Measurements and uncertainties

What you'll learn

How to recognise random errors, systematic errors, and zero errors.
The difference between precision and accuracy.
How to calculate absolute and percentage uncertainties, then combine them in calculations.
How to use error bars, a line of best fit, a worst acceptable line, and percentage difference.

1. Measurements are estimates

In physics, a measurement is an estimate of a physical quantity, written with a number and a unit, such as 0.248 m or 12.6 V. No measurement is perfectly exact, because instruments have limited scale divisions and experiments are affected by variation.

The true value is the value you would obtain with a perfect measurement. In real experiments, you usually do not know it exactly.

Definition

Uncertainty

An uncertainty is an estimate of the range around a measured value within which the true value is likely to lie. A result written as $\pm \Delta x$ has measured value $x$ and absolute uncertainty $Δx\Delta x$ , in the same unit as $x$ .

The resolution of an instrument is the smallest change it can display or that you can read from its scale. As a simple first estimate:

an analogue scale is often read to about half the smallest division;
a digital meter is often taken as plus/minus one in the last displayed digit, unless the instrument specification says otherwise;
for repeated readings, the spread of values gives evidence about random uncertainty, often estimated using half the range.

The percentage uncertainty compares the absolute uncertainty with the measured value:

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

Example

Calculating percentage uncertainty

A length is measured as $\pm 0.002)\ \text{m}$ .

Identify the absolute uncertainty: $m\Delta l = 0.002\ \text{m}$ .
Divide by the measured value and multiply by 100%: $0.0020.834×100%=0.2398%\frac{0.002}{0.834} \times 100\% = 0.2398\%$ .
Quote sensibly: the percentage uncertainty is about 0.24%.

2. Random and systematic errors

An error is the difference between a measured value and the true value. This is not quite the same as uncertainty: the error is usually unknown, while the uncertainty is your estimate of how large the error might reasonably be.

Definition

Random error

A random error causes readings to vary unpredictably about a mean value. Repeating measurements and calculating a mean reduces the effect of random errors.

Examples include reaction-time variation when using a stopwatch, small fluctuations in a meter reading, or judging a scale from slightly different angles.

Definition

Systematic error

A systematic error shifts all readings in the same direction, so the measurements are consistently too high or too low. Repeats do not remove a systematic error.

Examples include a miscalibrated voltmeter, a ruler with a worn end, or reading a scale from a fixed poor viewing angle.

A zero error is a systematic error where an instrument does not read zero when the quantity being measured is actually zero. For example, a micrometer might read 0.03 mm even when fully closed.

Key Idea

Zero correction

For a zero error, use:

xcorrected=xobserved−xzerox_{\text{corrected}} = x_{\text{observed}} - x_{\text{zero}}

So if the zero reading is positive, subtract it from every observed reading.

Example

Correcting a positive zero error

A micrometer reads 0.03 mm when fully closed. When measuring a wire, it reads 4.86 mm.

The zero reading is positive, so the micrometer is reading too high by 0.03 mm.
Apply the zero correction: $mmx_{\text{corrected}} = 4.86\ \text{mm} - 0.03\ \text{mm}$ .
The corrected diameter is $mm4.83\ \text{mm}$ .

Common Mistake

Repeating does not fix everything

Taking more repeats helps with random errors, but it does not remove a systematic error. A badly zeroed instrument can give very precise repeated readings that are still inaccurate.

3. Precision and accuracy

Accuracy means closeness to the true value. A result is accurate if it is near the accepted or true value.

Precision means closeness of repeated readings to each other. A set of readings is precise if they have a small spread, even if they are all shifted away from the true value.

The target diagram is a useful mental model: the centre represents the true value, and each dot represents a repeated measurement.

Accuracy and precision target diagram

Key Idea

Accuracy versus precision

Random errors mainly reduce precision. Systematic errors mainly reduce accuracy.

4. Combining uncertainties

When you calculate a new quantity from measured values, the uncertainties must be combined too. At A Level, you use simple “worst-case” rules, not a full statistical treatment.

Definition

Absolute and percentage uncertainty

An absolute uncertainty has the same unit as the measurement, such as $m\pm 0.002\ \text{m}$ . A percentage uncertainty expresses this as a percentage of the measured value.

For the simple rules below, $ΔA\Delta A$ means the absolute uncertainty in $A$ .

Q=AB⇒ΔQQ=ΔAA+ΔBBQ=An⇒ΔQQ=∣n∣ΔAA\begin{aligned} Q &= A+B\ \text{or}\ Q=A-B \quad \Rightarrow \quad \Delta Q = \Delta A + \Delta B \\ Q &= AB\ \text{or}\ Q=\frac{A}{B} \quad \Rightarrow \quad \frac{\Delta Q}{Q} = \frac{\Delta A}{A} + \frac{\Delta B}{B} \\ Q &= A^n \quad \Rightarrow \quad \frac{\Delta Q}{Q}= |n|\frac{\Delta A}{A} \end{aligned}

Key Idea

Which uncertainty should you add?

For addition and subtraction, add absolute uncertainties. For multiplication, division, and powers, add percentage uncertainties.

Example

Subtracting two length readings

A marker position changes from $mx_1 = (0.112 \pm 0.001)\ \text{m}$ to $mx_2 = (0.284 \pm 0.001)\ \text{m}$ . Find the extension.

Calculate the extension: $x_2 - x_1 = 0.284\ \text{m} - 0.112\ \text{m} = 0.172\ \text{m}$ .
Because this is subtraction, add absolute uncertainties: $m\Delta e = 0.001\ \text{m} + 0.001\ \text{m} = 0.002\ \text{m}$ .
The result is $\pm 0.002)\ \text{m}$ , with percentage uncertainty $0.0020.172×100%=1.2%\frac{0.002}{0.172} \times 100\% = 1.2\%$ .

Example

Combining multiplication, division and powers

A cylinder has mass $\pm 0.001)\ \text{kg}$ , radius $\pm 0.0001)\ \text{m}$ , and length $\pm 0.001)\ \text{m}$ . Its density is $ρ=mπr2L\rho = \frac{m}{\pi r^2 L}$ .

Calculate the density: $m−3\rho = \frac{0.184}{\pi(0.0120)^2(0.150)} = 2.71 \times 10^3\ \text{kg m}^{-3}$ .
Add percentage uncertainties, remembering that $r^2$ doubles the percentage uncertainty in $r$ : $0.0010.184×100%+2(0.00010.0120×100%)+0.0010.150×100%=2.9%\frac{0.001}{0.184}\times100\% + 2\left(\frac{0.0001}{0.0120}\times100\%\right) + \frac{0.001}{0.150}\times100\% = 2.9\%$ .
Convert back to an absolute uncertainty: $0.029 \times 2.71 \times 10^3\ \text{kg m}^{-3} \approx 0.08 \times 10^3\ \text{kg m}^{-3}$ , so $m−3\rho = (2.71 \pm 0.08)\times10^3\ \text{kg m}^{-3}$ .

Tip

Quoting final answers

Usually quote an uncertainty to one significant figure, then quote the measured value to the same decimal place as the uncertainty. If the uncertainty begins with 1 or 2, two significant figures are often sensible.

5. Graphical treatment of uncertainties

Graphs are powerful because they show both the trend and the scatter. An error bar is a line through a plotted point showing the uncertainty in that value. At A Level, you are expected to know the basic use of error bars; you are not expected to do advanced statistics.

A line of best fit is a straight line or smooth curve that represents the overall trend. It should balance the data, not simply join the points. A worst acceptable line is the steepest or shallowest line that still reasonably fits the error bars.

This graph shows how vertical error bars, a best-fit line, and worst acceptable lines are used to estimate uncertainty in a gradient.

Graph with error bars, best-fit line and worst acceptable lines

For a straight-line graph, calculate the gradient using a large triangle on the line:

gradient=ΔyΔx\text{gradient} = \frac{\Delta y}{\Delta x}

To estimate uncertainty in the gradient, compare the best-fit gradient with the gradient of the worst acceptable line furthest from it:

Δm=∣mworst−mbest∣\Delta m = |m_{\text{worst}} - m_{\text{best}}|

Example

Estimating uncertainty in a gradient

From a graph, the best-fit gradient is $m−1m_{\text{best}} = 2.42\ \text{N m}^{-1}$ . The steepest acceptable gradient is $m−12.58\ \text{N m}^{-1}$ and the shallowest acceptable gradient is $m−12.29\ \text{N m}^{-1}$ .

Compare the two possible differences: $0.16\ \text{N m}^{-1}$ and $0.13\ \text{N m}^{-1}$ .
Choose the larger difference as the absolute uncertainty: $m−1\Delta m = 0.16\ \text{N m}^{-1}$ .
Calculate the percentage uncertainty: $0.162.42×100%=6.6%\frac{0.16}{2.42}\times100\% = 6.6\%$ , so $\pm 0.16)\ \text{N m}^{-1}$ .

Common Mistake

Forcing the line through the origin

Only force a graph through the origin if the physics requires it. Otherwise, draw the line that best represents the data and their error bars.

6. Percentage difference

A percentage difference is used to compare two values, often your experimental result with an accepted reference value.

value×100%\text{percentage difference} = \frac{|\text{experimental value} - \text{reference value}|}{\text{reference value}} \times 100\%

If there is no accepted reference value and you are comparing two experimental values, follow the question wording; some questions use the mean of the two values as the denominator.

Example

Checking agreement with an accepted value

An experiment gives $s−2g_{\text{exp}} = (9.67 \pm 0.15)\ \text{m s}^{-2}$ . The accepted value is $s−29.81\ \text{m s}^{-2}$ .

Calculate the percentage difference: $∣9.67−9.81∣9.81×100%=1.4%\frac{|9.67 - 9.81|}{9.81}\times100\% = 1.4\%$ .
Calculate the percentage uncertainty in the experimental value: $0.159.67×100%=1.6%\frac{0.15}{9.67}\times100\% = 1.6\%$ .
Since the percentage difference is smaller than the percentage uncertainty, the result is consistent with the accepted value.

Exam technique

In the exam

For error questions, state both the type of error and the remedy: repeat and average for random errors; calibrate or correct for systematic and zero errors.
For uncertainty calculations, choose the rule before calculating: absolute uncertainties for addition/subtraction; percentage uncertainties for multiplication/division/powers.
For graph questions, draw clear error bars, a balanced best-fit line, and a justified worst acceptable line; use a large gradient triangle and include units.

Self review

Check yourself

A micrometer reads 0.02 mm when fully closed. What correction should you apply to a measured reading?
For $\frac{V}{I}$ , should you add absolute uncertainties or percentage uncertainties?
How would you estimate the uncertainty in a gradient from a graph with error bars?

What you'll learn

How to recognise random errors, systematic errors, and zero errors.
The difference between precision and accuracy.
How to calculate absolute and percentage uncertainties, then combine them in calculations.
How to use error bars, a line of best fit, a worst acceptable line, and percentage difference.

1. Measurements are estimates

The true value is the value you would obtain with a perfect measurement. In real experiments, you usually do not know it exactly.

Definition

Uncertainty

The resolution of an instrument is the smallest change it can display or that you can read from its scale. As a simple first estimate:

an analogue scale is often read to about half the smallest division;
a digital meter is often taken as plus/minus one in the last displayed digit, unless the instrument specification says otherwise;
for repeated readings, the spread of values gives evidence about random uncertainty, often estimated using half the range.

The percentage uncertainty compares the absolute uncertainty with the measured value:

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

Example

Calculating percentage uncertainty

A length is measured as $\pm 0.002)\ \text{m}$ .

Identify the absolute uncertainty: $m\Delta l = 0.002\ \text{m}$ .
Divide by the measured value and multiply by 100%: $0.0020.834×100%=0.2398%\frac{0.002}{0.834} \times 100\% = 0.2398\%$ .
Quote sensibly: the percentage uncertainty is about 0.24%.

2. Random and systematic errors

Definition

Random error

A random error causes readings to vary unpredictably about a mean value. Repeating measurements and calculating a mean reduces the effect of random errors.

Examples include reaction-time variation when using a stopwatch, small fluctuations in a meter reading, or judging a scale from slightly different angles.

Definition

Systematic error

A systematic error shifts all readings in the same direction, so the measurements are consistently too high or too low. Repeats do not remove a systematic error.

Examples include a miscalibrated voltmeter, a ruler with a worn end, or reading a scale from a fixed poor viewing angle.

A zero error is a systematic error where an instrument does not read zero when the quantity being measured is actually zero. For example, a micrometer might read 0.03 mm even when fully closed.

Key Idea

Zero correction

For a zero error, use:

xcorrected=xobserved−xzerox_{\text{corrected}} = x_{\text{observed}} - x_{\text{zero}}

So if the zero reading is positive, subtract it from every observed reading.

Example

Correcting a positive zero error

A micrometer reads 0.03 mm when fully closed. When measuring a wire, it reads 4.86 mm.

The zero reading is positive, so the micrometer is reading too high by 0.03 mm.
Apply the zero correction: $mmx_{\text{corrected}} = 4.86\ \text{mm} - 0.03\ \text{mm}$ .
The corrected diameter is $mm4.83\ \text{mm}$ .

Common Mistake

Repeating does not fix everything

Taking more repeats helps with random errors, but it does not remove a systematic error. A badly zeroed instrument can give very precise repeated readings that are still inaccurate.

3. Precision and accuracy

Accuracy means closeness to the true value. A result is accurate if it is near the accepted or true value.

Precision means closeness of repeated readings to each other. A set of readings is precise if they have a small spread, even if they are all shifted away from the true value.

The target diagram is a useful mental model: the centre represents the true value, and each dot represents a repeated measurement.

Accuracy and precision target diagram

Key Idea

Accuracy versus precision

Random errors mainly reduce precision. Systematic errors mainly reduce accuracy.

4. Combining uncertainties

When you calculate a new quantity from measured values, the uncertainties must be combined too. At A Level, you use simple “worst-case” rules, not a full statistical treatment.

Definition

Absolute and percentage uncertainty

An absolute uncertainty has the same unit as the measurement, such as $m\pm 0.002\ \text{m}$ . A percentage uncertainty expresses this as a percentage of the measured value.

For the simple rules below, $ΔA\Delta A$ means the absolute uncertainty in $A$ .

Q=AB⇒ΔQQ=ΔAA+ΔBBQ=An⇒ΔQQ=∣n∣ΔAA\begin{aligned} Q &= A+B\ \text{or}\ Q=A-B \quad \Rightarrow \quad \Delta Q = \Delta A + \Delta B \\ Q &= AB\ \text{or}\ Q=\frac{A}{B} \quad \Rightarrow \quad \frac{\Delta Q}{Q} = \frac{\Delta A}{A} + \frac{\Delta B}{B} \\ Q &= A^n \quad \Rightarrow \quad \frac{\Delta Q}{Q}= |n|\frac{\Delta A}{A} \end{aligned}

Key Idea

Which uncertainty should you add?

For addition and subtraction, add absolute uncertainties. For multiplication, division, and powers, add percentage uncertainties.

Example

Subtracting two length readings

A marker position changes from $mx_1 = (0.112 \pm 0.001)\ \text{m}$ to $mx_2 = (0.284 \pm 0.001)\ \text{m}$ . Find the extension.

Calculate the extension: $x_2 - x_1 = 0.284\ \text{m} - 0.112\ \text{m} = 0.172\ \text{m}$ .
Because this is subtraction, add absolute uncertainties: $m\Delta e = 0.001\ \text{m} + 0.001\ \text{m} = 0.002\ \text{m}$ .
The result is $\pm 0.002)\ \text{m}$ , with percentage uncertainty $0.0020.172×100%=1.2%\frac{0.002}{0.172} \times 100\% = 1.2\%$ .

Example

Combining multiplication, division and powers

A cylinder has mass $\pm 0.001)\ \text{kg}$ , radius $\pm 0.0001)\ \text{m}$ , and length $\pm 0.001)\ \text{m}$ . Its density is $ρ=mπr2L\rho = \frac{m}{\pi r^2 L}$ .

Calculate the density: $m−3\rho = \frac{0.184}{\pi(0.0120)^2(0.150)} = 2.71 \times 10^3\ \text{kg m}^{-3}$ .
Add percentage uncertainties, remembering that $r^2$ doubles the percentage uncertainty in $r$ : $0.0010.184×100%+2(0.00010.0120×100%)+0.0010.150×100%=2.9%\frac{0.001}{0.184}\times100\% + 2\left(\frac{0.0001}{0.0120}\times100\%\right) + \frac{0.001}{0.150}\times100\% = 2.9\%$ .
Convert back to an absolute uncertainty: $0.029 \times 2.71 \times 10^3\ \text{kg m}^{-3} \approx 0.08 \times 10^3\ \text{kg m}^{-3}$ , so $m−3\rho = (2.71 \pm 0.08)\times10^3\ \text{kg m}^{-3}$ .

Tip

Quoting final answers

5. Graphical treatment of uncertainties

This graph shows how vertical error bars, a best-fit line, and worst acceptable lines are used to estimate uncertainty in a gradient.

Graph with error bars, best-fit line and worst acceptable lines

For a straight-line graph, calculate the gradient using a large triangle on the line:

gradient=ΔyΔx\text{gradient} = \frac{\Delta y}{\Delta x}

To estimate uncertainty in the gradient, compare the best-fit gradient with the gradient of the worst acceptable line furthest from it:

Δm=∣mworst−mbest∣\Delta m = |m_{\text{worst}} - m_{\text{best}}|

Example

Estimating uncertainty in a gradient

Compare the two possible differences: $0.16\ \text{N m}^{-1}$ and $0.13\ \text{N m}^{-1}$ .
Choose the larger difference as the absolute uncertainty: $m−1\Delta m = 0.16\ \text{N m}^{-1}$ .
Calculate the percentage uncertainty: $0.162.42×100%=6.6%\frac{0.16}{2.42}\times100\% = 6.6\%$ , so $\pm 0.16)\ \text{N m}^{-1}$ .

Common Mistake

Forcing the line through the origin

Only force a graph through the origin if the physics requires it. Otherwise, draw the line that best represents the data and their error bars.

6. Percentage difference

A percentage difference is used to compare two values, often your experimental result with an accepted reference value.

value×100%\text{percentage difference} = \frac{|\text{experimental value} - \text{reference value}|}{\text{reference value}} \times 100\%

If there is no accepted reference value and you are comparing two experimental values, follow the question wording; some questions use the mean of the two values as the denominator.

Example

Checking agreement with an accepted value

An experiment gives $s−2g_{\text{exp}} = (9.67 \pm 0.15)\ \text{m s}^{-2}$ . The accepted value is $s−29.81\ \text{m s}^{-2}$ .

Calculate the percentage difference: $∣9.67−9.81∣9.81×100%=1.4%\frac{|9.67 - 9.81|}{9.81}\times100\% = 1.4\%$ .
Calculate the percentage uncertainty in the experimental value: $0.159.67×100%=1.6%\frac{0.15}{9.67}\times100\% = 1.6\%$ .
Since the percentage difference is smaller than the percentage uncertainty, the result is consistent with the accepted value.

Exam technique

In the exam

For error questions, state both the type of error and the remedy: repeat and average for random errors; calibrate or correct for systematic and zero errors.
For uncertainty calculations, choose the rule before calculating: absolute uncertainties for addition/subtraction; percentage uncertainties for multiplication/division/powers.
For graph questions, draw clear error bars, a balanced best-fit line, and a justified worst acceptable line; use a large gradient triangle and include units.

Self review

Check yourself

A micrometer reads 0.02 mm when fully closed. What correction should you apply to a measured reading?
For $\frac{V}{I}$ , should you add absolute uncertainties or percentage uncertainties?
How would you estimate the uncertainty in a gradient from a graph with error bars?

What you'll learn

1. Measurements are estimates

2. Random and systematic errors

3. Precision and accuracy

4. Combining uncertainties

5. Graphical treatment of uncertainties

6. Percentage difference

Review Measurements and uncertainties by teaching Genie

Recap your knowledge with an interactive lesson

Remember key concepts with flashcards

Measurements and uncertainties

What you'll learn

1. Measurements are estimates

2. Random and systematic errors

3. Precision and accuracy

4. Combining uncertainties

5. Graphical treatment of uncertainties

6. Percentage difference

Review Measurements and uncertainties by teaching Genie

Recap your knowledge with an interactive lesson

Remember key concepts with flashcards

Measurements and uncertainties

What you'll learn

1. Measurements are estimates

2. Random and systematic errors

3. Precision and accuracy

4. Combining uncertainties

5. Graphical treatment of uncertainties

6. Percentage difference

Review Measurements and uncertainties by teaching Genie

Recap your knowledge with an interactive lesson

Remember key concepts with flashcards

Measurements and uncertainties Revision Guide

Measurements and uncertainties

What you'll learn

1. Measurements are estimates

2. Random and systematic errors

3. Precision and accuracy

4. Combining uncertainties

5. Graphical treatment of uncertainties

6. Percentage difference

Review Measurements and uncertainties by teaching Genie

Recap your knowledge with an interactive lesson

Remember key concepts with flashcards

Measurements and uncertainties Revision Guide