Limit Error

Knowledge of measurement uncertainty and error propagation in a series of calculations is crutial in evaluating the accuracy of a certain set of data. There are two major types of measurement uncertainties: random and systematic. In measuring the textbook's dimensions, for example, the uncertainty due to your limited ability to read the ruler to better than 1 mm is a random error. If, on the other hand, you mistakenly confuse the inch and the centimetre units, your measurements will include a systematic error of scale.

Returning to your exercise, the surface area ($A$) of the textbook is calculated using the equation ($A = L \times W$). If you multiply the measured values of length and width using a calculator you will get $A = 539.72 \; \text{cm}^2$. But how accurate is this value? How many significant figures are there in $A$, and what is its uncertainty? As a general rule for combining numbers by multiplication or division, the final result should not have more significant figures than the original value with the least number of significant figures. For addition and subtraction, however, the final result should not have more decimal places than the original value with the least number of decimal places. Therefore, since $L$ and $W$ have three significant figures each, $A$ should be rounded to three significant figures and written as $A = 540 \; \text{cm}^2$ (or $5.40 \times 10^2 \; \text{cm}^2$). Similarly, the volume of the textbook is written as $V = 2100 \; \text{cm}^3$ (or $2.1 \times 10^3 \; \text{cm}^3$), which should not include more than two significant figures.

What about the uncertainties of $A$ and $V$? Since the area and the volume of the textbook are calculated from the measured dimensions, the errors in $A$ and $V$ are propagation of the errors of $L$, $W$ and $T$. Table 0.2 lists the rules used to calculate the propagated errors in various arithmetic operations. Notice that whether we are adding or subtracting numbers, the combined error simply adds up. We use the same rule for combining error whether the operation is a multiplication or a division. Notice that in the last rule, $k$ is an exact constant factor that has no uncertainty (i.e. $\Delta k = 0$).

So, the uncertainty in area is given by \begin{align} \Delta A &= |A| \left( \frac{\Delta L}{L} + \frac{\Delta W}{W} \right) \nonumber\\ &= 540 \times \left( \frac{0.1}{26.2} + \frac{0.1}{20.6} \right) \nonumber\\ &= 4.7 \; \text{cm}^2 \end{align} which is reduced to $\Delta A = 5 \; \text{cm}^2$ because it is not an actual measurement. So, we see that the third digit in $A$ is uncertain, indicating that the number of significant figures in $A$ is indeed three. The final answer is then written as \begin{equation} A = (5.40 \pm 0.05)\times 10^2 \; \text{cm}^2 \end{equation} Similarly, we find that the volume of the textbook is equal to \begin{equation} V = (2.1 \pm 0.1)\times 10^2 \; \text{cm}^2 \end{equation}

It is often useful to speak of the relative error $\Delta x/x$ to compare the size of error to the size of the value measured. Another advantage is that the relative error of a product or dividend is just the sum of the relative errors of what makes them up.

Example: What is the uncertainty of the volume ($v$) of a cylinder in terms of its radius ($r$) and height ($h$)?

Solution: The volume of the cylinder is given by the formula $v = \pi r^2 h$. Using the rules in Table 0.2, the uncertainty in $v$ is calculated as follows \begin{align} \Delta v &= \pi \times \Delta (r^2 h) \nonumber\\ &= \pi \times |r^2 h| \left( \frac{\Delta r^2}{r^2} + \frac{\Delta h}{h} \right) \nonumber\\ &= (\pi r^2 h) \left( \frac{2 r \Delta r}{r^2} + \frac{\Delta h}{h} \right) \nonumber\\ &= v \left( \frac{2 \Delta r}{r} + \frac{\Delta h}{h} \right) \end{align}