Error Types and Propagation

Every measurement carries an uncertainty. To report a result honestly you must know how individual errors combine when measured quantities are added, multiplied or raised to powers. NEET tests this with one-step propagation problems and definitions of absolute, relative and percentage error.

Concept

If $a_1, a_2, \dots, a_n$ are repeated measurements of a quantity, the best estimate is the mean $\bar a$. The absolute error of the $i$-th reading is $\Delta a_i = |\bar a - a_i|$. The mean absolute error is $\overline{\Delta a}$. The relative error is $\overline{\Delta a}/\bar a$ and percentage error is $100 \times \overline{\Delta a}/\bar a$.

Formula Derivation

For a sum or difference $Z = A \pm B$, absolute errors add:

$\Delta Z = \Delta A + \Delta B$

For a product or quotient $Z = A B$ or $Z = A/B$, relative errors add:

$\frac{\Delta Z}{Z} = \frac{\Delta A}{A} + \frac{\Delta B}{B}$

For a power $Z = A^{p} B^{q} / C^{r}$:

$\frac{\Delta Z}{Z} = p\,\frac{\Delta A}{A} + q\,\frac{\Delta B}{B} + r\,\frac{\Delta C}{C}$

The exponents always enter as positive contributions to fractional error.

NEET-style Worked Example

The density of a solid is computed from $\rho = m/V$, where $m = (50.0 \pm 0.1)\,g$ and $V = (10.0 \pm 0.2)\,cm^{3}$. Find the percentage error in $\rho$.

$\frac{\Delta m}{m} = \frac{0.1}{50.0} = 0.2\%$

$\frac{\Delta V}{V} = \frac{0.2}{10.0} = 2\%$

$\frac{\Delta \rho}{\rho} = 0.2\% + 2\% = 2.2\%$

Common Confusions

• For sums, absolute errors add; for products, relative errors add.
• A subtraction of nearly equal numbers magnifies relative error dramatically.
• Squaring a quantity doubles its relative error; cubing it triples it.
• Percentage error already includes the factor of 100; don't multiply again.

Key Takeaways

• $\Delta(A \pm B) = \Delta A + \Delta B$.
• $\Delta(AB)/AB = \Delta A/A + \Delta B/B$.
• Powers multiply the fractional error.
• Always quote results to the same precision as the error.