Uncertainty and error analysis

Lesson 2 showed that every reading comes with a range of doubt. This lesson makes that doubt quantitative. Uncertainty analysis answers two questions that come up in every lab: how well do I know this measured value, and how well do I know a result calculated from several measured values? The method used here follows the international Guide to the Expression of Uncertainty in Measurement, usually called the GUM, which is the basis of calibration certificates and of uncertainty statements in standards and test reports in Canada, the United States and most other countries. You do not need its full machinery for a teaching lab, but its core ideas fit on a page and they are used everywhere.

Amara gives Leo a concrete job. The pedestrian controller drives each LED signal head through a 2.2 Ω current-sense resistor, and the firmware estimates the LED current from the voltage across it. Leo must measure that voltage repeatedly, compute the current and the power dissipated in the resistor, and state both with uncertainties so that Amara can decide whether the resistor's 0.5 W rating leaves enough margin.

Error and uncertainty

The error of a measurement is the difference between the measured value and the true value. We never know it exactly, because we never know the true value exactly; if we did, we would not need to measure. Uncertainty is our quantitative estimate of how large the error could reasonably be. A result written as 0.418 ± 0.004 V says that the true value very probably lies within that interval, and the statement should also say how probable. The GUM expresses each source of uncertainty as a standard uncertainty, written u, which has the meaning of a standard deviation. Standard uncertainties from different sources are then combined, and the result is multiplied by a coverage factor to give an interval with a stated level of confidence.

Type A: uncertainty from repeated readings

A Type A evaluation uses statistics on repeated measurements. Take n independent readings x₁ to xₙ. Their mean is the best estimate of the value, and their experimental standard deviation s describes the scatter of individual readings. The uncertainty of the mean is smaller than the scatter of single readings, because averaging reduces random error: it is the standard deviation of the mean, s/√n.

The divisor n − 1 rather than n in s corrects for the fact that the deviations are measured from the sample mean rather than the unknown true mean. For small n, the estimate s is itself uncertain, which is why a mean of only three readings deserves a larger coverage factor than a mean of fifty. With ten or more readings the difference is modest, and teaching labs usually ignore it, but it is worth knowing that a standard deviation from three readings can easily be off by a factor of two.

Type B: uncertainty from other information

A Type B evaluation uses everything except repeated readings: the instrument specification, a calibration certificate, a resistor tolerance, the resolution of the display, or experience. The trick is to turn each of those into a standard uncertainty. When a specification gives limits ±a and nothing else is known about where in that range the error lies, the GUM treats every value in the range as equally likely, a rectangular distribution, whose standard deviation is a/√3. A meter specified to ±0.017 V therefore contributes about 0.0098 V of standard uncertainty. A calibration certificate that reports an expanded uncertainty U with coverage factor k = 2 contributes U/2. The display's resolution, one count of width δ, is a rectangle of half-width δ/2, contributing δ/(2√3), which is usually negligible when the accuracy specification already covers it.

Combining uncertainties

When independent sources each contribute a standard uncertainty to the same quantity, they combine in quadrature, as the square root of the sum of the squares. Adding them directly would assume that every source is at its worst limit in the same direction at once, which is unlikely for independent errors. The root-sum-square, or RSS, gives the combined standard uncertainty u_c.

Finally, the expanded uncertainty U = k·u_c gives an interval x̄ ± U. With k = 2 and a roughly normal result, the interval has a coverage probability of about 95 %, and that is the usual choice on calibration certificates and in reports. Always state k. A reader cannot interpret ±0.004 V without knowing whether it is one standard uncertainty, a 95 % interval or a worst-case limit.

Propagating uncertainty through a calculation

Most lab results are calculated from several measurements: a current from a voltage and a resistance, a power from a current and a voltage, a time constant from a resistance and a capacitance. If y = f(x₁, x₂, …) and the inputs are independent, each input's standard uncertainty contributes to y in proportion to how sensitive y is to that input, which is the partial derivative ∂f/∂xᵢ, called the sensitivity coefficient. The contributions then combine in quadrature.

Two shortcuts cover most lab calculations. For sums and differences, y = x₁ ± x₂, the absolute uncertainties combine in quadrature. For products, quotients and powers, y = c·x₁ᵃ·x₂ᵇ, the relative uncertainties combine in quadrature, each multiplied by its exponent: (u_y/y)² = (a·u₁/x₁)² + (b·u₂/x₂)². A quantity that appears squared therefore contributes twice its relative uncertainty, which is why the power P = V²/R is more sensitive to the voltage than to the resistance.

One warning: the quadrature rule assumes the inputs are independent. If two readings are taken with the same meter on the same range, their calibration errors are correlated, and some of the error cancels in a ratio or adds in a sum. A classic example is measuring the gain of an amplifier as the ratio of two voltages read on the same meter and range: the meter's gain error divides out, and the uncertainty of the ratio is much smaller than the quadrature sum of the two readings would suggest. The GUM handles this with covariance terms. For lab work, it is enough to notice when correlation is likely and to say which way it pushes the result.

# Type A + Type B uncertainty for the sense-resistor voltage, then propagation to I and P
import math, statistics as st
v = [0.4184, 0.4179, 0.4188, 0.4181, 0.4186, 0.4183, 0.4177, 0.4185, 0.4182, 0.4180]
n = len(v); vbar = st.mean(v); s = st.stdev(v)
uA = s / math.sqrt(n)
a = 0.0005 * vbar + 5 * 0.0001          # meter spec limit on the 6 V range
uB = a / math.sqrt(3)
uV = math.hypot(uA, uB)
print(f"V = {vbar:.5f} V, s = {s*1e3:.3f} mV, uA = {uA*1e3:.3f} mV, uB = {uB*1e3:.3f} mV, uc = {uV*1e3:.3f} mV")

R = 2.2; uR = 0.01 * R / math.sqrt(3)   # 1 % tolerance resistor, rectangular
I = vbar / R
uI = I * math.hypot(uV / vbar, uR / R)
P = vbar**2 / R
uP = P * math.hypot(2 * uV / vbar, uR / R)
print(f"I = {I*1e3:.2f} mA +/- {2*uI*1e3:.2f} mA (k=2)")
print(f"P = {P*1e3:.2f} mW +/- {2*uP*1e3:.2f} mW (k=2); rating 500 mW, margin x{0.5/P:.1f}")

# Monte Carlo check of the propagation for P
import random
random.seed(1)
samples = []
for _ in range(100000):
    vv = random.gauss(vbar, uV)
    rr = R * (1 + random.uniform(-0.01, 0.01))
    samples.append(vv * vv / rr)
print(f"Monte Carlo: P mean {st.mean(samples)*1e3:.2f} mW, std {st.stdev(samples)*1e3:.3f} mW "
      f"vs propagated {uP*1e3:.3f} mW")

V = 0.41825 V, s = 0.337 mV, uA = 0.107 mV, uB = 0.409 mV, uc = 0.423 mV
I = 190.11 mA +/- 2.23 mA (k=2)
P = 79.52 mW +/- 0.97 mW (k=2); rating 500 mW, margin x6.3
Monte Carlo: P mean 79.52 mW, std 0.486 mW vs propagated 0.486 mW

The program confirms the hand result and adds the current and power. The 1 % resistor tolerance is now the largest single contributor to the current's uncertainty, ahead of the meter. The power is about 80 mW, so the 0.5 W resistor runs at about a sixth of its rating; Amara is satisfied. The Monte Carlo section draws a hundred thousand random combinations of voltage and resistance, computes the power for each, and finds the same spread as the formula. Monte Carlo simulation is the GUM's recommended method when the formula is non-linear or the inputs are far from normal, and with a few lines of code it is often easier than taking derivatives.

Uncertainty budgets

An uncertainty budget is a table that lists each source, its type, its standard uncertainty, its sensitivity coefficient and its contribution to the result. It shows at a glance which source dominates, and therefore where effort will pay. If the budget shows that the resistor tolerance dominates, buying a better meter is wasted money; measuring the resistor itself with a four-wire ohmmeter, as lesson 8 describes, would help far more.