Uncertainty and Its Propagation Through Calculations

Uncertainty and Its Propagation Through Calculations Valerie L. Young Department of Chemical & Biomolecular Engineering Ohio University Last modified 01 October 2010

Uncertainty • No measurement is perfect • The uncertainty (or error) is an estimate of a range likely to include the true value • Uncertainty in data leads to uncertainty in calculated results • Uncertainty never decreases with calculations, only with better measurements • Reporting uncertainty is essential • Knowing the uncertainty is critical to decision-making • Knowing the uncertainty is the engineer’s responsibility

Topics . . . • How to report uncertainty • The numbers • The text • Identifying sources of uncertainty • Estimating uncertainty when collecting data • Uncertainty and simple comparisons • Propagation of error in calculations

Reporting Uncertainty – The Numbers • Always show experimental data and results as xbest± x • Uncertainty gets 1 significant figure • Or maybe 2 sig figs if its value is 1 • Round the best estimate to be consistent with the uncertainty • Keep extra digits temporarily when calculating How much more or less than xbest the true value might reasonably be Best estimate of the true value

Examples One sig fig Two sig figs cuz the first is a 1 • Right (6050 ± 30) m/s (10.6 ± 1.3) gal/min (-16 ± 2) °C (1.61 ± 0.05)  1019 coulombs • Wrong (6051.78 ± 32.21) m/s (-16.597 ± 2) °C Scientific notation like this xbest and x should have the same units

Examples • Right (6050 ± 30) m/s (10.6 ± 1.3) gal/min (-16 ± 2) °C (1.61 ± 0.05)  1019 coulombs • Wrong (6051.78 ± 32.21) m/s (-16.597 ± 2) °C You can’t be this certain of the uncertainty Not rounded consistent with uncertainty

Reporting Uncertainty – The Text You must explain how you estimated each uncertainty. For example: The reactor temperature was (35 ± 2) °C. The uncertainty. . . . . .is estimated based on the thermometer scale. . . .is given by the manufacturer’s specifications for the thermometer. . . .is the standard deviation of 10 measurements made over the 30 minutes of the experiment. . . .represents the 95% confidence limits for 10 measurements made over the 30 minutes of the experiment.

Reporting Uncertainty – The Text You must explain how you estimated each uncertainty. For example: The reactor temperature was (35 ± 2) °C. The uncertainty. . . . . .is estimated based on the thermometer scale. . . .is given by the manufacturer’s specifications for the thermometer. These account for uncertainty due to the measurement technique, but do not account for any variability in the actual temperature of the reactor during the experiment. Reporting ONLY measurement uncertainty implies that the true variability is LESS than the measurement uncertainty. Is it?

Reporting Uncertainty – The Text • You must explain how you estimated each uncertainty. For example: The reactor temperature was (35 ± 2) °C. The uncertainty. . . . . .is the standard deviation of 10 measurements made over the 30 minutes of the experiment. . . .represents the 95% confidence limits for 10 measurements made over the 30 minutes of the experiment. These estimates of uncertainty include both the precision of temperature control on the reactor and the precision of the measurement technique. They do not account for the accuracy of the measurement technique.

Precision vs. Accuracy • Precision • Accuracy

Precision vs. Accuracy • Precision • How widely scattered multiple values are • Closely spaced (precise) values may not be clustered around the true value • Accuracy • How much the center of multiple values differs from the true value • Values centered on the true value (accurate) may be widely scattered

Estimating Uncertainty from Scales What uncertainty would you place on the length of the pencil? As a rule of thumb, people often use ½ the distance between measurement marks, but you may quote a larger uncertainty, based on your judgment.

Estimating Uncertainty from Scales An uncertainty of ± 0.5 V or ±0.3 V would probably be reasonable, if the needle is steady.

Graphical Display of Data and Results Error bars show uncertainty. Axes scaled so data fills plot. Caption for figure, NO title for figure. Figure 1. Cell reproduction declines exponentially as the mass of growth inhibitor present increases. Vertical error bars represent standard deviation of 5 replicate measurements for one growth plate. Caption interprets figure; doesn’t repeat axis labels. Caption explains estimate of uncertainty.

Experimental Results and Conclusions • A single measured number is uninteresting • An interesting conclusion compares numbers • Measurement vs. expected value • Measurement vs. theoretical prediction • Measurement vs. measurement • Do we expect exact agreement? • No, just “within experimental uncertainty”

Comparison and Uncertainty A and B are significantly different. C and D are NOT significantly different.

Comparison and Uncertainty Consistent with the accepted value. Significantly different from the accepted value. May be significantly different from the accepted value.

Comparison and Uncertainty • xbest± x means . . . xtrue is probably between xbest- x and xbest+ x (need a formal consideration of probability to make “probably” quantitative) • Two values whose uncertainty ranges overlap are not significantly different • They are “consistent with one another” • A value just outside the uncertainty range may not be significantly different • To decide, will need to consider probability

Propagation of Uncertainties • We often do math with measurements Density = (m ± m) / (V ± V) What is the uncertainty on the density? • “Propagation of Error” estimates the uncertainty when we combine uncertain values mathematically • Don’t use error propagation if you can measure the uncertainty directly (as variation among replicate experiments). • Do use error propagation to estimate uncertainty on a single measurement, or before you perform the experiment.

Simple Rules Absolute uncertainty • Addition / Subtraction, q = x1 + x2 – x3 – x4 q = sqrt((x1)2+(x2)2+(x3)2+(x4)2) • Multiplication / Division, q = (x1x2)/(x3x4) q/|q| = sqrt((x1/x1)2+(x2/x2)2+(x3/x3)2+(x4/x4)2) • 1-Variable Functions, q = ln(x) q = |dq/dx| x  |1/x| x Fractional uncertainty d(ln(x))/dx = 1/x

Simple Rules • Addition / Subtraction, q = x1 + x2 – x3 – x4 q = sqrt((x1)2+(x2)2+(x3)2+(x4)2) Uncertainty gets bigger even when you subtract • Multiplication / Division, q = (x1x2)/(x3x4) q/|q| = sqrt((x1/x1)2+(x2/x2)2+(x3/x3)2+(x4/x4)2) Uncertainty gets bigger even when you divide • 1-Variable Functions, q = ln(x) q = |dq/dx| x  |1/x| x

General Formula for Error Propagation • q = f(x1,x2,x3,x4) q = sqrt(((q/ x1) x1)2 + ((q/ x2) x2)2 + ((q/ x3) x3)2 + ((q/ x4) x4)2 ) Partial derivative of q wrt x3 Absolute uncertainty in x4

User Beware! • Error propagation assumes that the relative uncertainty in each quantity is small • Weird things can happen if it isn’t, particularly for nonlinear functions like ln • Example of weird thing: ln(0.5 ± 0.4) = -0.7 ± 0.8 • For nonlinear functions, I suggest assuming that the relative error in x is equal to the relative error in f(x) • Remember that the engineer’s goal is a reasonable estimate of uncertainty, not slavish compliance with an inappropriate procedure. • Don’t use error propagation if you can measure the uncertainty directly (as variation among replicate experiments)

Sample Calculation • You pour the following into a lab reactor: • (100 ± 1) ml of 1.00 M NaOH in water • (1000 ± 1) ml of water • (1000 ± 1) ml of water • What is the concentration of NaOH in the reactor?

Sample Calculation ([NaOH] VNaOH) / (VNaOH+Vwater+Vwater)) The concentration of NaOH in the reactor is (0.0476 ± 0.0007)M. The uncertainty was estimated by propagation of error, using the measurement uncertainties in the volumes added, and assuming an uncertainty of ± 0.01 M in the concentration of the 1.00 M NaOH solution. Note that writing 1.00 M implies an uncertainty of 0.01 M.

Uncertainty and Its Propagation Through Calculations