Introduction to Error Analysis:
Lecture 3: Combining measurements
Petar Maksimović
Feb 25 2003
Overview
propagation of errors
covariance
weighted average and its error
error on sample mean and sample standard deviation
Propagation of Errors
,
is a known function of
...
assume that most probable value for
is
is the mean of
by definition of variance
in Taylor series:
expand
Variance of
is COvariance. Describes correlation between errors on
and .
For uncorrelated errors
This is the error propagation equation.
Examples
const. Thus
where
where
const.
correlation can be negative, i.e.
if an error on counterballanced by a proportional error
can get very small!
on ,
More examples
etc., etc.
Weighted average
From lecture #2: calculation of the mean
minimizing
, but now
minimum at
const.
each measurement is weighted by
so-called weighted average is
!
Error on weighted average
points contribute to a weighted average
straight application of the error propagation equation:
putting both together
Example of weighted average
error
weighted average
morale:
result:
practically doesn’t matter!
Error on the mean
measurements from the same parent population ( ,
)
from Lecture #1: sample mean
and sample standard
deviation are best estimators of the parent population
but: more measurements still gives same :
our knowledge of shape of parent population improves
and thus of original true error on each point
but how well do we know the true value? (i.e. ?)
points from same population with
:
if
Standard deviation of the mean, or standard error.
Example: Lightness vs Lycopene content, scatter plot
Lycopene content
lyc:Light
120
110
100
90
80
70
60
50
30
40
50
60
Lightness
Example: Lightness vs Lycopene content: RMS as Error
Lycopene content
Lightness vs Lycopene content -- spread option
105
100
95
90
85
80
75
70
65
60
30
35
40
45
50
55
Lightness
Points don’t scatter enough
the error bars are too large!
Example: Lightness vs Lycopene content: Error on Mean
Lycopene content
Lightness vs Lycopene content
100
95
90
85
80
75
70
65
60
30
35
40
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Lightness
This looks much better!
A note on reporting measurement results
keep only one digit of precision on the error – everything
else is noise
Example:
exception: when the first digit is , keep two:
Example:
round off the final value of the measurement up to the
significant digits of the errors
Example:
kg
kg
rounding rules:
and above
round up
and below
round down
: if the digit to the right is even round down, else round
up
(reason: reduces systematic erorrs in rounding)