Errors in measurement

Weigh a coin twice on the same balance and you may get $2.13\ \text{g}$ the first time and $2.15\ \text{g}$ the second — the same coin, two different numbers. No measurement is ever perfectly exact; every reading of length, mass or time carries some uncertainty, and the honest scientist reports not just a value but how much that value can be trusted. This article builds the vocabulary and arithmetic for that trust: the difference between systematic errors (which push a result the same way every time) and random errors (which scatter it unpredictably), the difference between being accurate and being precise, and the standard recipe for turning a set of repeated readings into a mean, an absolute error, and a percentage error. It closes with the rule for combining the errors of two measured quantities that are multiplied together — the single most exam-tested calculation in this chapter — worked through with real numbers you can check by hand.

The core insight

Every measurement has an inherent uncertainty, which is classified as systematic (a consistent, correctable bias) or random (unpredictable scatter reducible only by averaging), and is reported as a mean plus a mean absolute error whose percentage size also governs how errors combine when measured quantities are multiplied or divided.

The core idea

An error is the difference between a measured value and the true (or accepted) value of a quantity. Errors are not "mistakes" in the everyday sense — they are an unavoidable feature of measurement, and the goal is to classify, quantify and report them honestly rather than to imagine they can be made to vanish.

Systematic errors push every reading the same way, because they have a definite, identifiable cause: a mis-calibrated or zero-error instrument (a worn metre scale, a balance that does not read zero unloaded), a flawed procedure (measuring a boiling point before thermal equilibrium), or a personal bias such as reading a scale from an angle (parallax). Because the cause is identifiable, systematic error can in principle be found and removed — correct the zero error, refine the technique, or use a better instrument.

Random errors scatter a reading unpredictably in both size and direction, from uncontrollable fluctuations or the limits of judgement when estimating a scale's last digit. They cannot be eliminated, only reduced, by taking several readings and treating the mean as the best estimate of the true value. A special case is the least count error: the smallest error possible when a reading is limited by an instrument's finest division, since the observer must estimate a fraction of it — a form of random error. Two further terms describe the quality of a measurement rather than its cause: accuracy is how close a measured value is to the true value, while precision is how close repeated measurements are to each other, regardless of whether they are close to the truth. A set of readings can be precise but inaccurate (tightly clustered, but off-target because of a systematic error), or accurate on average but imprecise (scattered around the right value).

Figure

Four target diagrams showing every combination of accuracy and precision from shots clustered on the bullseye to shots scattered wide of it

For $n$ repeated readings $a_1, a_2, \ldots, a_n$ of a quantity, the best estimate of the true value is the mean, and the spread around it is captured by the mean absolute error:

a_{mean} = \frac{a_1+a_2+\cdots+a_n}{n}, \qquad \overline{\Delta a} = \frac{|a_{mean}-a_1|+|a_{mean}-a_2|+\cdots+|a_{mean}-a_n|}{n}

where: $a_{mean}$ is the mean value taken as the best estimate of the true value, $\Delta a_i = |a_{mean}-a_i|$ is the absolute error of the $i$ -th reading, and $\overline{\Delta a}$ is the mean absolute error. The relative (fractional) error is $\overline{\Delta a}/a_{mean}$ , and multiplying it by $100\%$ gives the percentage error. A result is reported as $a = a_{mean}\pm\overline{\Delta a}$ , and the number of significant figures kept in the final answer should match the size of this uncertainty — not more, not fewer.

Errors in a combined (product) quantity

Many quantities are not measured directly but calculated from other measured quantities — an area from a length and a breadth, a density from a mass and a volume. When two measured quantities each carrying their own error are multiplied (or divided), the errors do not cancel; they add.

Derivation APercentage error in a product of two measured quantities

The goal is to show that when a calculated quantity $Z = A \times B$ is built from two independently measured quantities $A$ and $B$ , each known only to within a small error $\Delta A$ and $\Delta B$ , the maximum fractional error in $Z$ is simply the sum of the individual fractional errors — a rule that applies directly to any product or quotient of measured quantities.

Step 1. Write the true value of each factor as the measured value plus its (unknown, but bounded) error: $A + \Delta A$ and $B + \Delta B$ , where $\Delta A$ and $\Delta B$ may be positive or negative but are small compared with $A$ and $B$ .

Step 2. Multiply the two extreme values to get the extreme value of $Z$ . $Z + \Delta Z = (A+\Delta A)(B+\Delta B) = AB + A\,\Delta B + B\,\Delta A + \Delta A\,\Delta B$ . Since $\Delta A$ and $\Delta B$ are each small fractions of $A$ and $B$ , their product $\Delta A\,\Delta B$ is smaller still and is dropped.

Step 3. Subtract the central value $Z = AB$ from both sides to isolate the error itself: $\Delta Z = A\,\Delta B + B\,\Delta A$ .

Step 4. Divide through by $Z = AB$ to convert to a fractional error, since it is fractional (or percentage) error that is physically meaningful and comparable across different quantities: $\dfrac{\Delta Z}{Z} = \dfrac{A\,\Delta B}{AB} + \dfrac{B\,\Delta A}{AB} = \dfrac{\Delta A}{A} + \dfrac{\Delta B}{B}$ . The worst case — when the two individual errors happen to point the same way — is that they add rather than partially cancel, so the fractional error in a product is bounded by the sum of the fractional errors of its factors. The identical argument for $Z = A/B$ gives the same result, $\Delta Z/Z = \Delta A/A + \Delta B/B$ , because division is multiplication by a reciprocal and the same worst-case addition applies.

Worked example

Problem. The period of oscillation of a simple pendulum is measured five times: $2.63\ \text{s}$ , $2.56\ \text{s}$ , $2.42\ \text{s}$ , $2.71\ \text{s}$ , $2.80\ \text{s}$ . Find the mean, the mean absolute error, the percentage error, and report the result.

Step 1. Compute the mean of the five readings. $T_{mean} = \dfrac{2.63+2.56+2.42+2.71+2.80}{5} = \dfrac{13.12}{5} = 2.624\ \text{s} \approx 2.62\ \text{s}$ .

Step 2. Compute the absolute error of each reading against this mean. $|2.62-2.63|=0.01$ , $|2.62-2.56|=0.06$ , $|2.62-2.42|=0.20$ , $|2.62-2.71|=0.09$ , $|2.62-2.80|=0.18$ , all in seconds.

Step 3. Average these five absolute errors to get the mean absolute error. $\overline{\Delta T} = \dfrac{0.01+0.06+0.20+0.09+0.18}{5} = \dfrac{0.54}{5} = 0.108\ \text{s} \approx 0.11\ \text{s}$ .

Step 4. Convert to a percentage error and state the reported result. $\dfrac{\overline{\Delta T}}{T_{mean}}\times100\% = \dfrac{0.11}{2.62}\times100\% \approx 4.2\%$ . The period is reported as $T = (2.62 \pm 0.11)\ \text{s}$ , a percentage error of about $4.2\%$ .

Competitive stretch

If this pendulum's length is separately measured as $l = (98.0\pm0.5)\ \text{cm}$ (fractional error about $0.5\%$ ), the $T$ -error of $4.2\%$ still dominates any quantity built from $l$ and $T$ together — the largest percentage error usually controls a combined result.

Common traps

Treating precision and accuracy as synonyms — a tightly clustered (precise) set of readings can still sit far from the true value if a systematic error is present.
Believing random error can be eliminated entirely — it can only be reduced by taking more readings and averaging, never removed completely.
Assuming more readings fix a systematic (e.g. zero) error — every reading is biased the same way, so repetition only helps with random error, not systematic error.
Subtracting or dividing fractional errors of a product's factors instead of adding them — even for a quotient $A/B$ , the fractional errors of $A$ and $B$ still add.

Ask the tutor: want to practise combining errors for a quotient, a squared quantity, or work through more examples of spotting systematic versus random error in a lab setup — ask the AI tutor to walk through one with you.

Why do we add fractional errors even when a quantity is being divided, not multiplied?

Because division by $B$ is the same as multiplication by $1/B$ , and the fractional error of a reciprocal has the same size as the fractional error of the original quantity — only its sign in the algebra flips, not its magnitude. So the worst-case (same-direction) errors still add for $Z = A/B$ , exactly as they do for $Z = A\times B$ : whether a calculated quantity comes from multiplying or dividing measured quantities, add the fractional errors of the factors to get the fractional error of the result.

If random errors are unpredictable, how does taking more readings actually help?

A single reading can land anywhere in the random scatter, but the scatter above and below the true value tends to balance out across many readings, so the mean of several readings sits closer to the true value than any one reading typically does. This is why the standard procedure is to repeat a measurement several times and report the mean rather than trusting a single trial — it does not remove randomness from any one reading, but it improves the reliability of the reported estimate.

My reported error came out to 0.0006, but my mean value has only two decimal places — do I round the error too?

Yes, and the two need to match. Errors are conventionally rounded to one significant figure, and the mean value is then rounded to the same decimal place as that rounded error, not the other way round. Writing a mean to five decimal places alongside a coarse error, as in the pendulum example above, is inconsistent and loses marks even if the arithmetic itself is correct.