Download MiniChem1 Significant Figures and Error Analysis

1.
MiniChem
1:
Significant
Figures,
Rounding
and
Error Analysis
1.1 Significant Figures
1.1.1 Definition
• All non-zero digits are significant
823.47 → 5 significant figures
• Zeros between two non-zero digits are significant
702.1 → 4 sig. figs
• Leading zeros are not significant
0.0035 → 2 sig. figs
• Trailing zeros are significant*
1.34500 → 6 sig. figs.
*
Some authors (purists?) treat exact factors of 10 when written
explicitly as in 20 or 350, in the same way as if they were written 2 x 101 or 35
x 101. This means that these numbers have only 1 and 2 significant figures,
respectively; and only if the measurements are reported as 20. and 350. should
these numbers be considered to have 2 and 3 significant figures, respectively.
I believe that most people who report the value of a measurement as
20 or 350 do intend to inform the reader that their measurements were
performed with 2 and 3 significant figures, instead.
1.1.2 Arithmetic and Significant Figures
• Addition and subtraction: keep as many decimal places as there are in the
measured value with the smallest # of decimal places.
1.340 + 2.54 = 3.88
• Multiplication and division: keep as many sig figs as there are in the
measured value with the smallest # of sig figs.
2.34567 x 3.21 = 7.53
• Logarithm: keep as many decimal places as there are sig figs in the
number in the log (Ln, ln or log) function.
log(3.451) = 1.2387 (4 sig figs → 4 decimal places)
• Exponential: keep as many sig figs as there are decimal places in the
number in the exp function
exp(3.451) = 31.5
(3 decimal places → 3 sig figs)
Marco Ceruso, The C Research Lab, © 2009-2012
1.2.Rounding
1.2.1 Round-to-nearest
2.526 → 2.53
23.443 → 23.44
3.465 → 3.47 or 3.46*
-3.455 → -3.46 or -3.45*
* In other words decimal ending in 1 2 3 and 4 are rounded down, decimals
ending in 6 7 8 9 are rounded up and decimals ending exactly in 5 need a tiebreaking rule as the Round-half to even, Round-half to odd and Round-half
up rules below:
(a) Round-half-to-even: nearest even integer up or down
3.465 → 3.46
-3.455 → -3.46
(b) Round-half-to-odd nearest odd integer up or down
3.465 → 3.47
-3.455 → -3.45
(c) Round-half-up: round up
3.465 → 3.47 (since 3.47 > 3.46)
-3.455 → -3.45 (since -.345 > -3.46)
1.3 Error Analysis
1.3.1 Reporting a measurement and its associated error/uncertainty
xbest ± δx
xbest is the best estimate of the measurement (usually the average) and δx is
the uncertainty in the measurement. The statement xbest ± δx indicates that
the probable range is expected to fall within the range [x-δx, x+δx].
Note that experimental uncertainties should always be rounded to 1 significant
figure. Thus:
report 9.82 ± 0.02 instead of 9.8200 ± 0.0234
1.3.2 Propagation of uncertainties
• Addition and subtraction
E.g. if result = x1 + x 2 + ... + xN then ! result =
•
2
2
( ) ( )
! x1
+ ! x2
2
( )
+ ... + ! xN
Multiplication and division
E.g. if result = z1 ! z 2... ! zN then
! result
result
2
=
2
"!z %
"!z %
"!z
1
$
' + $ 2 ' + ... + $ N
$# z '&
$# z
$# z '&
1
2
N
Marco Ceruso, The C Research Lab, © 2009-2012
%
'
'&
2
•
If result=B.x, where B is known exactly, then δx=|B|.δx
1.3.3 Average and standard deviation
•
•
Average or mean: x =
1
N
Standard deviation ! x =
N
!x
i =1
1
N
i
N
=
#(
1
N
(x + x
1
xi " x
2
)
2
+ ... xN
or ! x =
)
1
N
#(
xi " x
2
)
N " 1 i =1
If one computes an average and a standard deviation from a series of
measurements it is common to report the best estimate and its associated
error as the average and the standard deviation, respectively:
i =1
xbest ± δx → x ± ! x
If the distribution of measurements is expected to be Gaussian this way of
reporting a measured value indicates that ~68% of the measured values fell
within the range # x ! " x , x + " x % . Note that ~95% of the measured values are
$
&
expected to fall within # x ! 2" x , x + 2" x % and ~99% of the measured values
$
&
within # x ! 3" x , x + 3" x % .
$
&
1.4 Additional Resources
•
•
•
•
•
•
•
John R. Taylor, An introduction to Error Analysis, 2nd edition, 1997,
University Science Books, Sausalito, CA.
http://www.chem.sc.edu/faculty/morgan/resources/sigfigs/index.html
http://www.wellesley.edu/Chemistry/stats/sigfig2.html
http://www.av8n.com/physics/uncertainty.htm
http://en.wikipedia.org/wiki/Significant_figures
http://en.wikipedia.org/wiki/Significance_arithmetic
http://en.wikipedia.org/wiki/Rounding
Marco Ceruso, The C Research Lab, © 2009-2012