Estimating and Plotting Logarithmic Error Bars
Estimating and Plotting Logarithmic Error Bars
Eric M. Stuve
Department of Chemical Engineering University of Washington
Box 351750, Seattle, WA 98195-1750, USA stuve@uw.edu
?2004?12
Absolute Error Bars
? Suppose that one has a sufficient number of measurements to make an estimate of a measured quantity y and report its absolute error, ?y.
? The absolute error ?y is represented on a Cartesian plot by extending lines of the appropriate size above and below the point y.
yi + y yi
yi ? y y
x
Absolute Error Bars on a log Plot
? If plotted on a logarithmic plot, however, absolute error bars that are symmetric on a y vs. x plot become asymmetric; the lower portion is longer than the upper portion.
log(yi + y) log(yi)
log(yi ? y)
log(y)
x
? This gives a misleading view of measurement precision, especially when measured quantities vary by several orders of magnitude.
Error in Logarithmic Quantities
? To represent error bars correctly on a log plot, one must recognize that the quantity being plotted, which we call z, is different than the measured quantity y.
z = log(y)
? The error z is
[ ] z = log(y)
log Error is Relative Error
? On the assumption of small errors, a differential analysis can be used
[ ] z dz = d log(y) = 1 dy 0.434 y
2.303 y
y
? The error z is thus given by the relative error in y
z 0.434 y y
zi + z zi
z = log(y)
zi ? z
? The error bars now display
correctly on a logarithmic
plot.
x
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