Difficulty: Intermediate | Prerequisites: Gravimetric analysis concepts (see companion notes), basic algebra, comfort with significant figures.
Every measurement in a chemistry lab has some degree of uncertainty. Reporting a result without its uncertainty is incomplete: the number looks precise, but the reader has no way of knowing how much to trust it. In gravimetric analysis, uncertainty enters through the balance (every mass reading), through the accepted values of atomic and molar masses, and through every arithmetic step that combines those readings. Propagation of uncertainty is the formal method for tracking how small errors in individual measurements accumulate into the overall error in the final answer. This topic sits at the intersection of analytical chemistry and statistics, and the skills transfer to every quantitative lab course you will take.
Measurement uncertainty tells you the range within which the true value likely sits. In this experiment, balance uncertainties (±0.0003 g at ~20 g, ±0.0002 g at ~1 g) and the uncertainty in AgCl's molar mass (±0.001 g/mol) were propagated through each arithmetic step to give a final uncertainty in percent chloride. The average absolute uncertainty in %Cl was ±0.099%, and the average relative uncertainty was ±0.138%.
Measurement uncertainty
A parameter that characterises the dispersion of values that could reasonably be attributed to the quantity being measured. It quantifies the doubt about the measurement result.
In simple terms, it is the "± number" after your measurement that tells you how far off the true value your reading could be.
Absolute uncertainty
The uncertainty expressed in the same units as the measurement itself (e.g. ±0.0003 g).
Think of it as the raw "wiggle room" around your number, stated in grams, moles, or whatever unit you measured in.
Relative uncertainty (fractional uncertainty)
The absolute uncertainty divided by the measured value, often expressed as a percentage. It tells you the size of the uncertainty relative to the measurement.
In simple terms, it answers "how big is my error compared to my number?" A ±0.0003 g uncertainty on a 20 g reading is tiny (0.0015%); the same ±0.0003 g on a 0.5 g reading is much larger (0.06%).
Propagation of uncertainty (error propagation)
The mathematical procedure for combining the individual uncertainties of measured quantities to estimate the overall uncertainty of a result calculated from those quantities.
Think of it as the accounting system that tracks how each small error feeds into the final answer's error.
Standard uncertainty
The uncertainty of a quantity expressed as a standard deviation. For values taken from reference tables (such as atomic masses from the periodic table), the standard uncertainty is typically published alongside the value.
In simple terms, it is the "official" ± value that comes with a reference number, telling you how precisely that number is known.
Analytical balance: Each mass reading carries an uncertainty. In this experiment, ±0.0003 g for readings near 20 g and ±0.0002 g for readings near 1 g.
Weighing by difference: The mass of AgCl is found by subtracting the empty crucible mass from the crucible-plus-AgCl mass. Because subtraction is involved, the uncertainties of both readings combine.
Molar mass of AgCl: Taken from accepted atomic masses of Ag and Cl, each with its own published standard uncertainty. The combined uncertainty in AgCl's molar mass is ±0.001 g/mol, giving 143.321 ± 0.001 g/mol.
Atomic mass of Cl: Also carries a standard uncertainty, which propagates into the mass-of-chloride calculation.
Step 1, mass of AgCl (subtraction): When you subtract two masses, the absolute uncertainties add in quadrature: u(mass AgCl) = sqrt(u₁² + u₂²), where u₁ and u₂ are the uncertainties of the two balance readings.
Step 2, moles of AgCl (division): When you divide mass by molar mass, relative uncertainties add in quadrature: u_rel(mol) = sqrt(u_rel(mass)² + u_rel(M)²).
Step 3, mass of Cl (multiplication): Multiplying moles by atomic mass of Cl follows the same rule as division: relative uncertainties add in quadrature.
Step 4, percent chloride (division): Dividing mass of Cl by mass of unknown, relative uncertainties add in quadrature again.
Mass of chloride per crucible: 0.1042 ± 0.0001 g (crucible 1), 0.1260 ± 0.0001 g (crucible 2), 0.1163 ± 0.0001 g (crucible 3).
Percent chloride values: 66.75%, 71.87%, 75.77%. Average: 71.46%.
Average absolute uncertainty in %Cl: ±0.099%.
Average relative uncertainty in %Cl: ±0.138%.
The spread between crucibles (~±6% from the average) is far larger than the propagated instrumental uncertainty, which tells you the dominant source of variation is not the balance, but procedural factors (incomplete precipitate collection, material left on glassware).
For addition or subtraction (e.g. weighing by difference):
u(result) = sqrt(u(a)² + u(b)²)
Absolute uncertainties combine in quadrature.
For multiplication or division (e.g. mass / molar mass, or moles × atomic mass):
u_rel(result) = sqrt(u_rel(a)² + u_rel(b)²)
Relative uncertainties combine in quadrature. To get the absolute uncertainty of the result, multiply the relative uncertainty by the result itself.
Relative uncertainty:
u_rel = u(x) / x
Or as a percentage: u_rel(%) = (u(x) / x) × 100
Molar mass of AgCl with uncertainty:
M(AgCl) = M(Ag) + M(Cl) = 143.321 ± 0.001 g/mol
(The uncertainty in the molar mass comes from adding the standard uncertainties of the two atomic masses in quadrature.)
Error propagation is routine in pharmaceutical quality control, where the uncertainty of an assay determines whether a drug batch meets specification. Environmental monitoring labs report pollutant concentrations with uncertainty bands so regulators can judge compliance. In engineering, uncertainty budgets feed into tolerance analysis for manufactured parts: if a component's measured dimension has too much uncertainty, you cannot guarantee it will fit.
Students often add absolute uncertainties directly when multiplying or dividing. For multiplication and division, you combine relative uncertainties in quadrature, not absolute ones.
A common mistake is treating the spread between replicate measurements and the propagated instrumental uncertainty as the same thing. They measure different aspects of error. In this experiment, the replicate spread (±6%) reflects procedural variability, while the propagated uncertainty (±0.1%) reflects only the instrument's precision.
Students sometimes assume that more decimal places mean more certainty. The number of meaningful decimal places is dictated by the uncertainty, not by what the calculator displays.
Another error is forgetting that subtraction can amplify relative uncertainty. Subtracting two large, similar numbers (e.g. crucible masses) produces a small result with a proportionally larger relative uncertainty.
⚠️ Know which rule applies to which operation: absolute uncertainties in quadrature for addition/subtraction, relative uncertainties in quadrature for multiplication/division.
⚠️ Be ready to propagate uncertainty through a two- or three-step calculation chain, starting from balance uncertainties and ending at percent composition.
⚠️ Understand the difference between precision (how close repeated measurements are to each other) and accuracy (how close they are to the true value). Propagation of uncertainty addresses precision.
⚠️ Know how to interpret a result where the replicate spread is much larger than the propagated uncertainty: the dominant error source is not the instrument but the procedure.
⚠️ Be able to explain why subtraction of two similar numbers is dangerous for relative uncertainty (loss of significance).
True or False: When dividing two quantities, you combine their absolute uncertainties in quadrature. (False. You combine relative uncertainties in quadrature for division.)
Fill in the blank: The relative uncertainty of a measurement is calculated as ___ divided by ___. (absolute uncertainty divided by the measured value)
True or False: If the spread between replicate measurements is much larger than the propagated instrumental uncertainty, the instrument is the main source of error. (False. The procedure or technique is the dominant source.)
Fill in the blank: Subtracting two large, similar numbers produces a small result with a proportionally ___ relative uncertainty. (larger)
True or False: The uncertainty in the molar mass of AgCl comes from adding the atomic mass uncertainties of Ag and Cl in quadrature. (True, because the molar mass is a sum.)
Q: A student measures the mass of a crucible as 20.4512 ± 0.0003 g and the mass of the crucible plus AgCl as 20.7389 ± 0.0003 g. What is the mass of AgCl and its absolute uncertainty?
A: Mass AgCl = 20.7389 - 20.4512 = 0.2877 g. Uncertainty = sqrt(0.0003² + 0.0003²) = sqrt(0.00000018) = 0.0004 g. So 0.2877 ± 0.0004 g.
Q: Using the mass of AgCl above (0.2877 ± 0.0004 g) and the molar mass of AgCl (143.321 ± 0.001 g/mol), calculate the moles of AgCl and the relative uncertainty.
A: mol AgCl = 0.2877 / 143.321 = 0.002008 mol. Relative uncertainty of mass = 0.0004 / 0.2877 = 0.00139. Relative uncertainty of molar mass = 0.001 / 143.321 = 0.0000070. Combined relative uncertainty = sqrt(0.00139² + 0.0000070²) ≈ 0.00139 (the molar mass uncertainty is negligible here). Absolute uncertainty = 0.002008 × 0.00139 = 0.000003 mol. So 0.002008 ± 0.000003 mol.
Q: In the experiment, the propagated uncertainty in percent chloride was about ±0.1%, but the spread between replicate measurements was about ±6%. What does this tell you about the dominant source of error?
A: The instrumental uncertainty (from the balance) is very small compared to the replicate variation. This means the dominant source of error is procedural, not instrumental. Likely causes include incomplete transfer of precipitate, material adhering to glassware, or inconsistent drying.
Q: Why is subtraction particularly problematic for uncertainty in the weighing-by-difference method?
A: When two large, similar numbers are subtracted, the result is small, but the absolute uncertainty stays roughly the same (the quadrature sum of two similar uncertainties). This makes the relative uncertainty of the result much larger than the relative uncertainty of either individual reading. The smaller the difference, the worse this effect.
This connects directly to statistics: standard deviation, variance, and the rules for combining independent random variables are the mathematical backbone of error propagation. In instrumental analysis courses, you will encounter calibration curves and regression, where uncertainty analysis becomes more sophisticated but rests on the same foundations.
The distinction between instrumental precision and procedural variability connects to the broader concept of random vs. systematic error, which is central to experimental design across all sciences.
Measurement uncertainty, error propagation, propagation of uncertainty, absolute uncertainty, relative uncertainty, fractional uncertainty, standard uncertainty, quadrature, weighing by difference, analytical balance precision, significant figures, loss of significance, random error, systematic error, precision vs accuracy, uncertainty budget, percent composition uncertainty, gravimetric analysis error