Why 2.50 cm and 2.5 cm Are Not the Same Measurement

Every ruler has a smallest division and every stopwatch depends on a human thumb, so every reading carries some doubt. Physics does not hide that doubt — it measures it, and knowing its size turns a rough number into a trustworthy result.

This lesson covers accuracy, precision and the two kinds of error, how errors are calculated and combined, and the rules for significant figures.

Accuracy is how close a measurement is to the true value and precision is how close repeated measurements are to each other, while systematic errors push every reading the same way and random errors scatter readings on both sides.

Accuracy and precision. For a rod whose true length is 10.0 cm:

- Readings of 9.6, 10.4 and 10.0 cm average to the true value but spread widely — accurate but not precise
- Readings of 10.51, 10.52 and 10.51 cm cluster tightly but miss the true value — precise but not accurate

Systematic errors — always in one direction, so repetition does not remove them:

- Instrumental errors, such as a zero error in a vernier caliper
- Imperfect technique, such as taking body temperature under the arm, which reads low

Random errors — irregular errors from unpredictable changes such as fluctuations in temperature or reaction time; some readings come out high and some low, so averaging many readings reduces them.

Least count. An instrument cannot read below its smallest division: 1 mm for a metre scale, 0.1 mm for a vernier caliper and 0.01 mm for a screw gauge.

Worked example — correcting a zero error. A vernier caliper reads +0.03 cm with its jaws closed, so a reading of 2.47 cm is corrected to

$$2.47-0.03=2.44\text{ cm}$$

An everyday example. A vegetable seller's scale that shows 50 g with an empty pan makes every weighing 50 g too heavy — a systematic error that repeated weighing will never remove.

The substance. Averaging removes random error but not systematic error — a thousand readings taken with a faulty zero are all wrong by the same amount.

Absolute error is the size of a reading's difference from the mean, relative error is the mean absolute error divided by the mean, and percentage error is relative error times 100; absolute errors add for sums and differences, while relative errors add for products, quotients and powers.

$$\Delta a_{\text{mean}}=\frac{\sum |a_{\text{mean}}-a_i|}{n}\text{relative error}=\frac{\Delta a_{\text{mean}}}{a_{\text{mean}}}$$

Worked example 1 — a set of readings. A pendulum's period is measured as 2.63, 2.56, 2.42, 2.71 and 2.80 s:

$$T_{\text{mean}}=\frac{13.12}{5}=2.62\text{ s}$$

The absolute errors are 0.01, 0.06, 0.20, 0.09 and 0.18 s, so

$$\Delta T_{\text{mean}}=\frac{0.54}{5}=0.11\text{ s}\frac{0.11}{2.62}\times 100=4.2\text{ per cent}$$

The result is written as $T=2.6\pm 0.1$ s.

Combining errors:

- Sum or difference, $Z=A\pm B$: $\Delta Z=\Delta A+\Delta B$
- Product or quotient, $Z=AB$ or $A/B$: $\frac{\Delta Z}{Z}=\frac{\Delta A}{A}+\frac{\Delta B}{B}$
- Power, $Z=A^n$: $\frac{\Delta Z}{Z}=n\frac{\Delta A}{A}$

Worked example 2 — powers. For $Z=\frac{A^2{B}^{1/2}}{C^3}$, with a 1 per cent error in each of A, B and C, the percentage error in Z is

$$2(1)+\frac{1}{2}(1)+3(1)=5.5\text{ per cent}$$

An everyday example. Measuring the side of a square plot of farmland with a 2 per cent error gives a 4 per cent error in its area, because area depends on the square of the side.

The substance. Errors never cancel when quantities are subtracted — the absolute errors still add, so a small difference between two large readings can carry a large percentage error.

Significant figures are the reliable digits of a measurement plus the first uncertain digit; products and quotients keep as many significant figures as the least precise factor, while sums and differences keep as many decimal places as the least precise term.

Counting significant figures:

- All non-zero digits count, as do zeros between them: 3.007 has four
- Leading zeros do not count: 0.0052 has two
- Trailing zeros after a decimal point count: 2.50 has three
- Scientific notation removes doubt: $4.600\times 10^3$ m has four

In calculations:

- Multiplication and division keep the fewest significant figures:

$$\text{density}=\frac{4.237\text{ g}}{2.51{\text{ cm}}^3}=1.688=1.69{\text{ g cm}}^{-3}$$

- Addition and subtraction keep the fewest decimal places:

$$436.32\text{ g}+227.2\text{ g}+0.301\text{ g}=663.821\text{ g}=663.8\text{ g}$$

Rounding off. Raise the previous digit if the dropped digit is more than 5, as 2.746 becomes 2.75, and leave it if less than 5; if the dropped digit is exactly 5, leave an even previous digit and raise an odd one, so 2.745 becomes 2.74 and 2.735 becomes 2.74.

An everyday example. A kitchen scale that reads 250 g and a jeweller's scale that reads 250.00 g say different things — the jeweller's value is known to five significant figures.

The substance. Exact numbers have unlimited significant figures — the 2 in $2\pi r$ never limits the precision of an answer.