The Central Idea
A model usually transforms inputs and parameters into an output. In a simple form, we may write
\[ y=f(x;\theta), \]where \(x\) represents observed inputs and \(\theta\) represents model parameters.
If either \(x\) or \(\theta\) is uncertain, then the predicted value \(y\) is also uncertain. The uncertainty in the prediction is therefore not an afterthought. It is part of the model itself.
The basic modeling question is not only What value does the model predict? It is also How much could that prediction change because the quantities entering the model are uncertain?
Exact Values and Model Values
In pure mathematics, a symbol may represent an exact quantity. In applied modeling, a number written in a formula often represents an approximation to a physical, economic, biological, or engineered quantity.
Suppose a measured length is reported as
\[ L=12.4\text{ cm}. \]The written value \(12.4\) does not usually mean that the true length is exactly \(12.400000\ldots\) centimeters. It means that the measurement process supports a value near \(12.4\) centimeters at some stated or implied level of precision.
A model should preserve that distinction between the number used in a calculation and the underlying quantity that number is intended to represent.
What Is Uncertainty?
Uncertainty describes the range of plausible values or the degree of incomplete knowledge associated with a quantity.
Uncertainty does not necessarily mean that a measurement or model is poor. Every empirical model operates with finite information. The important task is to identify uncertainty, represent it appropriately, and understand its effect on conclusions.
For example, a temperature might be represented as
\[ T=21.3\pm0.2^\circ\text{C}. \]This notation indicates that \(21.3^\circ\text{C}\) is the central reported value and \(0.2^\circ\text{C}\) describes an uncertainty scale around that value.
Two Major Sources of Uncertainty
In many models, it is useful to separate uncertainty into two broad categories.
- Measurement uncertainty concerns quantities observed or recorded from the world.
- Parameter uncertainty concerns quantities that control the model but are not known exactly.
The two can interact. Measurements are often used to estimate parameters, so uncertain measurements can lead directly to uncertain parameter values.
Measurement Uncertainty
A measurement is produced by an instrument, procedure, or observation. Each measurement process has limitations.
Common sources of measurement uncertainty include
- finite instrument resolution;
- calibration limits;
- environmental variation;
- sampling variation;
- observer or recording effects;
- rounding and digitization;
- natural short-term fluctuations in the quantity being measured.
Measurement uncertainty therefore reflects both the behavior of the system and the limitations of the process used to observe it.
Instrument Resolution
Resolution is the smallest change that an instrument can reliably distinguish or display.
If a ruler is marked only in millimeters, then a reported length such as
\[ 84\text{ mm} \]should not ordinarily be treated as though the measurement were known to the nearest micrometer.
The number of written digits should be consistent with the information actually supplied by the measuring process.
Repeated Measurements
Repeating a measurement provides information about variation in the observation process.
Suppose a quantity is measured five times:
\[ 10.2,\quad10.4,\quad10.3,\quad10.2,\quad10.5. \]A simple central estimate is the sample mean
\[ \bar{x}=\frac{10.2+10.4+10.3+10.2+10.5}{5}=10.32. \]The spread of the measurements contains additional information. Reporting only \(10.32\) hides the fact that repeated observations varied around that value.
Random and Systematic Effects
Measurement uncertainty is often discussed in terms of random and systematic effects.
Random effects produce variation from one measurement to another. Repeated observations may scatter above and below a central value.
Systematic effects shift measurements in a consistent direction. For example, an improperly calibrated scale might add approximately \(0.5\text{ kg}\) to every reading.
Repetition can reveal random variation, but repeating a biased measurement procedure does not automatically remove a systematic error.
Error and Uncertainty Are Not the Same
The words error and uncertainty are related, but they should not be treated as exact synonyms.
If the true value \(x_{\text{true}}\) were known, the measurement error of an observed value \(x_{\text{obs}}\) could be written as
\[ e=x_{\text{obs}}-x_{\text{true}}. \]In many real applications, however, the true value is not exactly known. Uncertainty describes what is known about the possible size or distribution of the error even when the actual error cannot be directly calculated.
Absolute Uncertainty
An uncertainty stated in the same units as the measured quantity is an absolute uncertainty.
For example,
\[ m=8.40\pm0.05\text{ g} \]has an absolute uncertainty of
\[ 0.05\text{ g}. \]A simple interval interpretation would place plausible values near
\[ 8.35\text{ g}\le m\le8.45\text{ g}, \]although the exact statistical meaning of the \(\pm\) notation depends on how the uncertainty was defined.
Relative and Percent Uncertainty
Absolute uncertainty is often easier to compare after scaling it by the size of the measured quantity.
If \(x\) has absolute uncertainty \(\Delta x\), then the relative uncertainty is approximately
\[ \frac{\Delta x}{|x|}. \]The corresponding percent uncertainty is
\[ 100\%\cdot\frac{\Delta x}{|x|}. \]For
\[ x=50.0\pm0.5, \]the relative uncertainty is
\[ \frac{0.5}{50.0}=0.01, \]or \(1\%\).
Parameter Uncertainty
A parameter is a quantity that helps define the behavior of a model. Parameters may represent rates, coefficients, material properties, probabilities, elasticities, growth constants, efficiencies, or other system characteristics.
Consider the exponential model
\[ P(t)=P_0e^{rt}. \]The parameter \(r\) controls the growth or decay rate. If available data support values of \(r\) near \(0.03\) but do not determine it exactly, then the model contains parameter uncertainty.
Fixed Does Not Mean Known Exactly
A parameter can be conceptually fixed for a particular system while still being uncertain to the modeler.
For example, suppose a material has a true heat-transfer coefficient \(k\). The value of \(k\) may be physically fixed under specified conditions, but if experiments estimate it only approximately, the model may use
\[ k=14.8\pm1.1. \]The uncertainty here represents incomplete knowledge about the parameter, not necessarily physical fluctuation in the parameter itself.
Parameters Estimated from Data
Many parameters are estimated by fitting a model to observed data.
Suppose a linear model has the form
\[ y=a+bx. \]A data set may produce estimates such as
\[ a=2.1, \qquad b=0.84. \]These are estimates, not automatically exact constants. Different samples could produce slightly different fitted values, so both \(a\) and \(b\) may carry uncertainty.
Later lessons will represent that uncertainty more formally with probability distributions, confidence intervals, Bayesian parameter distributions, and simulation methods.
Input Uncertainty and Parameter Uncertainty
It is useful to distinguish uncertain inputs from uncertain parameters.
Consider
\[ y=\theta x. \]The input \(x\) may be uncertain because it was measured. The parameter \(\theta\) may be uncertain because it was estimated from previous data.
Both uncertainties affect \(y\), but they may have different sources and may need different methods of analysis.
A Simple Range Model
One of the simplest ways to represent uncertainty is with a range.
Suppose
\[ 9.8\le x\le10.2 \]and the model is
\[ y=3x. \]Because multiplication by \(3\) preserves order,
\[ 29.4\le y\le30.6. \]The uncertainty in \(x\) has produced a corresponding uncertainty in the model output \(y\).
Parameter Range Example
Suppose a simple cost model is
\[ C=q p, \]where \(q=500\) units and the uncertain unit-cost parameter satisfies
\[ 18\le p\le22. \]Then
\[ 500(18)\le C\le500(22), \]so
\[ \boxed{9000\le C\le11000}. \]Even though the production quantity is fixed, uncertainty in the cost parameter creates a \(2000\)-unit range in predicted total cost.
Nominal Values
A nominal value is a central, standard, or reference value used for calculation.
If a parameter is represented as
\[ \theta=4.0\pm0.3, \]then \(4.0\) may be used as the nominal value.
A nominal model calculation might evaluate
\[ y=f(x;4.0), \]but a responsible uncertainty analysis also asks how \(y\) changes for plausible values of \(\theta\) around \(4.0\).
Precision and Accuracy
Precision concerns the degree to which repeated measurements agree with one another or the fineness with which a quantity is resolved.
Accuracy concerns closeness to the true or accepted value.
Measurements can be precise but inaccurate if they cluster closely around a biased value. They can also be relatively accurate on average while being individually imprecise.
Model uncertainty should therefore not be assessed from the number of decimal places alone.
Significant Digits and False Precision
A model can produce a calculator output with many digits even when the input information does not justify them.
Suppose
\[ r=2.1\pm0.1 \]and a calculation produces
\[ y=17.4382917. \]Reporting every displayed digit may create false precision. The uncertainty in \(r\) may make several of the trailing digits meaningless for decision-making.
Numerical formatting should reflect the quality of the underlying information, not merely the precision of the computing device.
Uncertainty Is Different from Model Structure
Uncertain inputs and parameters are not the only reasons a model may be imperfect. The model structure itself may simplify reality.
For example, a linear model
\[ y=a+bx \]may use well-measured data and well-estimated parameters but still perform poorly if the real relationship is strongly nonlinear.
This distinction is important:
- measurement uncertainty concerns observed quantities;
- parameter uncertainty concerns model coefficients or constants;
- structural uncertainty concerns the adequacy of the model form itself.
This lesson focuses primarily on the first two.
Two Interpretations of Uncertainty
In advanced uncertainty analysis, uncertainty is often separated into two conceptual types.
Epistemic uncertainty comes from incomplete knowledge. It may be reduced by better measurements, more data, improved calibration, or better parameter estimation.
Aleatory uncertainty comes from inherent variability or randomness in the system being modeled.
The distinction is not always perfectly sharp, but it helps modelers decide whether an uncertainty should be reduced through additional information or represented as an intrinsic source of variation.
A Basic Workflow for Uncertain Quantities
When measurement or parameter uncertainty is present, a useful modeling workflow is:
- identify the uncertain quantities;
- state how each uncertainty is represented;
- identify the source of the uncertainty;
- calculate a nominal model result;
- vary the uncertain quantities over plausible values;
- observe how much the model output changes;
- report both the prediction and the uncertainty relevant to the decision.
Later lessons in this section develop more formal versions of these steps.
Example: Distance from Speed and Time
Suppose distance is modeled by
\[ d=vt. \]Let the measured speed be
\[ v=20.0\pm0.5\text{ m/s} \]and the measured time be
\[ t=8.0\pm0.1\text{ s}. \]The nominal prediction is
\[ d=(20.0)(8.0)=160\text{ m}. \]But \(160\text{ m}\) should not be interpreted as exact because both inputs are uncertain.
One conservative range calculation uses the endpoint combinations
\[ d_{\min}=(19.5)(7.9)=154.05\text{ m} \]and
\[ d_{\max}=(20.5)(8.1)=166.05\text{ m}. \]Thus a simple range-based description is
\[ 154.05\text{ m}\le d\le166.05\text{ m}. \]More refined methods of error propagation will be developed later.
Example: Growth with an Uncertain Parameter
Suppose
\[ P(t)=1000e^{rt} \]and the growth-rate parameter is believed to lie between
\[ 0.02\le r\le0.04. \]At \(t=10\), the lower prediction is
\[ P_{\min}(10)=1000e^{0.2}\approx1221.4, \]while the upper prediction is
\[ P_{\max}(10)=1000e^{0.4}\approx1491.8. \]A parameter range that may appear small can create a substantial difference in long-term predictions because the parameter enters exponentially.
The Connection to Sensitivity
Uncertainty and sensitivity are closely related but distinct ideas.
Uncertainty asks how little we know about an input or parameter. Sensitivity asks how strongly the model output responds when that input or parameter changes.
A highly uncertain parameter may matter very little if the output is insensitive to it. A precisely estimated parameter may still deserve close attention if a very small change causes a large change in the model output.
Later uncertainty-quantification methods combine these two ideas.
A Local Derivative View
For a differentiable one-variable model
\[ y=f(x), \]a small input change \(\Delta x\) produces the local approximation
\[ \Delta y\approx f'(x)\Delta x. \]This gives a first indication of how uncertainty in \(x\) may influence uncertainty in \(y\).
For a multivariable model
\[ y=f(x_1,x_2,\ldots,x_n), \]small changes can be approximated by
\[ \Delta y\approx \frac{\partial f}{\partial x_1}\Delta x_1+ \frac{\partial f}{\partial x_2}\Delta x_2+ \cdots+ \frac{\partial f}{\partial x_n}\Delta x_n. \]This derivative-based idea becomes one foundation for formal error propagation.
An Uncertainty Range Is Not Automatically a Probability Distribution
Writing
\[ 4.8\le\theta\le5.2 \]states a range of plausible values, but it does not say that all values in the interval are equally likely.
Likewise, writing
\[ \theta=5.0\pm0.2 \]does not, by itself, specify a normal distribution, a confidence level, or a particular probability interpretation.
A complete uncertainty statement should make clear what the interval or \(\pm\) quantity means.
Uncertainty and Decision Thresholds
Uncertainty becomes especially important when a model output is compared with a threshold.
Suppose a design is considered acceptable only when
\[ y\le100. \]A nominal model prediction of
\[ y=98 \]may seem acceptable. But if plausible uncertainty places the output between
\[ 94\le y\le104, \]the decision is no longer clear from the nominal value alone.
This is one reason uncertainty analysis is central to engineering, forecasting, finance, public policy, reliability, and risk management.
Common Modeling Mistakes
Several recurring mistakes make uncertainty analysis less reliable.
- treating measured quantities as exact constants;
- reporting excessive digits that imply unsupported precision;
- using a best-fit parameter without reporting its uncertainty;
- assuming repeated measurements remove systematic bias;
- confusing a plausible range with a probability distribution;
- reporting only a nominal prediction when decisions depend on extremes;
- ignoring uncertainty because the underlying formula is exact.
An exact mathematical formula can still produce an uncertain applied prediction when the quantities inserted into that formula are uncertain.
Practice
Practice 1
A length is reported as
\[ L=24.6\pm0.2\text{ cm}. \]Identify the nominal value and the absolute uncertainty.
Practice 2
A mass is measured as
\[ 80.0\pm0.8\text{ g}. \]Find the relative uncertainty and percent uncertainty.
Practice 3
A model is
\[ y=5x \]with
\[ 3.8\le x\le4.2. \]Find the corresponding output range.
Practice 4
A revenue model is
\[ R=pq, \]where \(q=1200\) and the uncertain price parameter satisfies
\[ 14\le p\le16. \]Find the range of predicted revenue.
Practice 5
Explain why five nearly identical readings from a miscalibrated instrument can be precise without being accurate.
Practice 6
A fitted model uses the parameter estimate
\[ k=0.72\pm0.06. \]Is \(0.72\) best described as an exact constant, a nominal parameter value, or a probability distribution? Explain.
Practice 7
A model output is acceptable only if \(y<50\). The nominal prediction is \(y=47\), but uncertainty analysis gives
\[ 44\le y\le52. \]Explain why the nominal prediction alone is insufficient for the decision.
Practice 8
For
\[ y=x^2, \]estimate the change in \(y\) near \(x=10\) when \(x\) changes by \(\Delta x=0.1\), using
\[ \Delta y\approx f'(x)\Delta x. \]Practice Solutions
Practice 1
The nominal value is
\[ \boxed{24.6\text{ cm}}, \]and the absolute uncertainty is
\[ \boxed{0.2\text{ cm}}. \]Practice 2
The relative uncertainty is
\[ \frac{0.8}{80.0}=0.01. \]Therefore, the percent uncertainty is
\[ \boxed{1\%}. \]Practice 3
Multiply the entire input range by \(5\):
\[ 5(3.8)\le y\le5(4.2). \]Thus,
\[ \boxed{19\le y\le21}. \]Practice 4
The lower revenue is
\[ R_{\min}=14(1200)=16800, \]and the upper revenue is
\[ R_{\max}=16(1200)=19200. \]Therefore,
\[ \boxed{16800\le R\le19200}. \]Practice 5
The readings can be precise because they agree closely with one another. They can still be inaccurate if calibration bias shifts all of them away from the true value in the same direction.
Practice 6
The value \(0.72\) is best described as a nominal parameter value. The notation states an uncertainty scale around the estimate, but it does not by itself define a full probability distribution.
Practice 7
The nominal value \(47\) is below the threshold, but the plausible output range extends above \(50\). Because some plausible model outcomes fail the requirement, the nominal prediction alone cannot establish acceptability.
Practice 8
Since
\[ f'(x)=2x, \]at \(x=10\),
\[ f'(10)=20. \]Therefore,
\[ \Delta y\approx20(0.1)=2. \]So the estimated output change is
\[ \boxed{\Delta y\approx2}. \]What to Remember
- Measured values are usually approximations, not exact mathematical constants.
- Measurement uncertainty reflects limitations and variability in observation.
- Parameter uncertainty reflects incomplete knowledge of quantities controlling a model.
- Random variation and systematic bias affect measurements differently.
- Absolute uncertainty is expressed in the units of the quantity.
- Relative uncertainty compares the uncertainty with the size of the quantity.
- A nominal value is a central reference value, not a guarantee of exactness.
- Uncertain inputs and parameters create uncertain model outputs.
- Uncertainty is most important when output variation can change a decision.
A strong model does not pretend uncertainty is absent. It makes uncertainty visible enough that predictions can be interpreted and used responsibly.
Where We Go Next
Ranges and \(\pm\) notation provide useful first descriptions of uncertain quantities, but many models require a more detailed representation of which values are more or less plausible.
The next lesson, Probability Distributions in Models, treats uncertain quantities as distributions and develops the language needed to represent variation probabilistically inside a model.