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1
Nathan W. Moore
An Introduction to Error Analysis for
Chemical Engineers
Presented to:
Chemical Engineering Laboratory (ECH155A)
University of California at Davis
October 2005
Overview
Identifying error
Quantifying error
•
Confidence intervals
•
Error propagation
•
Uncertainties in bestfit lines
Reporting uncertainties
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The Sources of Error
The two kinds of error
Systematic error
...occurs in a predictable way.
...results from imperfect instrumentation or procedures.
...tells you how
accurate
your data is.
Random error
...occurs unpredictably.
...results from unrepresentative samples.
...tells you how precise your data is.
3
Sources of systematic error
•
Inaccurate calibration (or wrong model used)
•
Bias (measurement consistently high or low of actual
value)
•
Drift (operating parameters changing over time)
•
Hysteresis (reading depends on history of measurement,
i.e., from mechanical manipulation)
•
Incorrect model used for analyzing relationships
between data
•
Propagation of errors through calculation
Reducing systematic error
Goal: Make Systematic Error less than Random Error
•
Improve the skill of the person making measurements.
•
Compare measurements to standards and controls.
•
Measure bias, drift, and nonlinearity and adjust your data
accordingly.
•
Redesign the procedure to avoid known hysteresis.
•
If the experiment cannot be redesigned, randomize the
order of measurements.
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Sources of random error
Sample properties not uniform in time and space
•
usually from uncontrollable interactions with environment (e.g.
air temperature fluctuations)
Measurement device not consistent in time and space
•
e.g. voltage fluctuations in a thermocouple
Reducing random error
Goal: measure “representative samples”
•
Repeat measurements
•
Repeat measurements
•
Repeat measurements
5
Confidence Intervals
(a quick review)
Random errors
...occur from unrepresentative samples.
Measured values
Count
Guess for
true mean
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Random errors
...occur from unrepresentative samples.
Infinitely large samples usually have a Gaussian
distribution:
Measured values
Count or Probability
Student’s tdistributions
Better description when N<30
(sample size)
Still works when N>30
⇒
can use all the time!
Measured values
Probability
student tdistribution
is broader than Gaussian dist’
⇒
more uncertainty
in smaller sample sizes
Gaussian distribution
7
Confidence intervals
Given spread in data, what is the range of values in which
the true mean
is likely to be found?
Measured values
Probability
true mean is
somewhere in here
Estimating confidence intervals
..
t
CI
N
σ
⎛⎞
=
⎜⎟
⎝⎠
standard
deviation
of data
number of
data points
tvalue
Depends on:
confidence level
(usually 95%)
degrees of freedom
(N1 in this case).
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This note was uploaded on 09/30/2011 for the course ECH 155A taught by Professor Kuhl during the Summer '11 term at UC Davis.
 Summer '11
 Kuhl

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