Source: Physics 212 Lab Activity v1.1 (Appendices B & C), University of Illinois at Urbana-Champaign
Tags: uncertainty, error propagation, standard error, standard deviation, measurement uncertainty, accuracy rating, sample rate, quadrature, percent uncertainty, multimeter accuracy, IOLab, PHYS 212
Difficulty: Intermediate | Prerequisites: Basic statistics (mean, standard deviation), familiarity with RC circuit experiment from Lab 6 core notes.
Every measurement you take in a physics lab carries some uncertainty, and Physics 212 expects you to quantify it. In Physics 211, your main source of uncertainty was trial-to-trial variation. Here, measurements are often repeatable, so the uncertainty comes from subtler places: the accuracy rating of your instruments, the resolution of your sample rate, and the ambiguity of deciding exactly when an event starts or ends on a graph. This set of notes covers how to identify, quantify, and combine those uncertainties so your final result is reported as a value ± uncertainty. You will use these skills in every remaining lab.
Uncertainty comes from several sources (random variation, instrument accuracy, sample rate, and judgement calls on timing). You quantify each source separately, then combine them using quadrature addition when multiplying or dividing quantities. The final result should always be in the form measurement ± uncertainty.
Uncertainty (δx)
A quantitative estimate of how much a measured or calculated value might differ from the true value. Written with the lowercase Greek delta: δx. Every measured quantity in this course must carry one. In simple terms, it is the "± bit" that tells the reader how confident you are in the number.
Standard Deviation (σ)
A measure of how spread out a set of measurements is around the mean. Larger σ means more scatter. For a small number of trials, you can approximate it as (largest sample − smallest sample) / 2.
Standard Error on the Mean (δµ)
The uncertainty on the average of N measurements: δµ = σ / √N. As you take more samples, the standard error shrinks, meaning your average becomes a more precise estimate of the true value. Think of it as: more data points give you a tighter estimate of the average.
Accuracy Rating
A specification provided by the manufacturer of a measuring tool (such as a multimeter) that describes the expected error on any reading. Often expressed in the form ±(percent of reading + counts).
Counts (in Multimeter Accuracy)
An additional fixed uncertainty added to the last displayed digit. If the multimeter's lowest displayed digit represents 10 Ω and the accuracy spec says "+2 counts," that contributes 20 Ω of extra uncertainty on top of the percentage component.
Propagation of Uncertainty
The process of calculating how uncertainties in individual measurements combine to produce uncertainty in a calculated result. The method depends on the mathematical operation (addition, multiplication, logarithm, etc.).
Quadrature Addition
The method for combining percentage uncertainties when multiplying or dividing: take the square root of the sum of the squares of each fractional uncertainty. This is the standard rule for independent, random errors.
When you take many samples of the same quantity (through repeated trials or continuous sampling), the values will scatter randomly around the true value.
Report the result as: µ ± δµ, where µ is the mean and δµ = σ / √N.
The IOLab software displays µ and σ for any highlighted data region.
N (number of samples) = sample rate × Δt, where Δt is the highlighted time duration.
Worked example from the lab: a highlighted region with N = 195 samples, µ = 2.0109 V, σ = 0.0024 V gives δµ = 0.0024 / √195 ≈ 0.0002 V. Report: 2.0109 ± 0.0002 V.
When you are judging the moment an event begins or ends on a graph, there is often a range of plausible times rather than one sharp instant.
Method:
Zoom in as far as possible.
Identify the earliest plausible time for the event (t_early).
Identify the latest plausible time (t_late).
Report: midpoint ± half-range.
Formula: (t_early + t_late) / 2 ± (t_late − t_early) / 2
Worked example: voltage levels out somewhere between 4.970 s and 5.035 s. Report the event time as 5.0025 ± 0.0325 s.
If you need the duration between two uncertain events, the uncertainties add (not in quadrature, just plain addition): duration = (t_end − t_start) ± (δt_start + δt_end).
Sometimes the value you are looking for falls between two consecutive data points. You cannot know the exact moment it occurred.
Method: same midpoint ± half-range formula as above, but here t_early and t_late are the timestamps of the two adjacent samples that bracket the target value.
The half-range in this case equals half the sampling interval (i.e., 1 / (2 × sample rate)).
Worked example: looking for the moment voltage = 0.900 V, the last sample below is at 4.715 s and the first sample above is at 4.720 s. Report: 4.7175 ± 0.0025 s.
Determine which source is larger, the "fuzzy event boundary" uncertainty or the sample-rate uncertainty, and use that one.
If the start/end of the event spans multiple samples, the boundary uncertainty dominates; ignore the sample-rate contribution.
If the event is sharply defined within a single sample interval, the sample-rate uncertainty dominates.
Components have labelled tolerances: the resistors in the E&M kit are rated at ±5%.
A 10 kΩ resistor therefore has an uncertainty of ±0.5 kΩ straight from the label.
Multimeters have more detailed accuracy specs in the form ±(percent of reading + counts).
To apply the multimeter accuracy:
Step 1: Multiply the displayed reading by the percent figure to get the percentage component.
Step 2: Multiply the number of "counts" by the resolution of the lowest displayed digit to get the counts component.
Step 3: Add both components together for the total uncertainty.
Worked example (100 Ω resistor on the 200 Ω range, accuracy ±(0.8% + 5)):
Reading: 98.8 Ω
Percentage component: 98.8 × 0.008 = 0.79, rounded to 0.8 Ω
Counts component: 5 × 0.1 Ω (resolution on the 200 Ω range) = 0.5 Ω
Total uncertainty: 0.8 + 0.5 = 1.3 Ω
Report: 98.8 ± 1.3 Ω
Once you measure with a multimeter, its (smaller) uncertainty replaces the manufacturer's ±5% tolerance.
Range | Resolution | Accuracy |
|---|---|---|
200 Ω | 0.1 Ω | ±(0.8% + 5) |
2000 Ω | 1 Ω | ±(0.8% + 5) |
20 kΩ | 10 Ω | ±(0.8% + 2) |
200 kΩ | 100 Ω | ±(0.8% + 2) |
20 MΩ | 10 kΩ | ±(1% + 5) |
200 MΩ | 100 kΩ | ±(5% (reading−10) + 10) |
For Lab 6, you use the 20 kΩ range to measure your 10 kΩ resistor.
δµ = σ / √N
measurement = (t_early + t_late) / 2 ± (t_late − t_early) / 2
duration = (t_end − t_start) ± (δt_start + δt_end)
If D = AB / C, then:
δD = D × √( (δA/A)² + (δB/B)² + (δC/C)² )
Steps:
Convert each uncertainty to a fractional (percentage) form: δA/A, δB/B, δC/C.
Add them in quadrature (square root of the sum of squares).
Multiply the result by D to get δD as an absolute value.
If you take ln(A), the uncertainty on the result is approximately:
δ ln(A) ≈ δA / A
This comes up when using a curve-fit method (plotting ln(V) vs. time) to extract the time constant.
Uncertainty propagation is foundational to any field that relies on measurement: engineering tolerances, clinical diagnostics, environmental monitoring, financial modelling. When an engineer specifies that a bridge beam must support 10,000 ± 200 kg, the ± 200 is not decoration; it determines the safety margin. In this lab, the same logic tells you whether your measured capacitance is consistent with the labelled value or genuinely outside the expected range.
"If my measurements are repeatable, they have no uncertainty." Repeatability means low random variation, but the instrument itself still has an accuracy rating and finite resolution. Every measurement has uncertainty, even repeatable ones.
"I should always add uncertainties by simple addition." Simple addition applies when you are adding or subtracting measured quantities. When you multiply or divide, you must combine fractional (percentage) uncertainties in quadrature.
"More decimal places means more precision." Reporting 10.0000 kΩ when your multimeter resolution is 0.01 kΩ is misleading. Your reported value should reflect the resolution and uncertainty of the instrument.
"The ±5% on the resistor label is the same as the multimeter uncertainty." It is not. The label tolerance is the manufacturer's guaranteed range. The multimeter gives you a direct measurement with its own (typically much smaller) uncertainty, which replaces the label tolerance.
⚠️ You must be able to calculate standard error on the mean given σ and N.
⚠️ Know which uncertainty source dominates for a given type of measurement (timing ambiguity vs. sample rate vs. instrument accuracy).
⚠️ Be able to propagate uncertainty through a multiplication/division calculation using the quadrature formula.
⚠️ Understand the multimeter accuracy format ±(percent + counts) and be able to compute a total uncertainty from a reading.
⚠️ Know that uncertainties add directly for subtraction (durations) but add in quadrature for multiplication/division.
⚠️ The natural-log propagation rule (δ ln(A) ≈ δA / A) may appear if you use curve-fit methods.
True or false: Standard error on the mean decreases as the number of samples increases.
Fill in the blank: When combining uncertainties for the calculation D = A × B, you add the fractional uncertainties in ______.
True or false: If the start time of an event has an uncertainty of ±0.02 s and the end time has an uncertainty of ±0.03 s, the uncertainty on the duration is ±0.05 s.
Fill in the blank: The multimeter accuracy ±(0.8% + 2) means 0.8% of the ______ plus 2 units of the lowest displayed digit.
True or false: δ ln(A) ≈ δA / A is exact for all values of δA.
Q: You highlight a region of IOLab data and read µ = 1.800 V, σ = 0.005 V, with a sample rate of 800 Hz over Δt = 0.5 s. What is the standard error on the mean?
A: N = 800 × 0.5 = 400. δµ = 0.005 / √400 = 0.005 / 20 = 0.00025 V. Report: 1.8000 ± 0.0003 V (rounding δµ to one significant figure).
Q: You measure a 10 kΩ resistor on the 20 kΩ multimeter range and read 9.86 kΩ. The accuracy is ±(0.8% + 2). What is the total uncertainty?
A: Percentage component: 9.86 × 0.008 = 0.079 kΩ, round to 0.08 kΩ. Counts component: 2 × 0.01 kΩ = 0.02 kΩ. Total: 0.08 + 0.02 = 0.10 kΩ. Report: 9.86 ± 0.10 kΩ.
Q: Your capacitance is calculated as C = τ / R. You measured τ = 0.210 ± 0.005 s and R = 9.86 ± 0.10 kΩ. What is C and its uncertainty δC?
A: C = 0.210 / 9860 = 2.13 × 10⁻⁵ F = 21.3 µF. Fractional uncertainties: δτ/τ = 0.005/0.210 = 0.0238; δR/R = 100/9860 = 0.0101. δC/C = √(0.0238² + 0.0101²) = √(0.000566 + 0.000102) = √0.000668 ≈ 0.0258. δC = 21.3 × 0.0258 ≈ 0.55 µF. Report: 21.3 ± 0.6 µF.
Q: Why do uncertainties add directly (not in quadrature) when calculating the duration of an event from a start time and end time?
A: The duration is found by subtraction (t_end − t_start), and for addition/subtraction of quantities, absolute uncertainties add directly. Quadrature applies to multiplicative/divisive combinations of independent measurements.
Q: You are deciding between two sources of time uncertainty: the event boundary spans about 0.06 s, and the sample interval is 0.00125 s (800 Hz). Which uncertainty do you use?
A: The event-boundary uncertainty (±0.03 s, half the span) is far larger than the sample-rate uncertainty (±0.000625 s). Use the event-boundary uncertainty; it dominates.
Uncertainty propagation is not specific to capacitance measurements. You will use the same quadrature formula whenever you calculate a derived quantity from multiple measured inputs, in every remaining Physics 212 lab and in most experimental science courses. The standard-error concept reappears in statistics courses and in any discipline that averages repeated observations. The multimeter accuracy skills transfer directly to any benchtop electronics work.
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