Error Propagation: Addition & Subtraction

The absolute uncertainty $\Delta x$ of a measurement $x$ is the half-width of the range within which the true value is believed to lie. For example, if a ruler reads $12.4 \text{ cm}$ with an uncertainty of $\pm 0.1 \text{ cm}$, then $x = 12.4 \text{ cm}$ and $\Delta x = 0.1 \text{ cm}$. The full measurement is written $x \pm \Delta x$.

Suppose you measure two quantities $A = a \pm \Delta a$ and $B = b \pm \Delta b$, and you compute their sum $Z = A + B$. In the worst case, both measurements could simultaneously be at the high end of their ranges (or both at the low end). Therefore, the maximum possible deviation of $Z$ from its central value is $\Delta a + \Delta b$. The result is: $$Z = (a + b) \pm (\Delta a + \Delta b)$$

For subtraction $Z = A - B$, the same worst-case logic applies. If $A$ is at its maximum and $B$ is at its minimum (or vice versa), the deviation of $Z$ is again maximized at $\Delta a + \Delta b$. So: $$Z = (a - b) \pm (\Delta a + \Delta b)$$ Notice that even though we subtract the central values, we still ADD the uncertainties. Subtracting measurements never reduces the uncertainty — it always increases or maintains it.

This can be understood algebraically. If $A$ lies in the interval $[a - \Delta a,\; a + \Delta a]$ and $B$ lies in $[b - \Delta b,\; b + \Delta b]$, then $Z = A - B$ lies in the interval $[(a - b) - (\Delta a + \Delta b),\; (a - b) + (\Delta a + \Delta b)]$. The half-width of this interval is exactly $\Delta a + \Delta b$. This is the maximum (or 'worst-case') propagation rule used at high school level. In advanced statistics, if errors are random and independent, the combined uncertainty is $\sqrt{(\Delta a)^2 + (\Delta b)^2}$, which is smaller — but the linear addition rule is always safe and conservative.

The rule extends naturally to any number of terms. If $Z = A + B - C + D - \ldots$, then $\Delta Z = \Delta a + \Delta b + \Delta c + \Delta d + \ldots$ All absolute uncertainties are summed, regardless of whether the corresponding term is added or subtracted in the formula for $Z$.

After finding $\Delta Z$, you can compute the fractional (relative) uncertainty as $\frac{\Delta Z}{Z}$, often expressed as a percentage: $\frac{\Delta Z}{Z} \times 100\%$. This tells you how significant the uncertainty is relative to the measured value. Notably, when two nearly equal quantities are subtracted, $Z = a - b$ can be very small while $\Delta Z = \Delta a + \Delta b$ remains large — making the fractional uncertainty extremely large. This is called catastrophic cancellation and is a key reason why experimental physicists try to avoid subtracting nearly equal quantities.

• \Delta Z — Absolute uncertainty in the result (same as Z (e.g., m, kg, s)): The uncertainty associated with the calculated quantity Z
• \Delta a — Absolute uncertainty of measurement A (same as a): The uncertainty in the first measured quantity
• \Delta b — Absolute uncertainty of measurement B (same as b): The uncertainty in the second measured quantity

Problem. A student measures the length of two metal rods with a ruler that has an uncertainty of $\pm 0.1 \text{ cm}$. Rod A measures $23.5 \text{ cm}$ and Rod B measures $18.2 \text{ cm}$. (a) What is the total length of both rods placed end-to-end, with its uncertainty? (b) What is the difference in their lengths, with its uncertainty? (c) Calculate the fractional uncertainty in the difference.

Answer. Total length: 41.7 ± 0.2 cm; Difference: 5.3 ± 0.2 cm; Fractional uncertainty in difference ≈ 3.8%

• Types of Measurement Errors — Understanding systematic and random errors is prerequisite to knowing why uncertainties exist and must be propagated in the first place.
• Error Propagation: Multiplication and Division — The natural next step after addition and subtraction — multiplication and division use fractional (relative) uncertainties instead of absolute ones, requiring a different but complementary rule.
• Significant Figures — Significant figures are closely related to uncertainty: the number of significant figures in a result should reflect the precision of the measurement, which is determined by error propagation.
• Absolute and Fractional Uncertainty — Error propagation in addition and subtraction is expressed in terms of absolute uncertainties; understanding the distinction between absolute and fractional uncertainty is essential for applying the correct rule.