Error Propagation

Every measurement carries an uncertainty. Knowing how those uncertainties combine when quantities are added, multiplied or raised to powers is essential for JEE experimental questions and for choosing the most sensitive measuring instrument.

Concept

For independent random errors:

• Sum/difference $z = x \pm y$: $\Delta z = \Delta x + \Delta y$ (absolute errors add).
• Product/quotient $z = x y$ or $z = x/y$: $\dfrac{\Delta z}{z} = \dfrac{\Delta x}{x} + \dfrac{\Delta y}{y}$ (relative errors add).
• Power $z = x^{n}$: $\dfrac{\Delta z}{z} = |n|\dfrac{\Delta x}{x}$.

For a compound formula $z = \dfrac{a^{p} b^{q}}{c^{r}}$: $\frac{\Delta z}{z} = p\frac{\Delta a}{a} + q\frac{\Delta b}{b} + r\frac{\Delta c}{c}.$

The least count of an instrument fixes the minimum $\Delta x$; vernier callipers typically give $0.1\,\text{mm}$, screw gauges $0.01\,\text{mm}$.

Derivation

Consider $z = a b$. Differentiating: $\mathrm{d}z = b\,\mathrm{d}a + a\,\mathrm{d}b$. Dividing by $z = ab$: $\frac{\mathrm{d}z}{z} = \frac{\mathrm{d}a}{a} + \frac{\mathrm{d}b}{b}.$ Treating maximum errors as absolute values gives the additive rule for relative errors. For $z = a^{n}$, $\ln z = n \ln a$, so $\mathrm{d}z/z = n\,\mathrm{d}a/a$.

JEE Worked Example

Q. A physical quantity $X$ is given by $X = \dfrac{M^{2} L^{1/2}}{T^{3}}$. Measurements have errors: $\Delta M/M = 2\%$, $\Delta L/L = 4\%$, $\Delta T/T = 1\%$. Find the maximum percentage error in $X$.

Solution. $\frac{\Delta X}{X} = 2\cdot 2\% + \tfrac{1}{2}\cdot 4\% + 3\cdot 1\% = 4 + 2 + 3 = 9\%.$

So $X$ is determined to within $9\%$.

Traps

• Forgetting the absolute value on negative exponents — errors always add positively.
• Using absolute errors in product formulas (must use relative).
• Confusing least count with zero error; correct for zero error before applying least count.
• Mixing percentage and fractional errors in the same expression.

Key Takeaways

• Sum $\Rightarrow$ absolute errors add; product $\Rightarrow$ relative errors add.
• Powers multiply relative errors by their magnitude.
• Always report final results to the same precision as the least precise measurement.