Notes · PhysicsUK · A-Levels

Measurements and their errors

Physics is a measured science, so every number carries a unit and an uncertainty. This chapter builds the SI system and its prefixes, distinguishes random from systematic error and precision from accuracy, and develops the rules for combining uncertainties through a calculation and for extracting a gradient and its uncertainty from a graph. These skills run through every practical and every data-analysis question in the whole A-Level.

4 sections~14 min reading time3 competenciesLevel Foundation 2 · Standard 1 · Advanced 1

T·0111 / 13

Exam profile

AO1 · Know the SI base units and prefixes and define random and systematic error, precision, repeatability, resolution and accuracyAO2 · Convert units and prefixes and combine absolute, fractional and percentage uncertainties through a calculation, quoting results to an appropriate number of significant figuresAO3 · Evaluate data quality, plot error bars, draw worst-acceptable lines and determine the uncertainty in a gradient and intercept

Operators:stateestimatecalculatedetermineexplainevaluate

basic level

AS-Level requires the SI base units and prefixes, the distinction between random and systematic errors and between precision and accuracy, and the combination of uncertainties in simple sums and products.

higher level

The full A-Level applies these skills quantitatively throughout, especially in extracting a gradient and its uncertainty from error-bar graphs using worst-acceptable lines, and in propagating uncertainty through multi-step calculations.

Depth

Text

Contents · 4 sections

Measurements and their errors

SI units, prefixes and estimation#

FoundationAQA 7408 3.1.1AQA 7408 3.1.2DfE GCE Physics - measurements

Every physical quantity is a number multiplied by a unit, and in physics those units are built from the seven SI base units. Six are met at A-Level: the metre (m) for length, the kilogram (kg) for mass, the second (s) for time, the ampere (A) for electric current, the kelvin (K) for temperature and the mole (mol) for amount of substance. Every other unit is a derived unit expressible as a product of powers of these bases: the newton, for instance, is

1\,\text{N} = 1\,\text{kg}\,\text{m}\,\text{s}^{-2}

because

F = ma

, and the joule is

1\,\text{J} = 1\,\text{kg}\,\text{m}^2\,\text{s}^{-2}

because

W = Fs

Because both sides of any correct physical equation must describe the same kind of quantity, they must have the same base units. This gives a powerful check called homogeneity: reduce each term to base units and confirm they match. If the units do not balance, the equation is certainly wrong; if they do balance, the equation is dimensionally possible (though a dimensionless constant such as

\tfrac{1}{2}

2\pi

cannot be found this way). Checking homogeneity catches many algebraic slips before a single number is substituted.

Measurements in physics span from the sub-atomic to the astronomical, so we use prefixes that multiply a unit by a power of ten: tera (

10^{12}

), giga (

10^{9}

), mega (

10^{6}

), kilo (

10^{3}

), then centi (

10^{-2}

), milli (

10^{-3}

), micro (

10^{-6}

), nano (

10^{-9}

), pico (

10^{-12}

) and femto (

10^{-15}

). A capacitance of

4.7\,\mu\text{F}

4.7\times10^{-6}\,\text{F}

; a wavelength of

600\,\text{nm}

6.00\times10^{-7}\,\text{m}

. Converting cleanly into base units before substituting is the single most reliable way to avoid power-of-ten errors.

An estimate (a Fermi problem) asks for an answer to the nearest order of magnitude - the correct power of ten - from sensible rough figures rather than precise data. Estimating the number of breaths a person takes in a lifetime, or the number of atoms in a grain of sand, trains you to reason with orders of magnitude and to sense-check a calculated answer: if a car's kinetic energy comes out as

10^{9}\,\text{J}

you know at once that something is wrong.

1\,\text{N} = 1\,\text{kg}\,\text{m}\,\text{s}^{-2}

The newton in base units

From $F = ma$ : mass in kg times acceleration in $\text{m}\,\text{s}^{-2}$ .

1\,\text{J} = 1\,\text{N}\,\text{m} = 1\,\text{kg}\,\text{m}^2\,\text{s}^{-2}

The joule in base units

From $W = Fs$ : force in newtons times distance in metres.

Worked example

Checking an equation by its units

A student writes the pressure due to a liquid column as $p = \rho g h$ . Show that this equation is homogeneous in SI base units.

Base units of the left side
Pressure is force per area, $p = F/A$ , so its base units are $\text{kg}\,\text{m}\,\text{s}^{-2} \div \text{m}^2 = \text{kg}\,\text{m}^{-1}\,\text{s}^{-2}$ .
Base units of the right side
Density $\rho$ is $\text{kg}\,\text{m}^{-3}$ , $g$ is $\text{m}\,\text{s}^{-2}$ and height $h$ is $\text{m}$ .
$\rho g h : \text{kg}\,\text{m}^{-3} \times \text{m}\,\text{s}^{-2} \times \text{m}$
Simplify the right side
Collecting the powers of the metre: $-3 + 1 + 1 = -1$ , giving $\text{kg}\,\text{m}^{-1}\,\text{s}^{-2}$ .

Result: Both sides reduce to $\text{kg}\,\text{m}^{-1}\,\text{s}^{-2}$ , so the equation is homogeneous and dimensionally possible.

Typical mistakes

Treating the gram as the SI base unit of mass - it is the kilogram, so a mass given in grams must be divided by 1000 before use.
Forgetting to cube or square a prefix: $1\,\text{cm}^3 = (10^{-2}\,\text{m})^3 = 10^{-6}\,\text{m}^3$ , not $10^{-2}\,\text{m}^3$ .

Active revision

Show that the equation $v^2 = u^2 + 2as$ is homogeneous, and convert a density of $8.0\,\text{g}\,\text{cm}^{-3}$ into $\text{kg}\,\text{m}^{-3}$ .

Active recall

Recall the key points — then reveal.

Sources: GCE AS and A level subject content for the sciences (Department for Education) · AQA A-level Physics 7408 specification (AQA)

Errors, precision and accuracy#

FoundationAQA 7408 3.1.3DfE GCE Physics - errors and uncertainty

A measurement quoted with its uncertainty

Fig. 1A length of

24.0 \pm 0.5\,\text{mm}

: the best estimate is the central mark and the true value is expected to lie anywhere in the shaded interval.

No measurement is exact; every reading differs from the true value by an error. Errors come in two families. A random error varies unpredictably from reading to reading - the fluctuation of a stopwatch caught by human reaction time, or electrical noise - and scatters results above and below the true value. Because random errors are as likely to be positive as negative, they can be reduced by taking repeat readings and averaging: the mean of many trials is closer to the true value than any single reading.

A systematic error shifts every reading in the same direction by the same sort of amount, so it cannot be averaged away. The commonest kind is a zero error, where an instrument does not read zero when it should (a top-pan balance reading

0.02\,\text{g}

with nothing on it, or a micrometer with a non-zero closed reading). Systematic errors also arise from a mis-calibrated scale or from a flawed method, such as parallax when the eye is not square to a scale. They are found and removed by checking calibration and improving technique, not by repeating.

Precision and accuracy describe different things and must not be confused. A precise set of readings is closely bunched - it has small random error and small spread - regardless of whether it is near the true value. An accurate reading is close to the true value. A rifle grouping tightly but off the bullseye is precise but not accurate (a systematic error); a loose scatter centred on the bullseye is accurate on average but not precise. The best data are both. Related terms: repeatability (same experimenter, same apparatus, close results), reproducibility (a different experimenter or method, still close), and resolution (the smallest change an instrument can register).

The uncertainty of a measurement states the range within which the true value is expected to lie, written as value

\pm

absolute uncertainty. For a single reading from a scale the uncertainty is usually taken as half the smallest division (the resolution); for a set of repeated readings it is taken as half the range,

\tfrac{1}{2}(\text{max} - \text{min})

, about the mean. A length quoted as

24.0 \pm 0.5\,\text{mm}

means the true value almost certainly lies between

23.5\,\text{mm}

and

24.5\,\text{mm}

\text{uncertainty on a mean} = \tfrac{1}{2}\,(x_{\max} - x_{\min})

Half the range

For a set of repeated readings, the uncertainty is taken as half the spread about the mean.

\% \text{ uncertainty} = \dfrac{\Delta x}{x} \times 100\%

Percentage uncertainty

The absolute uncertainty $\Delta x$ divided by the value $x$ , expressed as a percentage.

Repeated readings scattered about a mean

Fig. 2Five timings of one pendulum swing, each shown with its reading uncertainty; the readings scatter about the mean of

2.01\,\text{s}

- a random error.

Worked example

Uncertainty on a mean

The five timings 2.01, 1.98, 2.05, 1.99 and 2.02 s are recorded for one swing of a pendulum. Find the mean period and its absolute and percentage uncertainty.

Mean
Add and divide by five: $(2.01 + 1.98 + 2.05 + 1.99 + 2.02)/5 = 10.05/5 = 2.01\,\text{s}$ .
Uncertainty from the range
The maximum is 2.05 s and the minimum 1.98 s, so the uncertainty is half the range.
$\Delta T = \tfrac{1}{2}(2.05 - 1.98) = 0.035 \approx 0.04\,\text{s}$
Percentage uncertainty
Divide the absolute uncertainty by the mean: $0.035/2.01 \times 100\% = 1.7\%$ .

Result: $T = 2.01 \pm 0.04\,\text{s}$ , a percentage uncertainty of about $2\%$ ; the scatter is a random error.

Typical mistakes

Believing that taking more repeat readings reduces a systematic error - it does not; only the random scatter is reduced.
Confusing precision with accuracy: closely bunched readings can still be inaccurate if a zero error shifts them all.

Active revision

Five timings of a pendulum swing give 2.01, 1.98, 2.05, 1.99 and 2.02 s. State the mean and the uncertainty on the mean, and identify whether reaction-time scatter is a random or a systematic error.

Active recall

Recall the key points — then reveal.

Sources: AQA A-level Physics 7408 specification (AQA)

Combining uncertainties#

StandardAQA 7408 3.1.3DfE GCE Physics - propagation of uncertainty

When measured quantities are combined in a calculation, their uncertainties propagate to the result, and the rule depends on how the quantities are combined. For a sum or a difference, the absolute uncertainties add. If

z = x + y

z = x - y

, then

\Delta z = \Delta x + \Delta y

. Note that subtracting two similar numbers is dangerous: the absolute uncertainty stays the same size while the result shrinks, so the percentage uncertainty can become very large - a good reason to avoid measuring a small quantity as the difference of two large ones.

For a product or a quotient, the fractional (or percentage) uncertainties add. If

z = xy

z = x/y

, then

\dfrac{\Delta z}{z} = \dfrac{\Delta x}{x} + \dfrac{\Delta y}{y}

. This is the workhorse rule of A-Level data analysis: a resistance found from

R = V/I

has a percentage uncertainty equal to the percentage uncertainty in

V

plus that in

I

A power raised in a formula multiplies the fractional uncertainty by that power. If

z = x^{n}

then

\dfrac{\Delta z}{z} = |n|\,\dfrac{\Delta x}{x}

. A volume found from

V = L^3

therefore carries three times the percentage uncertainty of the side length; a quantity under a square root (

n = \tfrac{1}{2}

) carries half. Because the power multiplies the uncertainty, the quantity raised to the highest power usually dominates and is worth measuring most carefully.

The final answer should be quoted to a number of significant figures consistent with its uncertainty - typically the uncertainty to one significant figure, and the value to the same decimal place. Writing

\rho = 8.0 \pm 0.2\,\text{g}\,\text{cm}^{-3}

is honest; writing

\rho = 8.0341 \pm 0.2\,\text{g}\,\text{cm}^{-3}

claims a precision the data do not support. Round only at the very end, keeping guard figures through the working.

z = x \pm y \;\Rightarrow\; \Delta z = \Delta x + \Delta y

Sums and differences

Absolute uncertainties add for addition and subtraction.

z = \dfrac{x\,y}{w} \;\Rightarrow\; \dfrac{\Delta z}{z} = \dfrac{\Delta x}{x} + \dfrac{\Delta y}{y} + \dfrac{\Delta w}{w}

Products and quotients

Fractional (percentage) uncertainties add for multiplication and division.

z = x^{n} \;\Rightarrow\; \dfrac{\Delta z}{z} = |n|\,\dfrac{\Delta x}{x}

Powers

A power multiplies the fractional uncertainty by that power.

Worked example

Density of a cube

A cube has side $L = 2.00 \pm 0.01\,\text{cm}$ and mass $m = 64.0 \pm 0.5\,\text{g}$ . Find its density and the uncertainty in the density.

Volume and density
$V = L^3 = (2.00)^3 = 8.00\,\text{cm}^3$ and $\rho = m/V = 64.0/8.00 = 8.00\,\text{g}\,\text{cm}^{-3}$ .
Percentage uncertainties
In $L$ : $0.01/2.00 = 0.50\%$ . In $V = L^3$ : $3 \times 0.50\% = 1.5\%$ . In $m$ : $0.5/64.0 = 0.78\%$ .
Combine for the quotient
$\rho = m/V$ so the percentage uncertainties add.
$\dfrac{\Delta \rho}{\rho} = 0.78\% + 1.5\% = 2.3\%$
Absolute uncertainty
$2.3\% \times 8.00 = 0.18 \approx 0.2\,\text{g}\,\text{cm}^{-3}$ .

Result: $\rho = 8.0 \pm 0.2\,\text{g}\,\text{cm}^{-3}$ (the side length dominates because it is cubed).

Typical mistakes

Adding percentage uncertainties for a sum or absolute uncertainties for a product - the rule depends on the operation.
Forgetting the factor from a power: for $V = L^3$ the percentage uncertainty is tripled, not left unchanged.

Active revision

A metal wire has resistance found from $R = V/I$ with $V = 6.0 \pm 0.1\,\text{V}$ and $I = 1.50 \pm 0.05\,\text{A}$ . Calculate $R$ and its absolute uncertainty.

Active recall

Recall the key points — then reveal.

Sources: GCE AS and A level subject content for the sciences (Department for Education)

Graphs, error bars and gradient uncertainty#

AdvancedAQA 7408 3.1.3DfE GCE Physics - graphical analysis

Best-fit and worst-fit lines through data points

Fig. 3The best-fit line (solid) balances the points; the steeper worst-acceptable line (dashed) still passes near every point. Half the difference of their gradients is the gradient's uncertainty.

A straight-line graph is the physicist's most powerful tool because a linear relationship

y = mx + c

lets a gradient and intercept carry the physics. When a law is not linear, it is linearised: to test

T = 2\pi\sqrt{L/g}

you plot

T^2

against

L

, giving a straight line of gradient

4\pi^2/g

through the origin, from which

g

is found. Choosing the right axes so that the theory predicts a straight line is a standard exam skill.

Each plotted point carries an uncertainty, drawn as an error bar: a vertical bar of total length twice the uncertainty in

y

, and where relevant a horizontal bar for

x

. The best-fit line is then drawn to pass as close as possible to the points, balancing those above and below, and ideally passing through every error bar. The gradient and intercept are read from this best-fit line using a large triangle spanning most of the data to keep the reading uncertainty small.

The uncertainty in the gradient is found from the worst-acceptable lines - the steepest and the shallowest straight lines that still pass through (or touch) all the error bars. Their gradients

m_{\text{max}}

and

m_{\text{min}}

bracket the true gradient, and the uncertainty is taken as half their difference,

\Delta m = \tfrac{1}{2}(m_{\text{max}} - m_{\text{min}})

. The same pair of lines gives the uncertainty in the intercept. This is exactly the analysis rewarded in the practical-skills paper.

A percentage difference between an experimental result and an accepted value should be compared with the experiment's percentage uncertainty. If the accepted value lies within the uncertainty range, the result is consistent with theory; if it lies well outside, a systematic error is suspected and the method should be scrutinised. Honest evaluation - stating limitations and the dominant source of uncertainty - is what separates top answers from arithmetic alone.

\Delta m = \tfrac{1}{2}\left(m_{\max} - m_{\min}\right)

Gradient uncertainty from worst lines

Half the difference between the steepest and shallowest lines that still pass through all the error bars.

T^2 = \dfrac{4\pi^2}{g}\,L

Linearised pendulum law

Plotting $T^2$ (y) against $L$ (x) gives a straight line of gradient $4\pi^2/g$ .

Worked example

Determining g and its uncertainty from a graph

A plot of $T^2$ against $L$ for a simple pendulum has a best-fit gradient $4.02\,\text{s}^2\,\text{m}^{-1}$ . The steepest and shallowest acceptable lines have gradients $4.20$ and $3.86\,\text{s}^2\,\text{m}^{-1}$ . Find $g$ and its uncertainty.

Relate gradient to g
The gradient equals $4\pi^2/g$ , so $g = 4\pi^2/\text{gradient}$ .
$g = \dfrac{4\pi^2}{4.02} = 9.82\,\text{m}\,\text{s}^{-2}$
Gradient uncertainty
Half the difference of the worst gradients: $\tfrac{1}{2}(4.20 - 3.86) = 0.17\,\text{s}^2\,\text{m}^{-1}$ , a percentage of $0.17/4.02 = 4.2\%$ .
Uncertainty in g
Since $g = 4\pi^2/\text{gradient}$ , the percentage uncertainty in $g$ equals that in the gradient: $4.2\% \times 9.82 = 0.4\,\text{m}\,\text{s}^{-2}$ .

Result: $g = 9.8 \pm 0.4\,\text{m}\,\text{s}^{-2}$ , consistent with the accepted $9.81\,\text{m}\,\text{s}^{-2}$ .

Typical mistakes

Reading a gradient from too small a triangle, or from the plotted points rather than from the best-fit line.
Taking the gradient uncertainty from only one worst line rather than from the spread between the steepest and shallowest acceptable lines.

Active revision

Readings of $T^2$ against $L$ for a pendulum give a best-fit gradient of $4.02\,\text{s}^2\,\text{m}^{-1}$ . Determine $g$ and, given worst-line gradients of $4.20$ and $3.86$ , state its uncertainty.

Active recall

Recall the key points — then reveal.

Sources: AQA A-level Physics 7408 specification (AQA)

References & sources

Sources

Department for Education

GCE AS and A level subject content for the sciences

AQA

AQA A-level Physics 7408 specification

Measurements and their errors

4 sections~14 min reading time3 competenciesLevel Foundation 2 · Standard 1 · Advanced 1

Measurements and their errors

Key points

Checking an equation by its units

Key points

Uncertainty on a mean

Key points

Density of a cube

Key points

Determining g and its uncertainty from a graph

Sources

Measurements and their errors

Key points

Checking an equation by its units

Key points

Uncertainty on a mean

Key points

Density of a cube

Key points

Determining g and its uncertainty from a graph

Sources