The shape of the lesson
By the end of the lesson, learners can
- Shift the cluster, or spread it: The bullseye is the true value.
- How uncertainty combines: Pick the rule by the operation, not the quantity.
- What a repeat can and cannot fix.
- Random vs systematic: averaging cures random scatter, not a fixed bias.
- Sum or difference: add absolute uncertainties.
- Product, quotient or power: add percentage uncertainties, times the power.
One picture every idea returns to
A systematic error shifts every reading the same way and is not reduced by repeating; a random error scatters readings and is reduced by averaging. Accuracy is closeness of the mean to the true value; precision is closeness of repeats to each other. When measurements combine, absolute uncertainties add for a sum or difference, and percentage uncertainties add for a product or quotient, multiplied by the power.
Forty-five minutes, phase by phase
| Time | Phase | What happens in the room | Grouping |
|---|---|---|---|
| 0 to 5 min | Hook: predict | Open with the question in the title, No measurement is exact.. Every reading you take is a best estimate wrapped in a band of doubt. Put the figure on the board: it is the one picture every idea in the lesson returns to. | Whole class, then pairs |
| 5 to 13 min | Teach and check: Shift the cluster, or spread it | The bullseye is the true value. Project the simulation and let a student drive it while the class predicts. | Whole class, then pairs |
| 13 to 21 min | Teach and check: How uncertainty combines | Pick the rule by the operation, not the quantity. | Pairs |
| 21 to 30 min | Teach and check: What a repeat can and cannot fix | What a repeat can and cannot fix. | Anticipation / Reaction Guide |
| 30 to 36 min | Check | The four Stage 1 questions from the student lesson, answered on mini whiteboards or by show of fingers. Re-teach the one most get wrong. | Individual, then whole class |
| 36 to 43 min | Plenary | The exam-style questions from Stage 2, one at a time: thirty seconds alone, then a random call. The feedback lines in the key give the examiner's wording. | Individual, random call |
| 43 to 45 min | Exit and homework | Set the worksheet below (6 questions, 14 marks). Learners self-assess against the objectives before they leave. | Individual |
Protect the parts that carry the learning
The sequence carries about fifty minutes of material, so the lab is driven by one student while the class predicts rather than explored freely. Times are a guide.
Protected: the check at 30 minutes. It is the only individual read on whether the key idea landed, and it is not cut.
Cut first if the build overruns: the last check in What a repeat can and cannot fix, never the check or the plenary.
In a 60 minute block: run worksheet questions 1 to 3 in class as a Rally Coach set and model both the hardest worked example and its check live.
Anticipation / Reaction Guide
Before the lesson, students mark five statements true or false. After the teaching they mark them again and justify any change of mind in one sentence each. The statements come from the examiner traps below.
The full structure, with its variations and the other twelve, is on the cooperative-learning page.
What to head off, and how
| Trap learners fall into | Teaching move that pre-empts it |
|---|---|
| Do not claim that "taking more readings improves accuracy". | More repeats reduce the effect of random error and so improve precision, but they do nothing to a systematic error, which is exactly what limits accuracy. The other classic slip is mixing the rules: add absolute uncertainties for a sum or difference, but add percentage uncertainties for a product or quotient. And for a power such as r², multiply the percentage uncertainty by the power before adding. |
Errors and uncertainties: worksheet
6 questions, 14 marks. Written in the style of the papers; no past paper question is reproduced.
Distinguish between a random error and a systematic error, and state which one is reduced by averaging repeated readings.
A micrometer reads 0.04 mm when its jaws are fully closed. The diameter of a ball bearing is then read as 8.36 mm. State the type of error and give the corrected diameter.
The resistance of a component is found from a potential difference of (6.0 ± 0.1) V and a current of (0.50 ± 0.02) A, using R = V / I. Find R and its absolute uncertainty.
The period T of a simple pendulum is used to find g from g = 4π²L / T². The length L has a 1% uncertainty and the period T has a 2% uncertainty. Determine the percentage uncertainty in g.
Two readings of a temperature are taken from a thermometer as (84.0 ± 0.5) °C and (21.0 ± 0.5) °C. Find the temperature rise and explain why its percentage uncertainty is larger than that of either reading.
A student suggests that switching from a stopwatch read by hand to a light gate would "make the experiment more accurate". Comment on whether this is the right word, and explain what actually improves.
Answer key and marking notes
- A random error scatters readings unpredictably about the true value. A systematic error shifts every reading by a fixed amount in the same direction. ✓
- Averaging repeated readings reduces the random error; it does not affect the systematic error. ✓
- This is a systematic (zero) error: every reading is 0.04 mm too high. ✓
- Corrected diameter = 8.36 − 0.04 = 8.32 mm. ✓
- R = 6.0 / 0.50 = 12 Ω. ✓
- Percentage uncertainties: V gives (0.1/6.0) = 1.7%, I gives (0.02/0.50) = 4.0%. For a quotient they add: 1.7 + 4.0 = 5.7%. ✓
- Absolute uncertainty = 5.7% of 12 = 0.68, so R = (12.0 ± 0.7) Ω. ✓
- g depends on L to the first power and on T to the power 2 (in the denominator). ✓
- %Δg = %ΔL + 2 × %ΔT = 1% + 2 × 2% = 5%. ✓
- The 4π² is an exact constant and contributes nothing. ✓
- Rise = 84.0 − 21.0 = 63.0 °C; absolute uncertainties add, so it is (63.0 ± 1.0) °C. ✓
- The absolute uncertainty has grown to 1.0 while the value (63.0) is smaller than either reading, so the percentage uncertainty (about 1.6%) exceeds that of each original reading. Subtracting close values inflates relative uncertainty. ✓
- The main gain is the removal of the human reaction-time scatter, which is a random error, so it improves precision (repeatability) of the timing. ✓
- It improves accuracy only if reaction time was also acting as a consistent bias; in general, calling it simply "more accurate" is loose. The precise statement is that the random timing uncertainty is greatly reduced. ✓
17 multiple-choice questions on this lesson
From the topic question bank. Assign them in the app for a marked score in your gradebook, or run the topic as a class game.
A balance that reads a value when nothing is on it has a:
- Agross mistake only
- Bzero error, which is systematic
- Creading error that averages out
- Drandom error
Taking many readings and averaging mainly reduces the effect of:
- Acalibration error
- Bsystematic error
- Czero error
- Drandom error
Accuracy describes how close a result is to:
- Athe true value
- Bthe mean of the data
- Cthe instrument resolution
- Dother repeated readings
A set of readings that agree closely with one another but sit far from the true value is:
- Aboth accurate and precise
- Baccurate but not precise
- Cprecise but not accurate
- Dneither
A digital voltmeter displays 2.47 V. The reading uncertainty is usually taken as:
- A±0.1 V
- B±0.01 V
- C±0.5 V
- D±0.005 V
A ruler is graduated in millimetres. The reading uncertainty in a single position is usually:
- A±0.1 mm
- B±2 mm
- C±1 mm
- D±0.5 mm
Two masses are (50 ± 1) g and (30 ± 1) g. The uncertainty in their total mass is:
- A±2 g
- B±1 g
- C±0.5 g
- D±0 g
A quantity y = ab. If a has a 3% uncertainty and b has a 2% uncertainty, the uncertainty in y is:
- A6%
- B2.5%
- C5%
- D1%
A quantity y = a³. If a has a 2% uncertainty, the uncertainty in y is:
- A8%
- B6%
- C0.67%
- D2%
A quantity y = a / b². If a has a 1% uncertainty and b has a 2% uncertainty, the uncertainty in y is:
- A3%
- B1.5%
- C9%
- D5%
A length is measured as (2.50 ± 0.05) m. Its percentage uncertainty is:
- A2%
- B5%
- C20%
- D0.5%
Which change would reduce the random error in a measurement?
- ARecalibrating against a standard
- BCorrecting a known zero error
- CTaking several readings and averaging
- DSubtracting a constant offset
A result is calculated as 3.46 with a percentage uncertainty giving an absolute uncertainty of 0.2. The best way to quote it is:
- A3.46 ± 0.20
- B3.5 ± 0.2
- C3.5 ± 0.20
- D3.46 ± 0.2
The percentage uncertainty in a measurement is defined as:
- Aabsolute uncertainty × 100%
- B(value / absolute uncertainty) × 100%
- Cvalue × absolute uncertainty
- D(absolute uncertainty / value) × 100%
Subtracting two nearly equal measured lengths tends to:
- Aincrease the percentage uncertainty of the result
- Bdecrease the percentage uncertainty
- Cleave it unchanged
- Dremove all uncertainty
Using g = 4π²L / T², L has a 2% uncertainty and T has a 1% uncertainty. The percentage uncertainty in g is:
- A3%
- B6%
- C4%
- D2%
An empty balance reads −0.2 g. With an object on the pan it reads 25.4 g. The true mass of the object is:
- A25.2 g
- B25.6 g
- C0.2 g
- D25.4 g
Quiz key
A non-zero reading with nothing on the pan biases every measurement by the same amount: a systematic zero error.
Averaging cancels the two-sided scatter of random error. Systematic errors bias every reading and survive averaging.
Accuracy is closeness to the true value. Closeness of repeats to one another is precision.
Close agreement means high precision; being off the true value means low accuracy. This signals a systematic error.
A digital scale is read to its last displayed digit, so the uncertainty is ±0.01 V.
For an analogue scale the reading uncertainty is half the smallest division, so ±0.5 mm.
Absolute uncertainties add for a sum: 1 + 1 = 2 g, giving (80 ± 2) g.
Percentage uncertainties add for a product: 3% + 2% = 5%.
For a power, multiply the percentage uncertainty by the power: 3 × 2% = 6%.
%Δy = %Δa + 2×%Δb = 1% + 2×2% = 5%. It does not matter that b is on the bottom.
(0.05 / 2.50) × 100% = 2%.
Averaging repeated readings reduces random scatter. The other options all address systematic errors.
The uncertainty is known to one decimal place, so the value is rounded to match: 3.5 ± 0.2.
Percentage uncertainty = (absolute uncertainty / measured value) × 100%.
Absolute uncertainties still add, but the difference is small, so the percentage uncertainty grows.
%Δg = %ΔL + 2×%ΔT = 2% + 2×1% = 4%. The 4π² is exact.
The balance reads 0.2 g too low, so the true mass is 25.4 + 0.2 = 25.6 g.
What to project
The student lesson embeds these tabs. Open them full screen on the projector; every slider has a drag handle on the stage.
Slides for this lesson
A projectable deck for every pack is on the way. Until then, present the student lesson full screen: its sections, figure and lab tabs follow the same order as the plan above.