📅 Pre-lab Due: Attempt all seven problems before 1 pm on the scheduled meeting on this topic (see syllabus agenda). Your pre-lab assignment will be graded based on evidence of sincere effort, and work shown, but not on accuracy.
📅 Final Problem Set Due: The final problem set is due at 1 pm one week after our scheduled meeting on this topic (see syllabus agenda). The final problem set will graded based on work shown, thorough and accurate responses to questions, accuracy of calculation, and correct reporting (including units).
🎯 Purpose: Learn how to correctly estimate experimental uncertainty, and how to correctly report a value and its estimated uncertainty. This practice will get you ready to perform uncertainty analysis for your own experimental measurements this semester (and in the future).
Each practice problem below requires that you correctly apply either statistical analysis or propagation of uncertainty to estimate the uncertainty in a calculated value. Each problem also asks you to correctly report calculated values and their uncertainties.
Uncertainty analysis
This is the expected workflow for analysis of experimental values in P-Chem Lab:
Calculate → estimate uncertainty → report value ± uncertainty appropriately → justify the reporting → interpret what the uncertainty tells you.
Python (using Jupyter Notebooks in CoLab) is the recommended tool for completing this problem set. However, you may complete this assignment using any tool including pencil and paper, excel, etc. You must understand the work you do, thus you may not use AI to complete this assignment.
After submitting your initial attempts before the meeting, you will work with other students during our meeting to reach consensus about the strategies used to answer each question, and how to report each value and its uncertainty. Your instructors will be with you to answer your questions and to help guide your discussions. After the meeting, you will submit your revised work and final answers.
Your answers to these problems may be hand-written or typed. However, any submissions that are deemed illegible, unclear, or poorly organized will not be accepted.
Note
(1) We recommend that you use python to complete this problem set – we have provided some CoLab notebooks to help guide you. However, you may use whatever technology (calculators, software, etc.) you like to aid in working these problems.
(2) No matter what tools you use, it is important to show how your calculations are set-up and, where appropriate, give a sample calculation. Simply giving the final answer is not acceptable.
(3) In reporting all results, you should always round your reported answer to be consistent with your estimated uncertainty. For example: a calculated result of 56.2189 ± 0.0386 would be reported as 56.20 ± 0.04 where 0.04 is the standard deviation or propagated uncertainty. See reading assignments for details about reporting.
The seven problems are below. Note that there are custom CoLab tutorials to help you through this problem set. See Canvas and/or your Google Classroom to find the tutorials.
PROBLEM 1: Estimate Uncertainty using Statistics
A student makes five independent measurements of each of the linear dimensions of a box. The results are tabulated as follows:
| Measurement |
Length (cm) |
Width (cm) |
Height (cm) |
| 1 |
1.010 |
5.376 |
2.001 |
| 2 |
1.015 |
5.339 |
2.105 |
| 3 |
1.013 |
5.385 |
1.985 |
| 4 |
0.999 |
5.375 |
1.989 |
| 5 |
1.009 |
5.369 |
2.008 |
- Summarize the measurements: calculate the mean (\(\bar{x}\)), sample standard deviation (\(s\)), and relative standard deviation (\(\%RSD = \frac{s}{\bar{x}} \times 100\)). Report \(\bar{x} \pm s\), and \(\%RSD \) for each dimension (length, width, height) using appropriate units and appropriate precision.
- Compare the precision of the measurements: Which dimension was most precise? ...least precise? Explain what statistical parameters from calculations in (A) are the most appropriate for comparing the precision of measurements that have different magnitudes. Support your answers using calculated values from (A).
- Calculate the volume: Each row of the table represents one set of independent measurements of the box.
- Calculate volume for each trial. This should produce five calculated volumes from the independent measurements of length, width, and height.
- Then calculate the mean (\(\bar{x}\)), sample standard deviation (\(s\)), and relative standard deviation (\(\%RSD\)).
- Report \(\bar{x} \pm s\), and \(\%RSD \) for volume using appropriate units and appropriate precision.
- Interpret results and justify choices for reporting: Write a short paragraph that addresses the following points.
- Which measurement (length, width, height) contributes the greatest uncertainty to the calculated volume?
- Does the calculated volume appear to be more or less precise that the individual measurements? Explain, using evidence from your calcuations to support your argument.
- Justify your choices in reporting to appropriate precision; in other words, what justifies "appropriate precision" of these values.
PROBLEM 2: A Simple Case for Propogation of Uncertainty
A student weighs a sample using an analytical balance, with an estimated uncertainty of ±0.5 mg in each measurement. Using the data below, determine the mass of the sample and the propagated uncertainty in that mass.
| Item measured |
Mass (g) |
| Sample and flask |
5.1647 ± 0.0005 |
| Empty flask (tare) |
2.3712 ± 0.0005 |
- Calculate the mass of the sample.
- Calculate the propagated uncertainty in the sample mass.
- Report the sample mass \(\pm\) uncertainty: report using appropriate units and precision.
- Explain why the propagated uncertainty is larger than the uncertainty in either individual mass measurement.
- Justify the number of decimal places used in your final reported answer.
PROBLEM 3: Statistics and Rejection of Discordant Data
An investigator measures photon emission from the reaction shown below. The emitted photons (at 589 nm) are detected using a photomultiplier. For each run, the investigator measures both the total counts (signal + background) and the background counts.
\( Na^{+} + O^{-} \rightarrow Na^{*} + O \overset{h\nu }{\rightarrow} Na + O \)
|
Run #
|
Signal + Background
(counts/sec)
|
Background
(counts/sec)
|
| 1 |
1500 |
1000 |
| 2 |
1200 |
1250 |
| 3 |
1375 |
1100 |
| 4 |
1425 |
900 |
| 5 |
1280 |
1200 |
| 6 |
1490 |
1363 |
The signal of interest is obtained by subtracting the background from the signal + background measurement.
- Determine the signal for each run. Examine the resulting values before doing any further calculations. What do you notice about the data?
- Calculate the mean and standard deviation of the signal measurements. Your software will probably display more digits than are meaningful. Do not simply copy the numbers produced by the computer. Decide how the mean and standard deviation should be reported, and explain your reasoning.
- One or more of these measurements might look suspicious. Should any measurement be removed simply because it looks unusual? Explain briefly.
- Use the Q-test at the 90% confidence level to determine whether there is statistical justification for rejecting the suspicious measurement(s) as discordant. Clearly identify the value(s) you test and compare Qcalc with Qcrit.
- Based on your statistical analysis, report your best estimate of the signal and its uncertainty. Explain what the standard deviation tells you about the precision of this measurement, and is the standard deviation sufficient for estimating uncertainty? What other ideas might guide your decision for how to report the final value?
PROBLEM 4: Least Squares Analysis and Rejection of Discordant Data
The following experimental data are expected to show an approximately linear relationship between x and y.
| Data Point |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
| x |
26 |
30 |
44 |
50 |
62 |
68 |
74 |
| y |
92 |
85 |
78 |
81 |
54 |
51 |
40 |
Your goal is to determine whether a linear model reasonably describes these data and to investigate whether there is statistical justification for rejecting any observation as an outlier.
- Plot and examine the data: Create a scatterplot of y versus x. Before performing a regression, describe what you see.
- Does a linear relationship appear reasonable?
- Is the relationship positive or negative?
- Does any point look unusual? If so, identify it.
-
Fit a linear model: Perform a linear least-squares regression and determine the slope, intercept, and correlation coefficient, \(r\).
- Add the best-fit line to your plot.
- Report the equation of the fitted line on the plot, using an appropriate number of digits.
- Report both the value and uncertainty for the slope and intercept. Justify your choices in reporting to appropriate precision; in other words, what justifies "appropriate precision" of these values.
- Then interpret your result:
- What does the sign of the correlation coefficient (\(r\)) tell you?
- What does the magnitude of \(r\) tell you?
- Does a large value of |\(r\)|, by itself, demonstrate that every observation is well described by the model? Explain.
-
Examine the residuals: For each observation, calculate the y-residual from the equation below (\ref{residuals}) and create a plot of residual versus x.
\[\text{residual} =y_i −y_{predicted,i} \label{residuals}\]
-
If the linear model adequately describes the data, what would you expect the residuals to look like?
-
Examine your residual plot. Do you see evidence of a systematic pattern, or does one observation appear particularly unusual? Identify the observation you would investigate further and explain your choice.
-
Test the suspicious data point: A point that looks unusual should not be discarded simply because removing it might improve the fit. Apply the Q-test to the y-residuals to determine whether there is statistical justification for rejecting the questionable observation. Compare \(Q_{expt}\) with the appropriate \(Q_{crit}\) at the 90% confidence level. Based on this comparison, is there statistical justification for rejecting the observation? Explain your decision.
-
What happens if the point is removed? Regardless of your decision in Part D, temporarily remove the questionable observation and repeat the linear regression (Parts A, B, C above). This is an exploration, not permission to discard the measurement. Compare the regression with and without the suspicious data point. Consider slope, intercept, correlation coefficient r, and residuals.
-
Make a scientific argument. Did removing the questionable point substantially change the relationship between x and y? Support your conclusion with quantitative evidence from the two regression analyses.
PROBLEM 5: Propagation of Uncertainty
In a chemistry lab, students obtain the following data and are asked to find the molecular weight (MW) of methanol using the ideal gas law, \(PV = nRT\).
| |
Pressure (methanol) |
Temperature |
Mass (of methanol) |
Volume |
| Trial 1 |
72.5 ± 0.5 mm Hg |
295.10 ± 0.05 K |
0.1331 ± 0.0002 g |
1.05678 ± 0.00005 L |
| Trial 2 |
72.0 ± 0.5 mm Hg |
295.05 ± 0.05 K |
0.1329 ± 0.0002 g |
1.05678 ± 0.00005 L |
- Develop the function and determine MW. Use the ideal gas law and the measured quantities to write a single expression for the molecular weight of methanol in grams per mole. Then calculate the molecular weight independently for each trial.
Reminder: pay attention to units, and perform unit conversions as necessary.
- Propagate the uncertainty. Determine the uncertainty in the molecular weight calculated for each trial. Clearly identify the uncertainty associated with each measured quantity and show how those uncertainties enter your calculation.
- Analyze the sources of uncertainty. Which measured quantity contributes most strongly to the uncertainty in the calculated molecular weight? Support your answer quantitatively.
- Determine the average molecular weight and its uncertainty. Calculate the average molecular weight determined from the two independent trials. Treat the two molecular wights as independently measured quantities and propagate their uncertainties through the calculation of the average. Report the average molecular weight and its uncertainty using appropriate units and precision.
- Justify your choices in reporting. Explain how you determined the appropriate precision for the reported average molecular weight and its uncertainty.
PROBLEM 6: Propagation of Uncertainty
The Arrhenius relationship allows us to determine the activation energy for a reaction by measuring the rate constant at two different temperatures, according to the following expression.
\[ \large {ln\left ( \frac{k_{2}}{k_{1}} \right )=-\frac{E_{act}}{R}\left (\frac{1}{T_{2}} -\frac{1}{T_{1}} \right )} \label{arrhenius} \]
The rate constants for the reaction below was measured at two temperatures.
\( 2DI \rightarrow D_{2} + I_{2} \)
\( k_{660}=(1.2 \pm 0.2)\times 10^{-3}\frac{liter}{mol\; sec} \) at \( 660\pm 2 K \)
\( k_{720}=(1.8 \pm 0.2)\times 10^{-3}\frac{liter}{mol\; sec} \) at \( 720\pm 2 K \)
- Develop the function and determine the activation energy. Use the Arrhenius expression (eq. \ref{arrhenius}) to develop a function for calculation of \(E_{act}\). Then use the two independent measurements of \(k\) and \(T\) to calculate the \(E_{act}\).
- Propagate the uncertainty in \(E_{act}\). Develop the function for calculation of uncertainty and determine the uncertainty in the value of \(E_{act}\).
- Report and interpret the results. Report \(E_{act}\) and its uncertainty using appropriate units and precision. Justify your choices in reporting.
- Analyze the sources of uncertainty. Compare the contributions from \(k_1\), \(k_2\), \(T_1\), and \(T_2\). Which measurement(s) have the greatest influence on the uncertainty in \(E_{act}\)? Support your answer quantitatively.
PROBLEM 7: Propagation of Uncertainty
In the method of Clement and Desormes, the heat capacity (\(C\)) ratio of an ideal gas is determined from the manometrically determined values of P1, the gas pressure at temperature T1; P2, the gas pressure immediately after a reversible adiabatic expansion when the temperature is T2; and, P3, the gas pressure after the gas has warmed back up to temperature T1. The relationship used to calculate the heat capacity ratio is given by equation \ref{CD1}.
\[ \large {\gamma=\frac{C_{P}}{C_{V}}=\frac{\frac{C_{V}}{R}+1}{\frac{C_{V}}{R}}} \label{CD1}\]
where
\[ \large {\frac{C_{V}}{R}=\frac{P_{2}\left [ 1-\left ( \frac{P_{3}}{P_{1}} \right ) \right ]}{P_{3}-P_{2}}} \label{CD2} \]
The following pressure data were collected:
P1 = 763.1 ± 0.2 mmHg
P2 = 757.0 ± 0.1 mmHg
P3 = 758.7 ± 0.2 mmHg
- Develop the function and determine the heat capacity ratio, \(\gamma\). Use the equations above (eqs. \ref{CD1} and \{refCD2}) to develop a function to calculate \(\gamma\) from the three pressure readings given above. Then use the function to calculate the value of \(\gamma\).
- Propagate the uncertainty in each measured pressure to determine uncertainty in \(\gamma\). Calculate the uncertainty in the heat capacity ratio, \(\gamma\).
- Report and interpret the result. Report \(\gamma\) and its uncertainty with appropriate precision and units, if applicable. Justify your choices in reporting. Based on your uncertainty analysis, what can you conclude about the precision with which this experiment determines \(\gamma\)?
