Errors and uncertainties

AS · 13 min

No measurement is perfect. This topic is about the two kinds of error that spoil measurements, the difference between a precise result and an accurate one, and how to work out the uncertainty in a quantity calculated from several measured values. Paper 1 almost always has one or two questions on combining uncertainties, Paper 2 tests the definitions, and Paper 3 is built around the same ideas (see Uncertainties in experiments).

Error and uncertainty

The error in a measurement is the difference between the measured value and the true value. You almost never know the true value, so you cannot know the error exactly. What you can estimate is the uncertainty: the range within which the true value is likely to lie.

Writing $l = 45.2 \pm 0.1\ \text{cm}$ says the true length is probably between $45.1\ \text{cm}$ and $45.3\ \text{cm}$ . The $0.1\ \text{cm}$ is the absolute uncertainty.

Systematic and random errors

Errors come in two kinds, and they behave completely differently.

Definition

A systematic error makes every reading differ from the true value by the same amount (or the same fraction) in the same direction. It cannot be reduced by repeating readings and averaging.

A random error makes readings scatter unpredictably above and below the true value. Its effect can be reduced by taking repeated readings and averaging, or by plotting a graph and drawing a line of best fit.

Sources of systematic error

A zero error: the instrument does not read zero when it should. A micrometer that reads $0.03\ \text{mm}$ with its jaws closed adds $0.03\ \text{mm}$ to every reading.
An instrument that is wrongly calibrated, such as a metre rule that has shrunk or a stop-watch that runs slow.
Reading a scale from the same wrong angle every time (consistent parallax).
Reaction time when starting and stopping a stop-watch, if the delay is always in the same direction.
A fixed effect the method ignores, such as heat lost to the surroundings in every heating experiment.

Sources of random error

Judging a reading between scale divisions.
Variations in reaction time when starting and stopping a stop-watch.
Fluctuations in conditions: draughts, small temperature changes, a flickering meter reading.
Difficulty judging the exact moment something happens (the bob passing the centre, a fringe passing a mark).

Watch out

Repeating readings and averaging does nothing to a systematic error: if every reading is $0.03\ \text{mm}$ too large, so is the mean. A systematic error is removed by finding and correcting its cause, for example checking the zero reading and subtracting it, or calibrating the instrument against a known standard.

A graph shows the difference clearly. Suppose $y$ should be proportional to $x$ . A zero error shifts every point by the same amount, so the line of best fit has the right gradient but the wrong intercept. Random error scatters points either side of the line without moving it.

y = 0.5 x y = 0.5 x + 1

Here the lower line is the true relationship $y = 0.5x$ , and the upper line is the same data with a zero error of $+1$ on every $y$ reading. The gradient is unaffected. This is why many experiments determine a constant from a gradient: it is immune to a constant zero error.

Correcting a zero error

A micrometer screw gauge reads $0.03\ \text{mm}$ when its jaws are closed with nothing between them. A wire placed between the jaws gives a reading of $0.31\ \text{mm}$ . (a) State the type of error. (b) Find the diameter of the wire.

Solution

(a) It is a zero error, which is a systematic error: it adds the same $0.03\ \text{mm}$ to every reading.

(b) Subtract the zero reading from the measured reading:

d = 0.31 - 0.03 = 0.28\ \text{mm}

If the zero reading had been $-0.02\ \text{mm}$ (the zero line below the datum), the diameter would be $0.31 - (-0.02) = 0.33\ \text{mm}$ .

Precision and accuracy

These two words mean different things, and examiners test the distinction directly.

Definition

A measurement is accurate if it is close to the true value.

A set of measurements is precise if the repeated readings are close to each other (small spread), whether or not they are close to the true value.

Poor precision is caused by random error. Poor accuracy can come from either kind, but a systematic error spoils accuracy even when the precision is excellent. Picture four sets of five readings of a length whose true value is $50.0\ \text{cm}$ :

Readings / cm	Spread	Mean close to $50.0$ ?	Description
$49.9,\ 50.1,\ 50.0,\ 49.9,\ 50.1$	small	yes	precise and accurate
$52.1,\ 52.0,\ 52.2,\ 52.1,\ 52.0$	small	no	precise but not accurate (systematic error)
$48.0,\ 52.5,\ 49.0,\ 51.5,\ 49.0$	large	yes, mean $50.0$	accurate (on average) but not precise
$51.0,\ 55.0,\ 53.5,\ 50.5,\ 54.0$	large	no	neither

The precision of a single reading is also limited by the instrument: a millimetre rule can give readings to the nearest millimetre, a micrometer to the nearest $0.01\ \text{mm}$ . That smallest division is often called the resolution of the instrument.

Watch out

Do not define precision as "how many decimal places a reading has" or accuracy as "how good a reading is". The mark is for "close to the true value" (accuracy) and "close to each other / small spread" (precision).

Absolute, fractional and percentage uncertainty

There are three ways to express the same uncertainty $\Delta x$ in a quantity $x$ .

Key result

\text{absolute uncertainty} = \Delta x

\text{fractional uncertainty} = \frac{\Delta x}{x}, \qquad \text{percentage uncertainty} = \frac{\Delta x}{x} \times 100\%

So $l = 45.2 \pm 0.1\ \text{cm}$ has fractional uncertainty $0.1/45.2 = 0.0022$ and percentage uncertainty $0.22\%$ . The absolute uncertainty carries the unit of the quantity; fractional and percentage uncertainties have no unit.

How big is the uncertainty of a single reading? As a starting point it is the smallest scale division, or half of it if you can genuinely judge halves. In practice it is often bigger: a stop-watch reads to $0.01\ \text{s}$ but human reaction time makes the realistic uncertainty around $0.1$ to $0.2\ \text{s}$ . For repeated readings, Cambridge accepts half the range as the absolute uncertainty in the mean.

Uncertainty from repeated readings

Four timings of an oscillation are $2.31\ \text{s}$ , $2.35\ \text{s}$ , $2.28\ \text{s}$ and $2.34\ \text{s}$ . State the mean time with its absolute uncertainty.

Solution

Mean:

\bar{t} = \frac{2.31 + 2.35 + 2.28 + 2.34}{4} = \frac{9.28}{4} = 2.32\ \text{s}

Half the range:

\Delta t = \frac{2.35 - 2.28}{2} = 0.035\ \text{s} \approx 0.04\ \text{s}

So $t = 2.32 \pm 0.04\ \text{s}$ . The mean is quoted to the same number of decimal places as the uncertainty.

Combining uncertainties

When you calculate a quantity from measured values, the uncertainties in the measurements carry through. The AS rules use simple addition. They give the largest likely uncertainty (the "worst case").

Key result

Adding or subtracting: add the absolute uncertainties.

\text{If } y = a + b \text{ or } y = a - b, \quad \Delta y = \Delta a + \Delta b

Multiplying or dividing: add the fractional (or percentage) uncertainties.

\text{If } y = ab \text{ or } y = \frac{a}{b}, \quad \frac{\Delta y}{y} = \frac{\Delta a}{a} + \frac{\Delta b}{b}

Powers: multiply the fractional uncertainty by the power (ignore its sign).

\text{If } y = a^n, \quad \frac{\Delta y}{y} = |n|\,\frac{\Delta a}{a}

Constants such as $2$ , $\pi$ and $\tfrac{1}{2}$ have no uncertainty and contribute nothing.

Why the power rule? $a^3 = a \times a \times a$ , and multiplying adds fractional uncertainties, so the fractional uncertainty is three times that of $a$ . A square root is the power $\tfrac{1}{2}$ , so it halves the fractional uncertainty. Dividing by $a^2$ is the power $-2$ , which still doubles it: uncertainties always add, never cancel.

Method

Finding the uncertainty in a calculated quantity

Calculate the value of the quantity itself.
Write the percentage (or fractional) uncertainty of each measured value.
Combine them with the rules: add absolute uncertainties for sums and differences; add percentage uncertainties (multiplied by any power) for products, quotients and powers.
Convert the total percentage uncertainty back into an absolute uncertainty: $\Delta y = y \times (\text{percentage}/100)$ .
Round the absolute uncertainty to one (or at most two) significant figures, then round the value to the same decimal place.

Resistance from a voltmeter and an ammeter

The potential difference across a resistor is $6.0 \pm 0.1\ \text{V}$ and the current in it is $0.50 \pm 0.02\ \text{A}$ . Calculate the resistance and its absolute uncertainty.

Solution

R = \frac{V}{I} = \frac{6.0}{0.50} = 12.0\ \Omega

Percentage uncertainties: $\dfrac{0.1}{6.0} \times 100 = 1.7\%$ and $\dfrac{0.02}{0.50} \times 100 = 4.0\%$ .

Division, so add percentages: $1.7 + 4.0 = 5.7\%$ .

\Delta R = 0.057 \times 12.0 = 0.68\ \Omega \approx 0.7\ \Omega

R = 12.0 \pm 0.7\ \Omega

A difference of two readings

The length of a spring is $25.0 \pm 0.1\ \text{cm}$ before loading and $42.5 \pm 0.1\ \text{cm}$ after. Find the extension with its absolute and percentage uncertainty.

Solution

x = 42.5 - 25.0 = 17.5\ \text{cm}

Subtraction, so add the absolute uncertainties: $\Delta x = 0.1 + 0.1 = 0.2\ \text{cm}$ .

x = 17.5 \pm 0.2\ \text{cm}, \qquad \text{percentage uncertainty} = \frac{0.2}{17.5} \times 100 = 1.1\%

Notice that the percentage uncertainty of the extension ( $1.1\%$ ) is much larger than that of either length (about $0.4\%$ and $0.2\%$ ). Subtracting two similar numbers always makes this worse; if the extension had been only $1.2\ \text{cm}$ , the same $\pm 0.2\ \text{cm}$ would be a $17\%$ uncertainty.

Density of a sphere

A metal sphere has diameter $d = 2.50 \pm 0.01\ \text{cm}$ and mass $m = 65.0 \pm 0.5\ \text{g}$ . Calculate its density in $\text{kg m}^{-3}$ and the absolute uncertainty.

Solution

Volume of a sphere in terms of diameter: $V = \dfrac{\pi d^3}{6} = \dfrac{\pi \times 2.50^3}{6} = 8.18\ \text{cm}^3$ .

\rho = \frac{m}{V} = \frac{65.0}{8.18} = 7.95\ \text{g cm}^{-3} = 7950\ \text{kg m}^{-3}

Percentage uncertainties: mass $\dfrac{0.5}{65.0} \times 100 = 0.77\%$ ; diameter $\dfrac{0.01}{2.50} \times 100 = 0.40\%$ , and $d$ is cubed so it contributes $3 \times 0.40 = 1.20\%$ .

Total: $0.77 + 1.20 = 1.97\% \approx 2.0\%$ .

\Delta\rho = 0.0197 \times 7950 = 157\ \text{kg m}^{-3} \approx 200\ \text{kg m}^{-3}

\rho = (7.9 \pm 0.2) \times 10^{3}\ \text{kg m}^{-3}

The $\pi$ and the $6$ contribute nothing.

Acceleration of free fall from a pendulum

The period of a simple pendulum is $T = 2\pi\sqrt{L/g}$ . A student measures $L = 0.800 \pm 0.002\ \text{m}$ and times 20 oscillations as $35.8 \pm 0.2\ \text{s}$ . Calculate $g$ and its absolute uncertainty.

Solution

Period: $T = 35.8/20 = 1.79\ \text{s}$ .

Rearrange: $g = \dfrac{4\pi^2 L}{T^2} = \dfrac{4\pi^2 \times 0.800}{1.79^2} = 9.86\ \text{m s}^{-2}$ .

Percentage uncertainty in $L$ : $\dfrac{0.002}{0.800} \times 100 = 0.25\%$ .

Percentage uncertainty in $T$ : the same as in the total time, $\dfrac{0.2}{35.8} \times 100 = 0.56\%$ . (Dividing by 20, an exact number, does not change the percentage.)

$T$ is squared, so it contributes $2 \times 0.56 = 1.12\%$ . Total: $0.25 + 1.12 = 1.37\%$ .

\Delta g = 0.0137 \times 9.86 = 0.13\ \text{m s}^{-2}

g = 9.9 \pm 0.1\ \text{m s}^{-2}

Timing 20 oscillations rather than one is the key technique: the absolute uncertainty of about $0.2\ \text{s}$ is spread over 20 periods, so the uncertainty in $T$ is only $0.01\ \text{s}$ .

Young modulus: which measurement matters most

The Young modulus of a wire is found from $E = \dfrac{4FL}{\pi d^2 x}$ . The measurements are $F = 50.0 \pm 0.5\ \text{N}$ , $L = 2.000 \pm 0.002\ \text{m}$ , $d = 0.40 \pm 0.01\ \text{mm}$ and $x = 4.0 \pm 0.1\ \text{mm}$ . (a) Calculate $E$ with its uncertainty. (b) State which measurement contributes most to the uncertainty and suggest how it could be reduced.

Solution

(a)

E = \frac{4 \times 50.0 \times 2.000}{\pi \times (0.40 \times 10^{-3})^2 \times 4.0 \times 10^{-3}} = 1.99 \times 10^{11}\ \text{Pa}

Quantity	Percentage uncertainty	Power	Contribution
$F$	$0.5/50.0 = 1.0\%$	1	$1.0\%$
$L$	$0.002/2.000 = 0.1\%$	1	$0.1\%$
$d$	$0.01/0.40 = 2.5\%$	2	$5.0\%$
$x$	$0.1/4.0 = 2.5\%$	1	$2.5\%$

Total $= 8.6\%$ , so $\Delta E = 0.086 \times 1.99 \times 10^{11} = 0.17 \times 10^{11}\ \text{Pa}$ .

E = (2.0 \pm 0.2) \times 10^{11}\ \text{Pa}

(b) The diameter contributes most ( $5.0\%$ ), because it is small and is squared. Reduce it by measuring $d$ with a micrometer at several points along the wire and in perpendicular directions, then averaging; or use a longer wire with a larger load so that $x$ is larger (this helps the second-largest term).

Exam tip

Quote the absolute uncertainty to one significant figure (two is accepted if the first digit is 1), and the value to the same number of decimal places. $9.86 \pm 0.13$ becomes $9.9 \pm 0.1$ ; writing $9.857 \pm 0.1$ loses the mark.
In Paper 1, uncertainty questions are quickest with percentages: write each one, multiply by the power, add. Never subtract uncertainties, even when the quantities are divided.
"Distinguish between precision and accuracy" is a two-mark question: one mark for each definition, each referring to the right thing (true value; spread of readings).
"Explain why the mean of repeated readings does not remove a zero error" needs the idea that a systematic error shifts every reading by the same amount, so the mean is shifted too.

Summary

Systematic errors shift every reading the same way; they are not reduced by averaging. Zero errors are systematic.
Random errors scatter readings either side of the true value; averaging repeats or using a best-fit line reduces their effect.
Accurate means close to the true value; precise means repeated readings are close to each other.
Absolute uncertainty has a unit; fractional and percentage uncertainties do not.
For repeated readings, absolute uncertainty $\approx$ half the range.
Sums and differences: add absolute uncertainties. Products and quotients: add percentage uncertainties. Powers: multiply the percentage uncertainty by the power.
Give the uncertainty to 1 s.f. and the value to the same decimal place.

Practice

Question

State whether each is mainly a systematic or a random error: (a) a voltmeter reads $0.05\ \text{V}$ with no p.d. across it; (b) the reading on an ammeter flickers between two values; (c) a student always reads a measuring cylinder from above the meniscus; (d) estimating when a rolling ball crosses a line.
A length is $45.2 \pm 0.1\ \text{cm}$ . Calculate the percentage uncertainty.
A lamp has $V = 2.40 \pm 0.05\ \text{V}$ and $I = 0.36 \pm 0.01\ \text{A}$ . Calculate its power with absolute uncertainty.
A trolley of mass $0.250 \pm 0.005\ \text{kg}$ moves at $3.0 \pm 0.1\ \text{m s}^{-1}$ . Calculate its kinetic energy and the absolute uncertainty.
The true value of $g$ is $9.81\ \text{m s}^{-2}$ . Student A obtains $9.52, 9.55, 9.53\ \text{m s}^{-2}$ ; student B obtains $9.4, 10.3, 9.8\ \text{m s}^{-2}$ . Comment on the accuracy and precision of each set.
A newton meter reads $0.2\ \text{N}$ when nothing hangs from it. It reads $3.8\ \text{N}$ when a stone hangs from it. State the weight of the stone and the type of error involved.
The resistivity of a wire is $\rho = \dfrac{R\pi d^2}{4L}$ . Given $R = 5.2 \pm 0.1\ \Omega$ , $d = 0.36 \pm 0.01\ \text{mm}$ and $L = 0.800 \pm 0.001\ \text{m}$ , calculate $\rho$ and its absolute uncertainty.
A quantity $X$ is calculated from $X = \dfrac{a b^2}{\sqrt{c}}$ . The percentage uncertainties in $a$ , $b$ and $c$ are $2\%$ , $3\%$ and $4\%$ . Find the percentage uncertainty in $X$ .
A mass $m = 200 \pm 2\ \text{g}$ oscillates on a spring. Ten oscillations take $8.4\ \text{s}$ , $8.6\ \text{s}$ and $8.5\ \text{s}$ in three trials. Using $T = 2\pi\sqrt{m/k}$ , determine the spring constant $k$ with its absolute uncertainty.
A student finds the height of a step by measuring the height of the floor and of the top of the step from the ceiling: $h_1 = 241.0 \pm 0.5\ \text{cm}$ and $h_2 = 253.0 \pm 0.5\ \text{cm}$ . (a) Calculate the height of the step and its percentage uncertainty. (b) Explain why this method is poor, and suggest a better one.

Answers

(a) Systematic (zero error). (b) Random. (c) Systematic (consistent parallax, always the same direction). (d) Random.
$\dfrac{0.1}{45.2} \times 100 = 0.22\%$ .
$P = VI = 2.40 \times 0.36 = 0.864\ \text{W}$ . Percentages: $2.1\% + 2.8\% = 4.9\%$ , so $\Delta P = 0.042\ \text{W}$ . $P = 0.86 \pm 0.04\ \text{W}$ .
$E_K = \tfrac{1}{2} \times 0.250 \times 3.0^2 = 1.125\ \text{J}$ . Percentages: $m$ gives $2.0\%$ ; $v$ gives $3.3\%$ , doubled to $6.7\%$ . Total $8.7\%$ , so $\Delta E_K = 0.098\ \text{J}$ . $E_K = 1.1 \pm 0.1\ \text{J}$ .
A: readings close together (precise) but all well below $9.81$ (not accurate; suggests a systematic error). B: large spread (not precise), but the mean $9.83$ is close to $9.81$ (accurate on average).
$3.8 - 0.2 = 3.6\ \text{N}$ . A zero error, which is systematic.
$\rho = \dfrac{5.2 \times \pi \times (0.36 \times 10^{-3})^2}{4 \times 0.800} = 6.62 \times 10^{-7}\ \Omega\ \text{m}$ . Percentages: $R$ : $1.9\%$ ; $d$ : $2.8\% \times 2 = 5.6\%$ ; $L$ : $0.1\%$ . Total $7.6\%$ , so $\Delta\rho = 0.50 \times 10^{-7}\ \Omega\ \text{m}$ . $\rho = (6.6 \pm 0.5) \times 10^{-7}\ \Omega\ \text{m}$ .
$2 + 2 \times 3 + \tfrac{1}{2} \times 4 = 2 + 6 + 2 = 10\%$ .
Mean time for ten $= 8.5\ \text{s}$ , uncertainty $=$ half range $= 0.1\ \text{s}$ ; $T = 0.85\ \text{s}$ . $k = \dfrac{4\pi^2 m}{T^2} = \dfrac{4\pi^2 \times 0.200}{0.85^2} = 10.9\ \text{N m}^{-1}$ . Percentages: $m$ : $1.0\%$ ; $T$ : $0.1/8.5 = 1.2\%$ , doubled to $2.4\%$ . Total $3.4\%$ , so $\Delta k = 0.37\ \text{N m}^{-1}$ . $k = 10.9 \pm 0.4\ \text{N m}^{-1}$ .
(a) $h = 253.0 - 241.0 = 12.0\ \text{cm}$ , $\Delta h = 0.5 + 0.5 = 1.0\ \text{cm}$ , so the percentage uncertainty is $8.3\%$ . (b) The step height is found as a small difference between two large readings, so the absolute uncertainties add to something large compared with the result. Measure the step height directly with a metre rule held vertically (uncertainty about $\pm 0.1\ \text{cm}$ , under $1\%$ ).

Error and uncertainty

Systematic and random errors

Precision and accuracy

Absolute, fractional and percentage uncertainty

Combining uncertainties

Summary

Practice

Console