Combining errors in calculations

The error travels into the answer

You measure the sides of a table and multiply them to get its area. Each side has an error. So the area has an error too. How big?

The rules below give the largest error the result can have. They are what your board exam and the JEE use.

Sums and differences: the absolute errors add

Two lengths, $A \pm \Delta A$ and $B \pm \Delta B$. In the worst case both errors push the same way, so

Watch out. Even for a difference the errors add. Subtracting never makes a result more certain. Worse: subtract two nearly equal numbers, and the small difference can have a huge percentage error.

Products and quotients: the relative errors add

For $Z = A \times B$ or $Z = A / B$:

See why with a rectangle. The extra area from an error in each side is a thin strip, and the two strips add.

You should find the rule and the drawing agree closely. The tiny difference is the corner square, $\Delta L \times \Delta B$, which the rule leaves out because it is so small in a real measurement.

Powers: multiply by the power

$Z = A^{n}$ is $A \times A \times \cdots$, so

All together, for $Z = \dfrac{A^{p} B^{q}}{C^{r}}$:

A square root is the power $\frac{1}{2}$, and it halves the relative error.

Which measurement matters most?

The quantity with the highest power, or the largest relative error, controls the result. In the volume of a sphere, $V = \frac{4}{3}\pi r^3$, a $1\%$ error in $r$ becomes $3\%$ in $V$. So measure $r$ with the best instrument you have.

A worked example

A sheet of A4 paper measures $(29.7 \pm 0.1)\ \text{cm}$ by $(21.0 \pm 0.1)\ \text{cm}$. Find its area with the error.

1. Area. $29.7\ \text{cm} \times 21.0\ \text{cm} = 623.7\ \text{cm}^2$.
2. Relative errors. $\dfrac{0.1\ \text{cm}}{29.7\ \text{cm}} = 0.34\%$ and $\dfrac{0.1\ \text{cm}}{21.0\ \text{cm}} = 0.48\%$.
3. Their sum. $0.81\%$, so $\Delta A = 0.0081 \times 623.7\ \text{cm}^2 \approx 5\ \text{cm}^2$.
4. Result. $(624 \pm 5)\ \text{cm}^2$.

Think first. Why does the shorter side give the bigger share? The same $0.1\ \text{cm}$ is a larger part of a shorter length.

Practise with new numbers

Try it in Python

The program finds the area of the A4 sheet with the rule, and checks it the slow way, with the largest and smallest areas.

Remember these

• Sum or difference: add the absolute errors.
• Product or quotient: add the relative errors.
• Power $n$: multiply the relative error by $n$ (a square root halves it).
• Errors always add; the highest power and the largest relative error matter most.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Two masses, $(4.2 \pm 0.1)\ \text{kg}$ and $(2.6 \pm 0.1)\ \text{kg}$. Find their sum with its error.

2. A speed is found as $v = \dfrac{d}{t}$, with $d$ known to $1\%$ and $t$ to $3\%$. What is the error in $v$?

3. The radius of a sphere is measured to $0.5\%$. What is the percentage error in its volume?

4. Why is the fall in a water level, found as one reading minus another, often very uncertain?