Propagation of uncertainty

A calculation made from measured inputs inherits their uncertainty. Reporting only the central output hides how strongly the result depends on those inputs. Analytic formulas can propagate small uncertainties through simple functions; Monte Carlo sampling offers a flexible numerical approach when the function or input distribution is more complex.

Define what the uncertainty means

An expression such as

\(v=(5.0\pm0.1)\,\mathrm{m/s}\)

is incomplete until the plus-minus value is interpreted. It might be an instrument resolution, a standard deviation, a standard uncertainty, or a coverage interval. These mean different things and cannot be sampled interchangeably.

This lesson treats input uncertainties as standard deviations of a stated joint probability model. That is a modeling assumption, not a universal meaning of error bars. Systematic effects, calibration uncertainty, finite resolution, and sampling variation may require different models.

A distance calculated from speed and time

For constant speed, distance is

\[d=vt\]

where \(v\) is in metres per second, \(t\) is in seconds, and \(d\) is in metres. Let the central inputs be

\[v=5.0\,\mathrm{m/s},\qquad t=12.0\,\mathrm{s}\]

so the central distance is \(60.0\,\mathrm{m}\). Suppose their standard uncertainties are

\(\sigma_v=0.1\,\mathrm{m/s}\)

and \(\sigma_t=0.2\,\mathrm{s}\).

For small uncertainties, first-order propagation for a function of two inputs is

\[\sigma_d^2\approx\left(\frac{\partial d}{\partial v}\right)^2\sigma_v^2+\left(\frac{\partial d}{\partial t}\right)^2\sigma_t^2+2\frac{\partial d}{\partial v}\frac{\partial d}{\partial t}\operatorname{cov}(v,t)\]

For \(d=vt\), the partial derivatives are \(\partial d/\partial v=t\) and \(\partial d/\partial t=v\). If the input errors are independent, their covariance is zero and

\[\sigma_d\approx\sqrt{t^2\sigma_v^2+v^2\sigma_t^2}\]

For the stated values,

\[\sigma_d\approx\sqrt{(12.0\times0.1)^2+(5.0\times0.2)^2}\,\mathrm{m}=1.56\,\mathrm{m}\]

A suitable rounded report might be

\(d=(60.0\pm1.6)\,\mathrm{m}\)

with an explicit statement that the plus-minus value is a standard uncertainty under the independent-input model.

Why covariance matters

Independence cannot be assumed merely because two values have different variable names. If speed and duration are estimated from the same position-time data, their errors may be correlated. The covariance term can increase or decrease output variance depending on its sign.

Sampling each input independently would force covariance to zero and understate or overstate uncertainty. A Monte Carlo model must sample from the joint distribution, perhaps using paired observations, a covariance matrix, or a shared underlying error source.

Monte Carlo propagation

The numerical procedure samples plausible input pairs and evaluates the original physical relationship for each pair.

import numpy as np

rng = np.random.default_rng(42)
sample_count = 100_000
speed_metres_per_second = rng.normal(3.0, 0.1, sample_count)
duration_seconds = rng.normal(2.0, 0.05, sample_count)
# Independent, narrow normal distributions are assumed here.
distance_metres = speed_metres_per_second * duration_seconds
print(f"Mean: {distance_metres.mean():.3f} m")
print(f"Standard deviation: {distance_metres.std(ddof=1):.3f} m")
print("Central 95% interval (m):", np.quantile(distance_metres, [0.025, 0.975]))

For independent normal inputs, a compact implementation might look like this:

from collections.abc import Callable
from dataclasses import dataclass


@dataclass(frozen=True)
class NormalInput:
    mean: float
    standard_deviation: float


def sample_distances(
    speed: NormalInput,
    duration: NormalInput,
    sample_count: int,
    sample_normal: Callable[[NormalInput], float],
) -> list[float]:
    distances: list[float] = []

    for _ in range(sample_count):
        speed_value = sample_normal(speed)
        duration_value = sample_normal(duration)
        distances.append(speed_value * duration_value)

    return distances

The random generator is passed in so tests can provide a seeded implementation. Units remain part of the surrounding contract: the returned numbers represent metres only if the sampled speed and duration use compatible units.

Validate the propagation code

Start with cases whose answer is known. When both input standard deviations are zero, every output sample must equal the deterministic distance. If only duration varies and speed is fixed, multiplying all sampled durations by speed should multiply both their mean and standard deviation by that speed.

For small independent normal uncertainties in the worked example, the simulated output standard deviation should approach the first-order value of about \(1.56\,\mathrm{m}\) as the sample count grows. It will not match exactly in a finite run because the Monte Carlo summary has sampling variability.

Run several sample counts and several seeds. The estimated output statistics should stabilize within an expected Monte Carlo error. Reusing one favorable seed is not evidence of convergence.

Probability models must respect physics

A normal distribution extends without bound in both directions. It assigns some probability to negative durations and, for a speed magnitude, negative values that may not match the intended quantity. When the mean is many standard deviations from zero, that probability may be negligible; otherwise a normal model is inappropriate.

Possible alternatives include a truncated distribution, a lognormal distribution for a positive multiplicative quantity, a uniform distribution for bounded resolution, or direct resampling of observed data. Choose from knowledge of the measurement process, not from convenience or habit.

Hard-clamping invalid samples to zero changes the distribution by accumulating probability at the boundary. Rejecting and resampling creates a truncated distribution. Those are different models and should be named.

Quantiles and intervals

If the output distribution is skewed, a mean plus-or-minus one standard deviation may describe it poorly. Report quantiles such as the median and a central interval, or the probability of crossing a decision threshold. State whether an interval is central, equal-tailed, or defined in another way.

A Monte Carlo interval reflects only uncertainties included in the input model. It does not automatically include systematic bias, model-form error, or numerical error in the physical calculation. Those components need separate assessment or explicit sampling models.

Sensitivity and uncertainty are related, not identical

Changing the time standard deviation from \(0.2\,\mathrm{s}\) to \(0.4\,\mathrm{s}\) while holding the speed model fixed increases the distance spread. The analytic expression predicts how much under the independence and linearization assumptions.

That sensitivity study does not prove the larger uncertainty is realistic. It answers a conditional question: how would the output respond if this input uncertainty changed? Evidence about the instrument or measurement process determines which input model is credible.

Uncertainty propagation is therefore not a machine for manufacturing confidence intervals. It is a transparent calculation that connects an explicit model of input knowledge to an explicit model of output knowledge. Its credibility depends as much on the input assumptions and correlations as on the number of samples.

Preserve a reproducible record

Record the input distribution parameters, units, correlation model, random-number generator, seed, sample count, rejected-sample policy, and summary definition. Also record the version of the physical function evaluated by each sample. Without that information, another run may produce a different interval for reasons that cannot be separated from ordinary sampling variability.

Keep enough digits internally to reproduce the summary, but report only precision supported by the uncertainty estimate and Monte Carlo error. Saving every sample is not always necessary; for large runs, validated streaming statistics or selected quantiles may be sufficient. The retention choice should still allow important claims to be checked.

A final report should name exclusions. If calibration bias, model-form uncertainty, or input correlation was not quantified, say so beside the interval. A narrow numerical distribution is not a complete statement of confidence when important uncertainty sources were never included.

A worked Python notebook

Examine how uncertainty in a speed measurement changes the distance covered in two seconds. The question for this calculation is precise: how does travel distance change when we change measured speed, while keeping the other choices fixed? The earlier sections explain the physical relationship. Here we turn one part of that relationship into a calculation that you can run, inspect, and change in Google Colab.

The duration is held fixed here; correlated uncertain inputs need a joint distribution. Keep that sentence beside the results. A formula can return a number outside its useful range, and a plotting library will usually draw that number without asking whether the physical assumptions still hold. The selected interval is from 2.5 to 3.5 m/s. It gives us a specific domain to discuss instead of leaving the graph's range to a software default.

Use the notebook link above the lesson, run the setup cell, and then run the cells in order. The code below is also included there. NumPy handles the arrays of numerical values, SymPy keeps the equation symbolic, and Matplotlib draws the results. If these libraries or notebook cells are unfamiliar, start with Your first physics notebook.

State the relationship before choosing a graph

In this small example, q stands for measured speed, expressed in m/s. The output is travel distance, expressed in m. The equation below uses the numerical values in those units. Any fixed coefficients therefore carry the units needed to make the relation consistent. A value supplied in another unit must be converted before it enters this expression.

\[f(q) = 2 q\]

This is a function of one changing input. It is not a statement that the physical system has only one variable. We deliberately hold the other inputs fixed so that one cause of variation is visible. Changing several inputs at once would make it harder to tell which change produced the output difference. Write down the held-fixed choices before trying a different experiment.

At 3 m/s, the expression gives 6 m. This is our reference calculation. It is a prediction of the stated example, not a new measurement. Its displayed digits help us compare implementations. They should not be read as a claim that physical inputs are known to six significant figures.

import numpy as np
import sympy as sp
import matplotlib.pyplot as plt

q = sp.Symbol("q", real=True)
expression = sp.sympify("2*q", locals={"q": q})
reference_input = 3.0
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "m/s")
print("Predicted travel distance:", reference_output, "m")

sp.Symbol declares a mathematical symbol. It does not assign a numerical measurement to that symbol. subs then replaces the symbol with our chosen input, and float converts the resulting symbolic number into a standard Python numerical value. Keeping these steps separate lets us inspect the equation before numerical substitution. It also prevents a later plot from becoming the only surviving record of the model.

Compare a few cases you can explain

Before drawing hundreds of points, calculate three. The lower endpoint gives 5 m. The midpoint, at 3 m/s, gives 6 m. The upper endpoint gives 7 m. Read these together with the assumptions: they describe different choices of measured speed within the same example, with the other inputs held fixed.

These three values are a useful first check, but they do not establish the shape between them. Two endpoints can be equal even when the curve bends or oscillates in between. Likewise, a large change between endpoints does not tell us whether the relationship is linear. We need the equation and intermediate samples to answer those questions.

evaluate = sp.lambdify(q, expression, modules="numpy")
inputs = np.array([2.5, 3.0, 3.5])
outputs = np.asarray(evaluate(inputs), dtype=float)
outputs = np.broadcast_to(outputs, inputs.shape)
for input_value, output_value in zip(inputs, outputs):
    print(f"{input_value:.6g} m/s -> {output_value:.6g} m")
assert np.all(np.isfinite(outputs))

lambdify turns the symbolic expression into a numerical function. We explicitly select NumPy as its numerical backend so it can accept an array. The same expression now has two useful forms: one for algebra and one for calculating many cases. The assertion checks that the chosen inputs produce finite outputs. It would catch an infinity or an undefined numerical result, but a finite value can still be physically inappropriate.

Make a graph that answers the question

The horizontal axis shows measured speed; the vertical axis shows travel distance. Every plotted point is a prediction of the same expression. We mark the reference case separately so that a reader can connect the earlier printed value to its place on the curve. The axis labels include units because a shape without a scale cannot tell us the size of a physical effect.

input_grid = np.linspace(2.5, 3.5, 301)
output_grid = np.asarray(evaluate(input_grid), dtype=float)
output_grid = np.broadcast_to(output_grid, input_grid.shape)
fig, ax = plt.subplots(figsize=(7, 4))
ax.plot(input_grid, output_grid, label="Stated model")
ax.scatter([reference_input], [reference_output], label="Reference case", zorder=3)
ax.set_xlabel("Measured speed (m/s)")
ax.set_ylabel("Travel distance (m)")
ax.legend()
ax.grid(alpha=0.25)
fig.tight_layout()
plt.show()

Look for a sign change, a turning point, or a region where the curve becomes steep. Connect each feature to the expression rather than to the colour of the line. If the curve has a minimum or maximum, changing the input in opposite directions may have similar effects on the output near that point. If it is nearly straight over a short interval, a local linear approximation may be useful even when the complete expression is nonlinear.

Increasing the number of samples makes the line smoother and may reveal features that a coarse grid missed. It does not improve the physical inputs or enlarge the model's validity range. For quantities that can take only discrete values, use the line as a guide to the relationship and interpret only the allowed input values. A plotting function does not enforce such restrictions for you.

Measure the local sensitivity

A derivative asks a more specific question than the whole graph: how quickly does travel distance change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 3 m/s is 2 in those ratio units. Locally, the output increases as the input increases. This statement is local; the sign or size of the slope can be different elsewhere on the curve.

\[f'(q) = 2\]

For a small input change, multiply the derivative by that change to estimate the output change. Then compare that estimate with a fresh evaluation of the full expression. The distinction between these two calculations matters: the derivative gives a tangent approximation, while substitution evaluates the chosen expression at the new point.

slope_expression = sp.diff(expression, q)
slope_at_reference = float(slope_expression.subs(q, reference_input))
changed_input = 3.005
linear_prediction = reference_output + slope_at_reference * (
    changed_input - reference_input
)
full_prediction = float(evaluate(changed_input))
print("Symbolic derivative:", slope_expression)
print("Tangent prediction:", linear_prediction, "m")
print("Full expression:", full_prediction, "m")
print("Approximation difference:", full_prediction - linear_prediction, "m")

Here the changed input is 3.005 m/s. The tangent estimate is 6.01 m, while the full expression gives 6.01 m. If the two values are close, it means the chosen input change is small enough for this local approximation at this point. It does not mean the entire model is linear. If the slope is zero, a quadratic or higher-order term may supply the leading change instead.

This is also the beginning of uncertainty propagation. If the input has a small stated standard uncertainty, the magnitude of the slope times that uncertainty estimates one contribution to output uncertainty. Do not automatically use the same rule for a wide interval, a discontinuity, a turning point with zero first derivative, or several correlated uncertain inputs. Those situations need more of the model than one slope.

Check the derivative with numerical differences

We can estimate the same slope from values on either side of the reference point. Subtract the lower output from the upper output and divide by twice the input step. Parentheses are essential: the entire output difference must be divided by the entire input interval. Without them, Python can execute a different mathematical expression while still producing a plausible number.

The calculation below uses three decreasing step sizes. Each estimate is compared with the symbolic derivative, so the reference is independent of the numerical difference formula. Near an endpoint, it is acceptable here to evaluate a small extension only because the displayed mathematical expression is smooth there; such a check does not assert that extrapolated values describe additional physical cases.

input_span = 1.0
for fraction in [1e-2, 1e-3, 1e-4]:
    step = input_span * fraction
    upper = float(evaluate(reference_input + step))
    lower = float(evaluate(reference_input - step))
    numerical_slope = (upper - lower) / (2 * step)
    difference = numerical_slope - slope_at_reference
    print(f"step={step:.6g}: slope={numerical_slope:.6g}, difference={difference:.6g}")

For a smooth function, reducing the step often reduces the approximation error at first. Very small steps can eventually make subtraction of nearly equal floating-point values less reliable. Therefore “smaller” is not an unlimited guarantee of “better.” Compare the observed behaviour over a useful range of steps instead of choosing the smallest number the computer can represent.

An absolute slope difference is useful when the expected slope is near zero. A relative difference would divide by a very small reference and could look enormous even when the absolute discrepancy is tiny. Away from zero, a relative difference can help compare quantities with very different scales. The tolerance should reflect the question and the numerical method, rather than a universal decimal threshold copied into every test.

Change the example and preserve the reasoning

Make one copy of the reference calculation before editing it. Choose a second value of measured speed inside the stated interval, predict whether travel distance will rise or fall, and then evaluate the expression. Place the prediction and the result in a text cell. If they disagree, identify whether the mistake was in the sign, the unit conversion, the algebra, or the assumption about how the curve behaves.

Next choose a shorter plotting interval around the reference point. Compare the local curve with its tangent approximation. Then widen the interval again. The difference shows why a useful local estimate need not describe the whole range. Keep the axis labels and numerical scale visible when comparing the plots; automatic axis limits can make two different effects appear equally large.

Finally, return to the physical limit stated at the start: The duration is held fixed here; correlated uncertain inputs need a joint distribution. Describe one change to the setup that would violate that limit. State what new input, equation, or observation would be needed before trusting an answer in the changed situation. This last step connects the Python experiment to the physics instead of treating successful execution as the only criterion for a useful result.

Before saving, restart the runtime and run all cells in order. A result that depends on an earlier, now-deleted input cell is not reproducible from the saved notebook. Keep unrounded values for later calculations, round only the reported output, and label any new data as measured, assumed, or generated. The graph and printed values should remain traceable to the same expression and the same choices of inputs.

Recover the output change by integrating the slope

There is another way to check that the expression and its derivative describe the same relationship. Add up the small output changes predicted by the slope across the chosen input interval. In the continuous limit, this integral equals the output at the upper endpoint minus the output at the lower endpoint. The result has units of m: the input unit in the integration cancels the input unit in the denominator of the slope.

This statement is about the mathematics of this particular relationship. It does not assign a new physical meaning to an arbitrary area under a graph. For example, an area under force versus distance represents work, while an area under a different pair of quantities may represent something else or have no standard name. Always carry the axis units through the calculation before naming an integral.

slope_function = sp.lambdify(q, slope_expression, modules="numpy")
endpoint_change = float(evaluate(3.5) - evaluate(2.5))
for points in [101, 201, 401]:
    grid = np.linspace(2.5, 3.5, points)
    slopes = np.asarray(slope_function(grid), dtype=float)
    slopes = np.broadcast_to(slopes, grid.shape)
    integrated_change = np.trapezoid(slopes, grid)
    print(
        points,
        "integrated change:",
        integrated_change,
        "endpoint change:",
        endpoint_change,
    )

The trapezoidal rule joins neighbouring slope samples with straight segments and adds their signed areas. A negative contribution remains negative; replacing every slope by its magnitude would calculate a different quantity. Increasing the sample count provides a practical refinement check. If the values do not settle toward the endpoint difference, first inspect the interval, singularities, and array ordering.

Do not conclude that two agreeing calculations prove the model is physically complete. Both use the same selected relationship, so an omitted effect would be omitted from both. Their agreement is evidence that the numerical tools implement that relationship consistently. Comparing with an independent experiment or a more appropriate physical model is a separate test.

Keep this comparison in the notebook when you change the expression. It is particularly useful after adding a coefficient, changing a sign convention, or converting an input unit. A code cell that produces a smooth graph can conceal these mistakes; a second calculation of the same output change gives you another place to notice them.

Sources

Share to: