Measurement uncertainty

You measure a desk twice and get slightly different answers. That is not automatically a failed experiment. It is a clue about the instrument, the procedure or both. The useful question is how much confidence each reported digit deserves.

This chapter separates several ideas that are often compressed into the word “error.” You will learn to distinguish a known correction from unknown error, repeatability from accuracy, and a stated bound from a statistical standard uncertainty. These distinctions prevent experimental questions from turning into the mechanical addition of every ± sign.

What you will learn to do

Interpret a reading with an uncertainty without inventing a probability statement.
Explain why repeated measurements cannot remove a shared calibration offset.
Use absolute and relative bounds consistently after unit conversion.
Correct a known zero error and check whether a claimed discrepancy is established by the stated bounds.

Useful before this chapterSignificant figures →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Measurement uncertainty
Learning goals:
- Interpret a reading with an uncertainty without inventing a probability statement.
- Explain why repeated measurements cannot remove a shared calibration offset.
- Use absolute and relative bounds consistently after unit conversion.
- Correct a known zero error and check whether a claimed discrepancy is established by the stated bounds.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

BEFORE YOU READ

Bring back one idea

Try a sentence or a sketch before opening the comparison. If this is new, use the explanation to get started.

A ruler has millimetre marks. Does a calculator result with six decimal places make its measurement more precise?

Compare your reasoning

No. Arithmetic cannot supply measurement information that the procedure did not provide. Resolution is one contribution to uncertainty; alignment and calibration may matter too.

Revisit the explanation →

01 / Measurement uncertainty

A measurement is an estimate, not an oracle

A ruler, balance or timer responds to a physical quantity through a procedure. The result depends on more than its display. The ruler may be tilted, the zero may be displaced, the temperature may differ from the calibration temperature, or the start and finish may be judged by eye. Writing a central value records the best estimate produced by that procedure.

The measurement error is the difference between the result and the relevant true value. Usually that true value is not known exactly, so the actual error cannot simply be read off the instrument. Uncertainty describes what remains unresolved about the result using a stated model and available information. It is not automatically the mistake someone made.

For introductory problems, a statement such as L = (24.6 ± 0.2) cm is often explicitly defined as a bound. Under that assumption, the model permits values from 24.4 to 24.8 cm. The same notation can mean something different in a laboratory report, so read the definition before calculating.

L=L_0\pm\Delta L,\qquad L_0-\Delta L\le L\le L_0+\Delta L

This interval interpretation applies when ΔL is stated to be a bound. It does not assign probabilities to positions inside the interval.

STOP & THINK

A result is 24.6 ± 0.2 cm, explicitly a bound. Which conclusion is justified?

The error is exactly 0.2 cm
The adopted model permits 24.4–24.8 cm
There is automatically a 95% confidence level

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Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Measurement uncertainty
Focus: A measurement is an estimate, not an oracle
Learning goals:
- Interpret a reading with an uncertainty without inventing a probability statement.
- Explain why repeated measurements cannot remove a shared calibration offset.
- Use absolute and relative bounds consistently after unit conversion.
- Correct a known zero error and check whether a claimed discrepancy is established by the stated bounds.
Equations in this section (LaTeX):
L=L_0\pm\Delta L,\qquad L_0-\Delta L\le L\le L_0+\Delta L
Meaning: This interval interpretation applies when ΔL is stated to be a bound. It does not assign probabilities to positions inside the interval.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

02 / Measurement uncertainty

Resolution is only one contribution

Resolution is the smallest change in an indication that can be distinguished under the stated conditions. A digital instrument may advance in steps of 0.01 V; a ruler may have marks 1 mm apart. That tells you something about reading granularity, but not the complete calibration uncertainty.

If a model says the displayed value is rounded to the nearest 0.01 V and has no other error, a half-step rounding bound of 0.005 V is appropriate. If a problem instead stipulates a one-division reading bound, use that. Neither half a division nor one full division is a universal law for every instrument.

When a length is found by subtracting two independently read endpoints, both endpoint readings contribute. The operation is a difference, but uncertainty bounds do not cancel merely because the readings are subtracted. Both readings could move in directions that enlarge the final error.

WORKED EXAMPLE

Reading both ends of a strip

A strip’s endpoints are read as 12.4 cm and 31.7 cm. Each reading has a stated independent admissible bound of ±0.05 cm. Find the length and a worst-case bound.

MEMBERSThe approach, step-by-step working and check are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Measurement uncertainty
Focus: Resolution is only one contribution
Learning goals:
- Interpret a reading with an uncertainty without inventing a probability statement.
- Explain why repeated measurements cannot remove a shared calibration offset.
- Use absolute and relative bounds consistently after unit conversion.
- Correct a known zero error and check whether a claimed discrepancy is established by the stated bounds.
Equations in this section (LaTeX):
Build the physical reasoning before choosing an equation.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

03 / Measurement uncertainty

A tight cluster can still be in the wrong place

Repeatability describes how closely repeated results agree under the same conditions. Accuracy concerns agreement with the relevant reference or true value. A group of nearly identical readings can be repeatable while all being shifted by the same calibration offset.

Random variation can make repeated readings scatter to either side of their underlying mean. A persistent systematic effect changes results in a related way across repetitions. The same effect need not be constant under every possible condition: for example, a temperature-dependent calibration error changes when the temperature changes.

The practical question is what repeating the experiment changes. Independent random contributions can average down. A common zero offset remains in every term of the mean. More repetitions of the same biased procedure cannot identify that offset without other information.

INTERACTIVE LAB

Move the whole cluster

Change the shared zero offset. Watch the mean and the spread separately. Explain why a smaller spread would not reveal a common offset.

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\frac1n\sum_{i=1}^{n}(x_i+b)=\bar x+b

A common additive offset b survives averaging. Increasing n does not divide that offset by n.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Measurement uncertainty
Focus: A tight cluster can still be in the wrong place
Learning goals:
- Interpret a reading with an uncertainty without inventing a probability statement.
- Explain why repeated measurements cannot remove a shared calibration offset.
- Use absolute and relative bounds consistently after unit conversion.
- Correct a known zero error and check whether a claimed discrepancy is established by the stated bounds.
Equations in this section (LaTeX):
\frac1n\sum_{i=1}^{n}(x_i+b)=\bar x+b
Meaning: A common additive offset b survives averaging. Increasing n does not divide that offset by n.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

04 / Measurement uncertainty

Averages and statistical uncertainty need assumptions

The arithmetic mean is a useful central estimate for repeated readings of the same stable quantity under comparable conditions. Before averaging, check that the procedure has not changed and the quantity is not drifting. A mean of measurements taken while the object is warming may hide a real trend.

A standard uncertainty has the form of a standard deviation. It is not a hard maximum possible error. For independent repeated readings with the same underlying distribution and a finite variance, the standard uncertainty of their mean is estimated by the sample standard deviation divided by the square root of the number of readings. A small sample makes that estimate itself uncertain.

This statistical rule does not justify dividing a stated worst-case bound by √n. Nor does it eliminate calibration uncertainty, shared timing bias or correlation. The distinction is the reason we name the uncertainty model before using a formula.

\bar x=\frac1n\sum x_i,\qquad s=\sqrt{\frac{\sum(x_i-\bar x)^2}{n-1}},\qquad u_{\bar x}=\frac{s}{\sqrt n}

The last expression models independent random scatter in the mean. It is not the total uncertainty if other contributions are present.

WORKED EXAMPLE

Understand what the spread estimate contains

Three readings are 2.98, 3.00 and 3.02 cm. Treat them as independent repeats of one stable quantity. Estimate their mean and the standard uncertainty due only to repeatability.

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STOP & THINK

A scale has a stable unknown +0.10 g offset. You average 100 readings. What happens to that offset in the mean?

It becomes +0.001 g
It becomes +0.010 g
It remains +0.10 g

MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Measurement uncertainty
Focus: Averages and statistical uncertainty need assumptions
Learning goals:
- Interpret a reading with an uncertainty without inventing a probability statement.
- Explain why repeated measurements cannot remove a shared calibration offset.
- Use absolute and relative bounds consistently after unit conversion.
- Correct a known zero error and check whether a claimed discrepancy is established by the stated bounds.
Equations in this section (LaTeX):
\bar x=\frac1n\sum x_i,\qquad s=\sqrt{\frac{\sum(x_i-\bar x)^2}{n-1}},\qquad u_{\bar x}=\frac{s}{\sqrt n}
Meaning: The last expression models independent random scatter in the mean. It is not the total uncertainty if other contributions are present.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Back to chapter contents ↑

05 / Measurement uncertainty

Correct a known offset with the right sign

Suppose an instrument reads too high by a known amount. Its corrected value must be lower. This plain-language check is often more reliable than memorizing a rule about positive or negative zero error.

Write a model of the indication. If indicated = quantity + offset, then corrected quantity = indicated − offset. The correction itself may have uncertainty because the offset was measured during calibration. Correcting the central value does not make the remaining uncertainty disappear.

WORKED EXAMPLE

Apply a calibration correction

A caliper indicates 18.42 mm. Its positive zero offset is measured as 0.08 mm. Reading and offset have stated bounds ±0.03 mm and ±0.02 mm, with no restriction preventing their worst-case combination. Find the corrected length and bound.

MEMBERSThe approach, step-by-step working and check are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Measurement uncertainty
Focus: Correct a known offset with the right sign
Learning goals:
- Interpret a reading with an uncertainty without inventing a probability statement.
- Explain why repeated measurements cannot remove a shared calibration offset.
- Use absolute and relative bounds consistently after unit conversion.
- Correct a known zero error and check whether a claimed discrepancy is established by the stated bounds.
Equations in this section (LaTeX):
Build the physical reasoning before choosing an equation.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

06 / Measurement uncertainty

Compare uncertainties on the same scale

An absolute uncertainty has the units of the quantity: millimetres for length, seconds for time. A relative uncertainty divides its magnitude by the magnitude of a nonzero central value. A percentage is one hundred times that dimensionless ratio.

The same absolute bound can matter very differently for two readings. A ±0.20 cm bound on 20.0 cm is 1%; the same bound on 2.0 cm is 10%. The shorter measurement is less tightly determined relative to its size, even though the instrument’s absolute bound is unchanged.

Convert both numerator and denominator to compatible units. A 0.5 mm bound divided by a 2.0 m reading is 0.00025, not 0.25. If the central value is zero, this relative ratio is undefined; use an absolute uncertainty or a separately justified reference scale instead.

r=\frac{\Delta x}{|x_0|},\qquad r_{\%}=100r

The ratio is dimensionless. The absolute value prevents a negative coordinate from producing a negative uncertainty.

STOP & THINK

A 50.0 cm length has a stated bound of ±0.10 cm. What is the percentage bound?

Answer in %.

MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Measurement uncertainty
Focus: Compare uncertainties on the same scale
Learning goals:
- Interpret a reading with an uncertainty without inventing a probability statement.
- Explain why repeated measurements cannot remove a shared calibration offset.
- Use absolute and relative bounds consistently after unit conversion.
- Correct a known zero error and check whether a claimed discrepancy is established by the stated bounds.
Equations in this section (LaTeX):
r=\frac{\Delta x}{|x_0|},\qquad r_{\%}=100r
Meaning: The ratio is dimensionless. The absolute value prevents a negative coordinate from producing a negative uncertainty.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

07 / Measurement uncertainty

Absolute error, relative error and deviations from a mean

When a reliable reference value is supplied, a signed error can be calculated: measured value minus reference value. Its magnitude is the absolute error. A relative error compares that magnitude with the magnitude of the nonzero reference value; a percentage error multiplies the ratio by one hundred. Always identify the denominator, because “relative uncertainty in an estimate” normally uses the estimate itself while “relative error from a reference” uses the reference.

Without a known reference, you cannot calculate the actual measurement error merely by inspecting the displayed digits. Repeated observations let you describe their scatter around their own mean, but the mean itself could be displaced from the true value. This distinction prevents a precise-looking set of repeated readings from being mistaken for proof of accuracy.

e=x_{\mathrm{meas}}-x_{\mathrm{ref}},\qquad |e|=|x_{\mathrm{meas}}-x_{\mathrm{ref}}|,\qquad r_e=\frac{|e|}{|x_{\mathrm{ref}}|}

The signed error and its magnitude have the measurement unit. The relative error is dimensionless and requires a nonzero reference.

WORKED EXAMPLE

Compare a measurement with a supplied reference

A measurement gives 9.74 m/s² for a quantity whose stipulated reference is 9.80 m/s². Find the signed error, absolute error and percentage error.

MEMBERSThe approach, step-by-step working and check are for members.Unlock →

A commonly used introductory scatter measure is the mean absolute deviation about the sample mean. First find the mean, then take the magnitude of each deviation, then average those magnitudes. Taking a signed mean of deviations would give zero by construction and conceal the scatter.

Mean absolute deviation, sample standard deviation and standard uncertainty of the mean are different statistics. They cannot be exchanged without changing the interpretation. Some examination exercises prescribe the mean absolute deviation as their reported “mean error”; follow the stipulated calculation but explain that it does not prove a hard bound on the unknown true value.

\bar x=\frac1n\sum x_i,\qquad D=\frac1n\sum|x_i-\bar x|

D describes the observed spread around the sample mean. A common additive offset shifts every reading and the mean together, leaving D unchanged.

WORKED EXAMPLE

A mean can be stable while the readings scatter

Three length readings are 9.9, 10.0 and 10.1 cm. Find their mean and mean absolute deviation. Does this establish the actual error relative to the true length?

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STOP & THINK

For repeated readings, the signed deviations from their arithmetic mean sum to zero. What does this establish?

All readings are perfectly accurate
Only an algebraic property of the chosen mean
The instrument has zero calibration uncertainty

MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Measurement uncertainty
Focus: Absolute error, relative error and deviations from a mean
Learning goals:
- Interpret a reading with an uncertainty without inventing a probability statement.
- Explain why repeated measurements cannot remove a shared calibration offset.
- Use absolute and relative bounds consistently after unit conversion.
- Correct a known zero error and check whether a claimed discrepancy is established by the stated bounds.
Equations in this section (LaTeX):
e=x_{\mathrm{meas}}-x_{\mathrm{ref}},\qquad |e|=|x_{\mathrm{meas}}-x_{\mathrm{ref}}|,\qquad r_e=\frac{|e|}{|x_{\mathrm{ref}}|}
Meaning: The signed error and its magnitude have the measurement unit. The relative error is dimensionless and requires a nonzero reference.
\bar x=\frac1n\sum x_i,\qquad D=\frac1n\sum|x_i-\bar x|
Meaning: D describes the observed spread around the sample mean. A common additive offset shifts every reading and the mean together, leaving D unchanged.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

08 / Measurement uncertainty

What does agreement within bounds establish?

Two central values can differ without the stated bounds establishing a contradiction. Compare their admissible intervals, or form a bound on their difference. If zero lies inside the permitted difference interval, the data do not establish a nonzero difference under that model.

This is a compatibility check, not proof that the quantities are identical. Likewise, overlapping one-standard-deviation intervals are not a universal statistical hypothesis test. Statistical conclusions require the distributions, correlations and chosen decision rule.

WORKED EXAMPLE

Avoid overstating a difference

Two independent procedures give 20.4 ± 0.2 cm and 20.7 ± 0.2 cm, where both ± terms are stated bounds. Is the second length established to be greater?

MEMBERSThe approach, step-by-step working and check are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Measurement uncertainty
Focus: What does agreement within bounds establish?
Learning goals:
- Interpret a reading with an uncertainty without inventing a probability statement.
- Explain why repeated measurements cannot remove a shared calibration offset.
- Use absolute and relative bounds consistently after unit conversion.
- Correct a known zero error and check whether a claimed discrepancy is established by the stated bounds.
Equations in this section (LaTeX):
Build the physical reasoning before choosing an equation.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

BRING THE IDEAS TOGETHER

What should stay with you

A measured central value is an estimate; uncertainty is not automatically the known error.
Resolution, repeatability and calibration answer different questions.
Independent random scatter in a mean can decrease with repetitions; a shared offset survives.
A bound is not a standard deviation, and the ± notation alone specifies no confidence level.
Correct the central value with a physical sign check; retain uncertainty in the correction.

RETRIEVE, THEN EXPLAIN

Try it without the examples

Work these out before checking. If an answer surprises you, identify which assumption or step needs another look. A correct choice is most useful when you can explain why the alternatives fail.

SELF CHECK 1

A display advances in 0.01 V steps. What is established?

Total uncertainty is exactly ±0.005 V
The display step is 0.01 V
Calibration error is zero

MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →

SELF CHECK 2

An instrument indicates 5.38 mm with a known zero offset of −0.04 mm. What is the corrected value?

Answer in mm.

MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →

SELF CHECK 3

A measured time is 8.0 s with a bound of ±0.2 s. What is its percentage bound?

Answer in %.

MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →

SELF CHECK 4

Which change directly addresses a suspected shared zero offset?

Repeat the unchanged reading many more times
Compare the instrument with an appropriate reference
Print more decimal places

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APPLY THE CHAPTER

Take the ideas into a problem

Try a complete problem next. Commit to a method before revealing a hint, then compare your reasoning with the worked solution.

COME BACK TO THE IDEA

What can you explain later?

Try these from memory at your next study session, then again a few days later. Say your reasoning aloud or sketch it before looking. Adjust the gap to what you can explain.

Next study session

A balance reads +0.30 g when empty. Explain the correction to a later reading without memorizing a sign rule.

Compare your reasoning

The balance adds 0.30 g to the true value, so subtract 0.30 g from the indicated reading. Check the rule on the empty balance: it should return zero.

Revisit the explanation →

A few days later: change the situation

You repeat a length measurement many times with the same wrongly calibrated ruler. What can improve, and what remains?

Compare your reasoning

Independent random scatter in the average can decrease. A shared calibration bias remains unless corrected; uncertainty in that correction also remains. Repetition is useful, but it cannot negotiate with the wrong scale.

Revisit the explanation →

Compare the physical reason, the assumptions and the conclusion. Your wording can differ. If one part is missing, revisit that section and try explaining it again after a break.

Back to chapter contents ↑