Beckwith, T. G., Marangoni, R. D., Lienhard, J. H.,
Mechanical Measurements, Addison-Wesley, Reading, MA,
1995.
Figliola, R. S., Beasley, D. E., Theory and Design for
Mechanical Measurements, 3rd ed., J. Wiley & Sons, New
York, 2000.
“Accuracy Specifications for Mensor Products,” Technical
Note 6, Mensor Corp., www.mensor.com, September, 2002.
NIST/SEMATECH e-Handbook of Statistical Methods,
http://www.itl.nist.gov/div898/handbook/, September, 2002.
Taylor, B. N., Kuyatt, C. E., Guidelines for Evaluating and
Expressing the Uncertainty of NIST Measurement Results,
NIST Technical Note 1297, 1994 edition.
BJ Furman SJSU MAE
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