Appendix 3: Uncertainty Propagation#
Uncertainty occurs in any scientific measurement and is often represented as the standard deviations, \(\sigma\), of measurements or the 95% confidence interval, 95% CI. When performing calculations containing values with uncertainty, the uncertainty needs to be propagated through the calculations, which is a tedious and error-prone task when done by hand. This appendix demonstrates how to use the Python uncertainties package to remove most of the pain from uncertainty propagation along with simulating uncertainty using a random number generator.
Uncertainties Package#
As of this writing, the uncertainties package can be installed using pip. We will then import a couple key functions, ufloat() and ufloat_fromstr(), along with the umath module which brings a range of math functions (e.g., log and sin). We will also import NumPy and matplotlib to use in the simulation section.
from uncertainties import ufloat, ufloat_fromstr
from uncertainties import umath
import numpy as np
import matplotlib.pyplot as plt
Uncertainties Variable#
Basic mathematical operations with the uncertainties package center around the uncertainties variable object. This is created using the ufloat() function accepts two important values - the first is the nominal value and the second is the standard deviation.
ufloat(nominal_value, std_dev)
For example, let’s say we have a value of 18.66 with a standard deviation of 0.03.
val = ufloat(18.32, 0.03)
We can access the nominal value or the standard deviation by themselves using the nominal_value or std_dev methods, respectively.
val.nominal_value
18.32
val.std_dev
0.03
Values from Strings#
If you are calculating uncertainties taken from a text problem, the uncertainties package provides a convenience function ufloat_fromstr() that allows you to copy-and-paste in values and their uncertainties all together. Below are acceptable formats.
ufloat_fromstr('0.011 ± 0.002')
0.011+/-0.002
ufloat_fromstr('0.172807(0.000008)')
0.172807+/-8e-06
ufloat_fromstr('0.172807 +/- 0.000008')
0.172807+/-8e-06
ufloat_fromstr('0.172')
0.172+/-0.001
The last one did not include an uncertainty, so the uncertainty was interpreted to be \(\pm\)1 of the least significant decimal place.
Simple Calculations#
Beyond this, we just need to carry out our mathematical operations. For example, let’s say we want to calculate the molar absorptivity constant using Beer’s law, \(A = \epsilon bC\), where A is absorbance, \(\epsilon\) is the molar absorptivity constant, \(b\) is the path length in cm, and \(C\) is concentration in molarity. If the A = 0.3822 \(\pm\) 0.0003, \(b\) = 1.00\(\pm\)0.01 cm, and \(C\)=0.0017\(\pm\)0.0001 M, we can calculate the molar absorptivity constant like below.
A = ufloat(0.3822, 0.0003)
b = ufloat(1.00, 0.01)
C = ufloat(0.0017, 0.0001)
E = A / (b * C)
E
224.8235294117647+/-13.415813085736838
This results in 225\(\pm\)13 cm\(^{-1}\)M\(^{-1}\).
If we multiply an uncertainties variable object by a regular int or float(), the int or float() is treated as having no uncertainty, so the uncertainty of the other value scales linearly with the nominal value. In the example below, both values triple.
3 * b
3.0+/-0.03
The umath module provides special mathematical functions like square root or sine. For example, if we want to calculate the pH of a solution with an [H\(_3\)O\(^+\)] = 6.33\(\times\)10\(^{-6}\) \(\pm\) 3\(\times\)10\(^{-7}\) M, or (6.33 \(\pm\) 0.3)\(\times\)10\(^{-6}\) M, we can carry out this calculation below which gives us a pH = 5.199\(\pm\)0.021.
H3O = ufloat(6.33e-6, 3e-7)
-umath.log10(H3O)
5.198596289982645+/-0.02058267686745269
Simulating Uncertainties#
We can also simulate uncertainties using Monte Carlo simulations as demonstrated below. Let’s say we want to calculate the molar absorptivity constant using the same nominal and standard deviation values as above. Using a random number generator, we can generate values for A, \(l\), and C with the given standard deviations using the normal(nominal_value, std_dev) function from the numpy.random module. We then carry out the calculation with all of these values. The molar absorptivity is the average of these values with and uncertainty calculated from the standard deviation of these calculated values.
import numpy as np
import matplotlib.pyplot as plt
A_nom, A_sig = 0.3822, 0.0003
l_nom, l_sig = 1.00, 0.01
C_nom, C_sig = 0.0017, 0.0001
N = int(1e7)
rng = np.random.default_rng(seed=21)
A = rng.normal(loc=A_nom, scale=A_sig, size=N)
l = rng.normal(loc=l_nom, scale=l_sig, size=N)
C = rng.normal(loc=C_nom, scale=C_sig, size=N)
E = A / (l * C)
plt.hist(E, bins=40)
plt.xlim(160, 300)
plt.xlabel('Molar Absorbtivity Constant (cm$^{-1}$M$^{-1}$)')
plt.ylabel('Count');
print(np.mean(E))
print(np.std(E, ddof=1))
225.63035520507208
13.600103576801123
This results in a value of 226\(\pm\)14 cm\(^{-1}\)M\(^{-1}\), which is close to what we calculated using the uncertainties library.
Further Reading#
Documentation for
uncertaintiespackage. https://uncertainties.readthedocs.io/en/latest/ (free resource)NumPy
polyfit()Documentation. https://numpy.org/doc/stable/reference/generated/numpy.polyfit.html (free resource)SciPy
curve_fit()Documentation. https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.curve_fit.html (free resource)Salter, C. Error Analysis Using the Variance-Covariance Matrix. J. Chem. Educ. 2000, 77 (9), 1239. https://doi.org/10.1021/ed077p1239.
Provides background and an example of using a variance-covariance matrix.