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,
(n:1)
xmeas- u ≤ xtrue ≤ xmeas+ u
™
™
Wider interval at higher odds
™
Narrower interval at lower odds
BJ Furman
SJSU MAE
1
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
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
Output, y units
30
25
K=sensitivity
20
15
10
5
0
0
5
10
15
20
25
Measurand input, x units
BJ Furman
SJSU MAE
3
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
Accuracy, Repeatability, and Systematic Error
Repeatability
BJ Furman
SJSU MAE
4
Calibration and Repeatability
Sensor Calibration Repeated Several Times
40
35
Output, y units
30
meas. 1
meas. 2
25
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
5
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
ng
as i
e
r
inc
Output, y units
25
20
15
10
d
g
sin
a
e
ecr
5
0
0
5
10
15
20
25
Measurand input, x units
BJ Furman
SJSU MAE
6
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
7