Measurement Errors: Absolute and Relative Uncertainties

No measurement is ever exact. A length read from a metre rule, a time read from a stopwatch, a current read from a digital ammeter — each returns a number that is close to the true value but never identical to it, and a physicist who quotes such a number without saying how close has not finished the measurement. This chapter is about that second half of every experimental result: the uncertainty.

The chapter matters twice over. It matters for Paper 1B (the data-based questions) and for the Internal Assessment, where uncertainties are examined directly and where a data table without them cannot score well. But it also matters conceptually: uncertainty is what turns a number into a claim about the world. Saying $g = 9.6~\dfrac{\text{m}}{\text{s}^{2}}$ is a bald assertion that happens to be wrong; saying $g = (9.6 \pm 0.4)~\dfrac{\text{m}}{\text{s}^{2}}$ is a defensible statement that happens to be right, because the accepted value $9.81~\dfrac{\text{m}}{\text{s}^{2}}$ lies inside the stated interval.

The core idea. A measurement is not a point, it is an interval. Everything in this chapter is machinery for finding how wide that interval is, and how to say so honestly.

1. Error and Uncertainty: Two Different Words

Everyday speech treats "error" and "uncertainty" as synonyms; physics does not.

An error is the difference between a measured value and the true value: $\text{error} = x_{\text{measured}} - x_{\text{true}}$ . Since the true value is generally unknown, the error of a single measurement is unknowable too.

An uncertainty is an estimate of the range within which the true value is believed to lie. It is not a mistake and it is not a failure of technique. Even a perfectly executed measurement with the finest apparatus has an uncertainty, because every instrument has a finite resolution and every physical quantity fluctuates at some level.

A mistake — misreading a scale, recording a value in the wrong column, writing $37$ instead of $73$ — is neither of these. Mistakes are eliminated by care and by repeating the measurement, not quantified by an uncertainty. (Forgetting to zero a balance is different: it shifts every reading the same way and is a systematic zero error, treated in the next section.)

2. Random and Systematic: The Two Kinds of Uncertainty

Uncertainties divide into two families, and the distinction governs what you can do about them.

A random uncertainty varies unpredictably in size and sign from one repeat to the next. Its sources are the ones that scatter data: human reaction time on a stopwatch, small draughts, the last flickering digit of a display, the judgement of exactly where a pointer sits between two graduations. Because random effects are as likely to push a reading up as down, repeating the measurement and averaging reduces them: for $N$ repeats the uncertainty in the mean falls roughly as $\dfrac{1}{\sqrt{N}}$ .

A systematic uncertainty shifts every reading in the same direction by a similar amount. Its sources are the ones that displace data: an ammeter that reads $0.02~\text{A}$ with nothing connected (a zero error), a ruler whose end is worn away, a thermometer calibrated against a bad standard, or the parallax introduced by always viewing a scale from the same wrong angle. Repeating a measurement does nothing to a systematic uncertainty; averaging a hundred readings from a mis-zeroed instrument simply gives a very precise wrong answer. Systematic effects are removed only by identifying and correcting their cause, or by a technique — such as taking a difference of two readings — that cancels them.

Precision and accuracy. An accurate result has a small systematic uncertainty: its mean sits close to the true value. A precise result has a small random uncertainty: repeats cluster tightly together. The two are independent — a warped rifle sight gives tightly grouped shots far from the bullseye (precise but inaccurate), while an unsteady hand with a good sight gives a widely scattered group centred on it (accurate but imprecise).

Graphically, on a plot of results against a controlled variable, a random uncertainty appears as scatter about the line of best fit, while a systematic uncertainty appears as a displaced intercept or a wrong gradient — a line that is beautifully straight but does not pass through the origin it ought to.

3. Absolute, Fractional and Percentage Uncertainty

Three ways of stating the same uncertainty are used throughout the course.

The absolute uncertainty, written $\Delta x$ , carries the same unit as the quantity itself and states the half-width of the interval:

x = x_{\text{best}} \pm \Delta x \qquad\text{e.g.}\qquad l = (24.6 \pm 0.1)~\text{cm}

This asserts that the true length is believed to lie between $24.5~\text{cm}$ and $24.7~\text{cm}$ .

The fractional uncertainty is the absolute uncertainty divided by the measurement, and is a pure number with no unit:

\text{fractional uncertainty} = \frac{\Delta x}{|x|}

The percentage uncertainty is the same thing expressed per hundred:

\text{percentage uncertainty} = \frac{\Delta x}{|x|}\times 100\%

For the length above, $\dfrac{0.1}{24.6} = 0.0041$ , or $0.41\%$ .

The fractional form is the one that matters when quantities are combined, as the next chapter shows in detail, and it is also the one that tells you whether a measurement was worth making. Measuring a $2~\text{mm}$ thickness with a millimetre rule ( $\Delta x = 0.5~\text{mm}$ ) gives a percentage uncertainty of $25\%$ — an almost worthless measurement — while the same rule measuring $200~\text{mm}$ gives $0.25\%$ . The instrument did not change; the fractional uncertainty did.

4. Where the Uncertainty of a Single Reading Comes From

For a single reading from an instrument, the uncertainty is estimated from the instrument itself.

Analogue scales (metre rule, thermometer, ammeter with a needle): take $\Delta x$ as half the smallest scale division. A metre rule graduated in millimetres therefore gives $\Delta x = 0.5~\text{mm}$ for a single reading.
Digital displays: take $\Delta x$ as $\pm 1$ in the last displayed digit. A balance reading $4.62~\text{g}$ carries $\Delta m = 0.01~\text{g}$ .
Manufacturer's tolerance, when quoted, overrides both: a resistor marked $470~\Omega \pm 5\%$ carries $\Delta R = 23.5~\Omega$ regardless of how the value was read.

Two refinements are examined regularly. First, a measurement of a length with a ruler requires two readings — one at each end — so the uncertainties add and $\Delta l = 1~\text{mm}$ for a millimetre rule, not $0.5~\text{mm}$ . Second, an uncertainty dominated by the experimenter rather than the instrument must be estimated realistically: timing ten swings of a pendulum with a stopwatch reading to $0.01~\text{s}$ has an uncertainty of about $0.2~\text{s}$ , set by human reaction time, and quoting $0.01~\text{s}$ would be dishonest.

5. Repeated Readings: Mean and Range

When a measurement is repeated, the best estimate is the arithmetic mean of the readings:

\bar{x} = \frac{1}{N}\sum_{i=1}^{N} x_i

and the uncertainty is estimated at IB level from the half-range — half the spread between the largest and smallest readings:

\Delta x = \frac{x_{\max} - x_{\min}}{2}

Worked example. Five timings of ten swings of a pendulum give $14.2$ , $14.6$ , $14.3$ , $14.5$ and $14.4~\text{s}$ . The mean is

\bar{t} = \frac{14.2+14.6+14.3+14.5+14.4}{5} = \frac{72.0}{5} = 14.4~\text{s}

The half-range is

\Delta t = \frac{14.6 - 14.2}{2} = 0.2~\text{s}

so the result is $t = (14.4 \pm 0.2)~\text{s}$ , a percentage uncertainty of $1.4\%$ . Note that the half-range, $0.2~\text{s}$ , is far larger than the stopwatch resolution of $0.01~\text{s}$ : the repeats have revealed the real (reaction-time) uncertainty that the instrument's display concealed. Whenever the half-range exceeds the instrument uncertainty, the half-range is the one to quote.

6. Quoting a Result Correctly

Three conventions govern how a result is written, and marks are awarded for each.

Quote the uncertainty to one significant figure. $\Delta t = 0.2~\text{s}$ , never $0.213\,4~\text{s}$ ; an uncertainty is an estimate and cannot itself be known precisely. (Two significant figures are tolerated when the leading digit is $1$ .)
Round the value to the same decimal place as the uncertainty. Writing $(14.372 \pm 0.2)~\text{s}$ is self-contradictory — the digits $7$ and $2$ are claimed and disclaimed in the same breath. Write $(14.4 \pm 0.2)~\text{s}$ .
Give both the same unit, and bracket them if the unit is written once: $(14.4 \pm 0.2)~\text{s}$ , or $14.4~\text{s} \pm 0.2~\text{s}$ .

Comparing with an accepted value. A measured result agrees with an accepted value when the accepted value lies within the measured interval. If a class measures $g = (9.6 \pm 0.4)~\dfrac{\text{m}}{\text{s}^{2}}$ , the interval runs from $9.2$ to $10.0~\dfrac{\text{m}}{\text{s}^{2}}$ and contains $9.81~\dfrac{\text{m}}{\text{s}^{2}}$ , so the experiment is consistent with theory. Had they measured $(9.6 \pm 0.1)~\dfrac{\text{m}}{\text{s}^{2}}$ , the interval $9.5$ – $9.7$ would exclude the accepted value, and the correct conclusion would be that an unidentified systematic effect is present — not that gravity is different in that laboratory.

The percentage difference from an accepted value,

\text{percentage difference} = \frac{|x_{\text{measured}} - x_{\text{accepted}}|}{x_{\text{accepted}}}\times 100\%

measures accuracy and should always be compared against the percentage uncertainty, which measures precision. A percentage difference much larger than the percentage uncertainty is the signature of a systematic error.

7. Uncertainty Bars on Graphs

Every plotted point should carry uncertainty bars: a horizontal bar of total length $2\Delta x$ and a vertical bar of total length $2\Delta y$ , so that the point sits at the centre of a small cross whose arms represent the intervals in which the true values lie.

Their purpose is not decoration. A straight line is an acceptable model of the data precisely when a single straight line can be drawn passing through every uncertainty bar. If no such line exists, the relationship is not linear, or the uncertainties have been underestimated. And the bars are what make a gradient uncertainty computable at all: drawing the steepest and the shallowest lines that still pass through all the bars gives a maximum and a minimum gradient, and

\Delta m = \frac{m_{\max} - m_{\min}}{2}

The same construction applied to the intercepts gives $\Delta c$ . This graphical method is the standard IB technique for extracting an experimental constant with its uncertainty, and it is why a well-drawn graph is worth more than a table of numbers.

8. Common Exam Pitfalls

Believing that repeating a measurement reduces every uncertainty. Averaging reduces random uncertainty only; a systematic offset survives any number of repeats untouched.
Confusing precision with accuracy. Tightly clustered readings prove precision, never accuracy — they are entirely consistent with a mis-zeroed instrument.
Quoting an uncertainty to several significant figures, or quoting a value to more decimal places than its uncertainty. $(14.372 \pm 0.2)~\text{s}$ loses the mark that $(14.4 \pm 0.2)~\text{s}$ earns.
Using half a scale division for a length measured with a ruler. Both ends of the object must be read, so the uncertainties combine: $\Delta l = 1~\text{mm}$ for a millimetre rule.
Quoting the instrument resolution when repeats scatter more widely. If the half-range exceeds the resolution, the half-range is the honest estimate.
Concluding that an experiment "disagrees with theory" without checking the interval. Agreement means the accepted value lies inside the measured range, not that the two numbers are identical.
Omitting uncertainty bars, or drawing them at a fixed pixel size. Their length must be $2\Delta y$ in the units of the axis, and it may differ from point to point.

Summary

An error is the (usually unknowable) difference from the true value; an uncertainty is the estimated range within which the true value lies; a mistake is neither and must simply be eliminated.
Random uncertainties scatter readings both ways and are reduced by repeating and averaging; systematic uncertainties displace every reading the same way and are unaffected by repetition.
Precision reflects small random uncertainty (tight clustering); accuracy reflects small systematic uncertainty (correct mean). They are independent.
Absolute uncertainty $\Delta x$ carries the unit of $x$ ; fractional uncertainty is $\dfrac{\Delta x}{|x|}$ ; percentage uncertainty is $\dfrac{\Delta x}{|x|}\times 100\%$ .
A single reading takes half a scale division (analogue) or $\pm 1$ in the last digit (digital); a length measured with a ruler takes twice this, because both ends are read.
Repeated readings give a best estimate $\bar{x} = \dfrac{1}{N}\sum x_i$ with uncertainty $\Delta x = \dfrac{x_{\max}-x_{\min}}{2}$ , and the half-range wins whenever it exceeds the instrument uncertainty.
Quote the uncertainty to one significant figure and round the value to the same decimal place: $t = (14.4 \pm 0.2)~\text{s}$ .
Uncertainty bars of total length $2\Delta x$ and $2\Delta y$ test linearity and yield $\Delta m = \dfrac{m_{\max}-m_{\min}}{2}$ from the steepest and shallowest acceptable lines.