Senior 2 practice book · Unit 1 of 10

Errors in Measurement of Physical Quantities

50 questions that complete the Senior 2 quiz for this unit: 24 core and 26 challenge. Easy and hard questions are mixed. Try each question, then tap “Show answer”.

Common misconceptions

Taking more readings and averaging removes every kind of error.Averaging only reduces random errors. A systematic error such as a zero error is in every reading, so it stays in the mean and must be found and corrected.
A reading with more decimal places, for example from a digital display, must be more accurate.Extra digits show precision (fineness of the scale), not accuracy. A badly calibrated digital balance can show 0.01 g steps and still be wrong by 2 g.
Zeros in a number are never significant.Only the leading zeros (as in 0.0045) are not significant. Zeros between digits (2006) and trailing zeros after a decimal point (3.20) are significant.
When two measured lengths are subtracted, their uncertainties are subtracted too.Uncertainties never cancel: for a sum or a difference the absolute uncertainties are added, so a small difference of two large readings can have a large percentage uncertainty.
The vernier division that lines up is added to the main-scale reading as it is (6 divisions = 0.6 cm).Each vernier division counts as one least count (0.01 cm on a normal calliper), so 6 divisions add 0.06 cm.

What this unit covers

Topics marked new are not tested much in the quiz, so this book gives them extra questions.

Types of error: systematic, random, zero and parallax errors
Reading a vernier calliper and a micrometer screw gauge (least count, scale readings)
Zero error and how to correct readings for it
Accuracy and precision
Significant figures and rounding, including results of calculations
Standard form and SI prefixes when recording measurements
Absolute, relative and percentage error
Uncertainty of a single reading and choosing a suitable instrument
Combining uncertainties in sums, differences, products and powers
Spread of repeated readings: mean absolute deviation as the uncertainty
Ways of reducing errors: repeating, timing many swings, measuring stacks
Errors and graphs: best-fit line, gradient, intercept caused by a zero error

Go to the questions

Questions (1–50)

Easier and harder questions are mixed, just like in a real exam. The 26 harder ones are marked ★ Challenge.

1True or false
A digital instrument that shows more decimal places is always more accurate.
Show answer
Answer: False
More decimal places means finer precision; accuracy also needs correct calibration.
Common mistakeLearners trust digital displays; a badly calibrated digital scale can be precise and still wrong.
2True or false · ★ Challenge
When a measured quantity is halved (divided by the exact number 2), its percentage uncertainty is also halved.
Show answer
Answer: False
Dividing by an exact number halves the value AND its absolute uncertainty, so the percentage uncertainty stays the same.
Common mistakeLearners confuse absolute and percentage uncertainty; only the absolute value is halved.
3True or false · ★ Challenge
If the mean absolute deviation of repeated readings is much larger than half the smallest scale division, the spread comes mainly from the method (random errors), not from the scale of the instrument.
Show answer
Answer: True
The scale alone could only cause a spread of about half a division; a bigger spread must come from random effects such as reaction time or a moving object.
Common mistakeLearners often blame the instrument's scale for all spread; buying a finer instrument will not help if the method causes most of the scatter.
4True or false
An instrument that was wrongly calibrated in the factory gives a systematic error.
Show answer
Answer: True
A calibration error shifts or stretches every reading in the same way, so it is systematic.
Common mistakeSome learners think only the user causes errors; instruments themselves can carry systematic errors.
5True or false
The small inside jaws of a vernier calliper are used to measure the internal diameter of a pipe.
Show answer
Answer: True
The inside (upper) jaws open outwards against the inner wall, so they measure internal diameters; the depth rod measures the depth of holes.
Common mistakeSome learners measure an inside diameter with the outside jaws, which only fit around objects and cannot reach the inner wall.
6Multiple choice · ★ Challenge
Which of these would cause a RANDOM error rather than a systematic error?
1. AA thermometer whose scale was printed 1 °C too high
2. BA balance that reads 0.2 g with an empty pan
3. CJudging when a swinging pendulum passes its centre
4. DA metre rule that has expanded in hot sunlight
Show answer
Answer: C. Judging when a swinging pendulum passes its centre
Judging the moment a moving bob passes a point is sometimes early, sometimes late; the other three shift every reading the same way.
Common mistakeThe expanded rule is tempting because heat changes things unpredictably, but an expanded rule reads every length too short by the same fraction: systematic.
7Fill in the blank · ★ Challenge
A micrometer has a zero error of +0.02 mm. To get the true value, the zero error must be ______ every reading.
Show answer
Answer: subtracted from
A positive zero error makes every reading 0.02 mm too large, so subtract it.
Common mistakeAdding a positive zero error doubles the mistake instead of removing it.
8Fill in the blank
For a scale with marks, the uncertainty in a single reading is usually taken as half of the ______ division.
Show answer
Answer: smallest
You can judge a reading to about half the gap between the finest marks.
Common mistakeTaking a whole division (or the largest labelled division) as the uncertainty overstates it.
9Multiple choice · ★ Challenge
A digital stopwatch shows times to 0.01 s, but a person's reaction time varies by about 0.2 s. What is a sensible uncertainty for one hand-timed reading?
1. A± 0.01 s
2. B± 2 s
3. C± 0.2 s
4. D± 0.02 s
Show answer
Answer: C. ± 0.2 s
The human reaction time (about 0.2 s) is far larger than the display step, so it decides the uncertainty.
Common mistakeChoosing ± 0.01 s trusts the display; the stopwatch cannot be more precise than the person pressing it.
10Fill in the blank
The prefix micro (symbol μ) stands for a factor of ______.
Show answer
Answer: 10⁻⁶ (one millionth)
1 μm = 10⁻⁶ m, one millionth of a metre.
Common mistakeMixing up micro (10⁻⁶) with milli (10⁻³) is common because both start with 'm'.
11Multiple choice · ★ Challenge
The volume of a cube must be known to within 3 %. What is the largest percentage uncertainty allowed in the measurement of its side?
1. A1 %
2. B3 %
3. C9 %
4. D1.5 %
Show answer
Answer: A. 1 %
V = side³, so % uncertainty in V = 3 × % in side. 3 × x = 3 % gives x = 1 %.
Common mistakeChoosing 9 % multiplies instead of dividing; cubing triples the percentage uncertainty, so the side needs one third of it.
12Fill in the blank
A reading that is far away from all the others in a set of repeated readings is called an ______ reading and is left out of the mean.
Show answer
Answer: anomalous
Anomalous (odd) readings come from a mistake such as miscounting swings, so they are checked, repeated or left out.
Common mistakeSome learners include every reading in the mean; one wrong reading can shift the mean a lot.
13Multiple choice · ★ Challenge
A shop's scale is checked five times with a standard 1.000 kg mass and reads 1.020, 1.021, 1.020, 1.019 and 1.020 kg. Which action would make the scale accurate?
1. ATake more readings and average them to remove the error
2. BAdjust it so that it reads 1.000 kg for the standard mass
3. CReplace it with a scale that shows more decimal places
4. DAlways read the display from directly in front of it
Show answer
Answer: B. Adjust it so that it reads 1.000 kg for the standard mass
The readings are precise (close together) but all about 0.020 kg too high: a systematic error, which is removed by recalibrating (adjusting the zero).
Common mistakeAveraging is tempting, but the mean of these readings is still 1.020 kg; averaging cannot remove a systematic error.
14Multiple choice
Which of these measurements has the smallest percentage uncertainty?
1. A(2.0 ± 0.1) m
2. B(40 ± 1) cm
3. C(100 ± 1) g
4. D(0.50 ± 0.01) mm
Show answer
Answer: C. (100 ± 1) g
Percentage uncertainties: A 5 %, B 2.5 %, C 1 % and D 2 %, so (100 ± 1) g is the smallest.
Common mistakeChoosing (0.50 ± 0.01) mm because ± 0.01 is the smallest absolute uncertainty ignores the size of the reading; divide by the value first.
15Multiple choice · ★ Challenge
A copper wire has a diameter of 0.45 mm. Its cross-sectional area A = πd²/4, in m², is about:
1. A1.6 × 10⁻¹ m²
2. B6.4 × 10⁻⁷ m²
3. C1.6 × 10⁻⁴ m²
4. D1.6 × 10⁻⁷ m²
Show answer
Answer: D. 1.6 × 10⁻⁷ m²
d = 0.45 mm = 4.5 × 10⁻⁴ m; A = π × (4.5 × 10⁻⁴)² ÷ 4 = 1.6 × 10⁻⁷ m².
Common mistakeChoosing 1.6 × 10⁻¹ m² uses d in mm (0.16 mm²) and calls it m²; convert d to metres BEFORE squaring.
16Fill in the blank
When measurements are multiplied or divided, the answer should have the same number of significant figures as the measurement with the ______ significant figures.
Show answer
Answer: fewest
The least precise measurement limits the precision of the result.
Common mistakeLearners often copy all the calculator digits, which claims more precision than the measurements have.
17Multiple choice · ★ Challenge
A thermometer reads 102 °C in pure water boiling at sea level (true value 100 °C). What is its percentage error?
1. A2.0 %
2. B0.02 %
3. C20 %
4. D98 %
Show answer
Answer: A. 2.0 %
Absolute error = 102 − 100 = 2 °C; percentage error = 2 ÷ 100 × 100 % = 2 %.
Common mistakeChoosing 0.02 % forgets to multiply the relative error (0.02) by 100 %.
18Multiple choice
The same pencil is measured with four different instruments. Which record shows the most precise instrument was used?
1. A15 cm
2. B15.0 cm
3. C1.5 × 10¹ cm
4. D15.00 cm
Show answer
Answer: D. 15.00 cm
15.00 cm has four significant figures, showing the instrument reads to 0.01 cm.
Common mistakeChoosing 1.5 × 10¹ cm is tempting because standard form looks scientific, but it has only 2 significant figures, the same as 15 cm.
19True or false · ★ Challenge
A zero error in an instrument changes the gradient of a straight-line graph drawn from its readings.
Show answer
Answer: False
A zero error adds the same amount to every reading, so all points move up (or down) equally: the intercept changes, the gradient does not.
Common mistakeLearners often think any error spoils the gradient; a constant shift leaves the slope unchanged.
20Multiple choice
A learner timing a race always presses STOP a little late, because he waits to see the runner clearly cross the line. This error is:
1. ASystematic, since every time is too long
2. BRandom, since it comes from his reactions
3. CA zero error of the stopwatch itself
4. DA parallax error from his eye position
Show answer
Answer: A. Systematic, since every time is too long
Pressing late every time makes all times too long by a similar amount, which is a systematic error.
Common mistakeChoosing random is tempting because reaction time is involved, but an error that always goes the same way is systematic.
21Short answer · ★ Challenge
Using an example from a school laboratory, explain how a set of readings can be precise but not accurate.
Show answer
Model answer: Example: a thermometer with a scale printed 2 °C too high reads 102.1, 102.0, 102.1 °C in boiling water at sea level. The readings agree closely (precise) but are all far from the true 100 °C (not accurate) because of a systematic error.
Common mistakeA common mistake is to say 'precise' means 'correct'; precision is about agreement between readings, accuracy is about closeness to the true value.
22True or false
1 cm³ is equal to 10⁻⁶ m³.
Show answer
Answer: True
1 cm = 10⁻² m, so 1 cm³ = (10⁻²)³ m³ = 10⁻⁶ m³.
Common mistakeLearners often write 1 cm³ = 10⁻² m³, converting the length once instead of cubing the conversion factor.
23Multiple choice · ★ Challenge
A learner times 10 swings of a pendulum four times: 15.2 s, 15.6 s, 15.4 s and 15.0 s. The mean is 15.3 s. What is the mean absolute deviation?
1. A0.2 s
2. B0.6 s
3. C0.3 s
4. D0.8 s
Show answer
Answer: A. 0.2 s
Deviations from 15.3 s: 0.1, 0.3, 0.1, 0.3 s; mean = 0.8 ÷ 4 = 0.2 s.
Common mistakeChoosing 0.6 s gives the range (15.6 − 15.0); 0.8 s is the sum of the deviations before dividing by the number of readings.
24Multiple choice
A sheet of exercise-book paper is 0.1 mm thick. What is this thickness in metres, in standard form?
1. A1 × 10⁻³ m
2. B1 × 10⁻⁴ m
3. C1 × 10⁻⁵ m
4. D1 × 10⁴ m
Show answer
Answer: B. 1 × 10⁻⁴ m
0.1 mm = 0.1 × 10⁻³ m = 1 × 10⁻⁴ m.
Common mistakeChoosing 1 × 10⁻³ m converts mm to m correctly but forgets that the value is 0.1 mm, not 1 mm.
25Multiple choice · ★ Challenge
A tap fills a (20.0 ± 0.2) L bucket in (40 ± 1) s. What is the percentage uncertainty in the flow rate (volume ÷ time)?
1. A1.5 %
2. B3.5 %
3. C2.5 %
4. D1.2 %
Show answer
Answer: B. 3.5 %
Volume: 0.2 ÷ 20.0 = 1 %; time: 1 ÷ 40 = 2.5 %. For a quotient the percentages add: 1 + 2.5 = 3.5 %.
Common mistakeChoosing 1.5 % subtracts the percentages because the quantities are divided; uncertainties always add.
26Fill in the blank
One full turn of a micrometer thimble moves the spindle 0.5 mm. The thimble has 50 equal divisions, so each division is worth ______ mm.
Show answer
Answer: 0.01
Value of one division = 0.5 mm ÷ 50 = 0.01 mm.
Common mistakeWriting 0.1 mm comes from dividing 0.5 by 5 instead of 50; count all the thimble divisions in one turn.
27Multiple choice · ★ Challenge
With its jaws closed, the zero of a calliper's vernier scale lies 3 divisions (0.03 cm) to the LEFT of the main-scale zero. A pipe then gives a reading of 2.45 cm. What is its true diameter?
1. A2.42 cm
2. B2.45 cm
3. C2.51 cm
4. D2.48 cm
Show answer
Answer: D. 2.48 cm
A vernier zero to the left is a negative zero error, −0.03 cm. True = reading − zero error = 2.45 − (−0.03) = 2.48 cm.
Common mistakeChoosing 2.42 cm subtracts 0.03 cm as if the error were positive; when the vernier zero is to the LEFT the reading is too small, so you add.
28Short answer · ★ Challenge
A micrometer screw gauge has a ratchet on the end of its thimble. Explain what the ratchet is for, and what goes wrong if a learner closes the jaws on a soft rubber tube by turning the thimble itself.
Show answer
Model answer: The ratchet slips once the jaws press on the object with a small, fixed force, so every reading is taken with the same gentle grip. Turning the thimble directly can squeeze the soft tube, so the reading is too small (and the screw can be damaged).
Common mistakeMany learners think the ratchet is only for turning faster; its job is to stop the object being over-tightened, which would give a reading that is too small.
29Multiple choice
A market trader measures a piece of cloth as 2.50 m long and 1.2 m wide. What area should she record?
1. A3.00 m²
2. B3.0 m²
3. C3 m²
4. D3.000 m²
Show answer
Answer: B. 3.0 m²
2.50 × 1.2 = 3.0 m²; the width has only 2 significant figures, so the answer has 2.
Common mistakeChoosing 3.00 m² follows the 3 significant figures of the length; the LEAST precise measurement (1.2 m) decides.
30Multiple choice · ★ Challenge
On a vernier calliper (least count 0.01 cm), the vernier zero lies just past the 2.3 cm mark of the main scale, and the 6th vernier line is the one that lines up exactly with a main-scale line. What is the reading?
1. A2.36 cm
2. B2.30 cm
3. C2.96 cm
4. D2.306 cm
Show answer
Answer: A. 2.36 cm
Reading = main scale + (coinciding division × least count) = 2.3 + 6 × 0.01 = 2.36 cm.
Common mistakeChoosing 2.96 cm comes from adding 6 × 0.1 cm; each vernier division is worth only one least count, 0.01 cm.
31Multiple choice
A Kigali radio station broadcasts at 96.1 MHz. What is this frequency in hertz?
1. A9.61 × 10⁴ Hz
2. B9.61 × 10⁷ Hz
3. C9.61 × 10⁸ Hz
4. D96.1 × 10³ Hz
Show answer
Answer: B. 9.61 × 10⁷ Hz
Mega (M) = 10⁶, so 96.1 MHz = 96.1 × 10⁶ Hz = 9.61 × 10⁷ Hz.
Common mistakeChoosing 96.1 × 10³ Hz treats mega as kilo (10³); mega means a million (10⁶).
32Multiple choice
A learner must measure out 25 cm³ of water with an uncertainty below 1 %. Which piece of apparatus is suitable?
1. AA 100 cm³ measuring cylinder, marks every 1 cm³
2. BA 250 cm³ beaker, marks every 50 cm³
3. CA 50 cm³ cylinder, marks every 1 cm³
4. DA 25 cm³ pipette, accurate to ± 0.06 cm³
Show answer
Answer: D. A 25 cm³ pipette, accurate to ± 0.06 cm³
Pipette: 0.06 ÷ 25 × 100 % = 0.24 %. The cylinders give ± 0.5 cm³ = 2 %, and the beaker much more.
Common mistakeThe 50 cm³ cylinder is tempting because it is small, but ± 0.5 cm³ in 25 cm³ is 2 %, double the limit.
33Short answer · ★ Challenge
A learner finds the period of a pendulum by timing ONE swing, once, using the second hand of a wall clock. Suggest three improvements and say why each helps.
Show answer
Model answer: Use a stopwatch, which reads more finely than a clock's second hand. Time 20 swings and divide by 20, so the reaction-time error is shared over 20 swings. Repeat the timing and average to reduce random errors (and use a marker at the centre of the swing to start and stop).
Common mistakeSaying only 'be more careful' is not an improvement; each suggestion must reduce a named error.
34Multiple choice
A vernier calliper has 10 vernier divisions that together span 9 mm of the main scale. What is its least count (precision)?
1. A0.9 mm
2. B1 mm
3. C0.1 mm
4. D0.01 mm
Show answer
Answer: C. 0.1 mm
One main division is 1 mm and one vernier division is 9 ÷ 10 = 0.9 mm, so the least count = 1 − 0.9 = 0.1 mm.
Common mistakeChoosing 0.9 mm gives the length of one vernier division; the least count is the DIFFERENCE between a main-scale division and a vernier division.
35Multiple choice · ★ Challenge
A learner checks a balance with known masses and plots balance reading (y) against known mass (x). The straight line has gradient 1 but cuts the y-axis at 0.4 g. The most likely cause is:
1. AA random error of about 0.4 g in each of the readings
2. BA parallax error when reading the scale
3. CA zero error: it reads 0.4 g with an empty pan
4. DThe balance being much too precise
Show answer
Answer: C. A zero error: it reads 0.4 g with an empty pan
Every reading is 0.4 g too high, so the whole line is shifted up by 0.4 g: a systematic zero error.
Common mistakeChoosing a random error is tempting, but random errors scatter points on both sides; a fixed shift of every point is systematic.
36Short answer · ★ Challenge
A learner in Kigali finds that pure water boils at 95 °C on her thermometer. She concludes the thermometer has a systematic error because 'water boils at 100 °C'. Is her conclusion justified? Explain.
Show answer
Model answer: Not necessarily. 100 °C is the boiling point at sea-level pressure. Kigali is about 1500 m high, where air pressure is lower, so water boils at about 95 °C. She should check the thermometer in melting ice (0 °C) or against a trusted thermometer.
Common mistakeThe mistake is treating 100 °C as a fixed fact everywhere; the boiling point depends on air pressure.
37Multiple choice
A spring balance reads 0.5 N before anything is hung on it. With a stone hanging from it, it reads 3.7 N. What is the weight of the stone?
1. A3.2 N
2. B4.2 N
3. C3.7 N
4. D3.5 N
Show answer
Answer: A. 3.2 N
True value = reading − zero error = 3.7 − 0.5 = 3.2 N.
Common mistakeChoosing 4.2 N adds the zero error; a balance that already reads 0.5 N with no load makes every reading 0.5 N too big.
38Short answer · ★ Challenge
When finding the gradient of a best-fit line, why should you use two points far apart on the line rather than two plotted points that are close together?
Show answer
Model answer: Points on the line already average the readings, while plotted points carry their own random errors. A large triangle makes the reading errors of the graph small compared with Δy and Δx, so the percentage uncertainty in the gradient is smaller.
Common mistakeUsing two nearby data points is a common mistake: small Δy and Δx make the gradient very sensitive to reading errors.
39Short answer
Describe how you would check a metre rule and a spring balance for zero error before using them.
Show answer
Model answer: Metre rule: look at the end; if it is worn, measure from the 1.0 cm (or 10.0 cm) mark and subtract that value. Spring balance: hang it up with nothing attached and check the pointer is on zero; adjust it to zero or note the reading and subtract it.
Common mistakeLearners often start measuring from a worn end of the ruler; the first marks may be missing, so every length is wrong by the same amount.
40Short answer · ★ Challenge
Explain why drawing a graph of several readings is a better way to find a quantity (such as the stiffness of a spring) than calculating it from one pair of readings.
Show answer
Model answer: The best-fit line averages out the random errors of all the readings, anomalous points can be seen and ignored, and a zero error only moves the intercept, so the gradient is still correct.
Common mistakeMany learners think a graph is only for display; it is also a way of averaging readings and of separating systematic from random errors.
41Multiple choice
A learner plots the extension of a spring against the load. The points scatter slightly about a straight line. Her best-fit line should:
1. AJoin each plotted point to the next one with short straight lines
2. BPass exactly through the first point and the last point only
3. CAlways be forced to go through the origin, whatever the data show
4. DPass close to all points, with about as many points above it as below
Show answer
Answer: D. Pass close to all points, with about as many points above it as below
A best-fit line averages out random scatter, so points are spread evenly on both sides of it.
Common mistakeJoining dot to dot (the tempting choice) follows every random error instead of averaging them out.
42Short answer · ★ Challenge
A learner finds the speed of a moto from 125 m ÷ 9.0 s and writes 13.888 m/s. Correct the answer and explain.
Show answer
Model answer: 125 ÷ 9.0 = 13.888… m/s. The time has only 2 significant figures, so the speed should be given to 2 significant figures: 14 m/s.
Common mistakeCopying the calculator display (13.888) claims a precision of 0.001 m/s that the 9.0 s timing cannot give.
43Multiple choice
A kitchen balance reads to the nearest 1 g. What is the best way to find the mass of one grain of rice?
1. AWeigh one grain ten times, then average all of the ten readings
2. BWeigh one grain once, reading the scale very carefully
3. CWeigh 1000 grains together and divide the total mass by 1000
4. DWeigh one grain with the balance's zero error removed
Show answer
Answer: C. Weigh 1000 grains together and divide the total mass by 1000
One grain (about 0.02 g) is far below the 1 g scale step; 1000 grains (about 20 g) can be read with a small percentage uncertainty.
Common mistakeRepeating a reading of one grain is tempting, but every reading is 0 g; averaging cannot reveal a mass below the scale step.
44Short answer · ★ Challenge
A car's speedometer shows 54 km/h when the true speed, measured by a police radar, is 50 km/h. Find the absolute error and the percentage error. Is this error random or systematic, and why does it matter for the driver?
Show answer
Model answer: Absolute error = 54 − 50 = 4 km/h. Percentage error = 4 ÷ 50 × 100 % = 8 %. It is systematic (the speedometer always over-reads), so the driver is actually slower than he thinks; if it under-read, he could break the speed limit without knowing.
Common mistakeDividing by the wrong value (4 ÷ 54) or forgetting × 100 % are the usual slips; percentage error is compared with the true value.
45Multiple choice
On a micrometer screw gauge, the sleeve scale shows 4.5 mm and line 37 of the thimble (divisions of 0.01 mm) is on the datum line. What is the reading?
1. A4.537 mm
2. B8.2 mm
3. C4.37 mm
4. D4.87 mm
Show answer
Answer: D. 4.87 mm
Reading = sleeve + thimble = 4.5 mm + 37 × 0.01 mm = 4.5 + 0.37 = 4.87 mm.
Common mistakeChoosing 4.37 mm forgets the half-millimetre mark on the sleeve; 4.537 mm treats the thimble divisions as 0.001 mm.
46Multiple choice · ★ Challenge
Two points read from a best-fit line are (2.0 s, 3.0 m) and (6.0 s, 11.0 m). What is the gradient of the line?
1. A1.8 m/s
2. B2.0 m/s
3. C1.5 m/s
4. D0.5 m/s
Show answer
Answer: B. 2.0 m/s
Gradient = Δy ÷ Δx = (11.0 − 3.0) ÷ (6.0 − 2.0) = 8.0 ÷ 4.0 = 2.0 m/s.
Common mistakeChoosing 1.8 m/s divides one y-value by one x-value (11 ÷ 6); a gradient needs the CHANGES in y and x.
47Multiple choice
Half the range of repeated readings can be used as a quick uncertainty. Four readings of a length are 4.8 cm, 5.2 cm, 5.0 cm and 4.9 cm. What uncertainty does this give?
1. A± 0.4 cm
2. B± 0.1 cm
3. C± 0.2 cm
4. D± 0.05 cm
Show answer
Answer: C. ± 0.2 cm
Range = 5.2 − 4.8 = 0.4 cm; half the range = ± 0.2 cm.
Common mistakeChoosing ± 0.4 cm uses the whole range; the readings lie within ± half the range of the middle value.
48Short answer · ★ Challenge
A learner writes: 'Length (6.0 ± 0.1) cm, width (3.0 ± 0.1) cm, so the area is 18 ± 0.01 cm², because I multiplied the uncertainties.' Find her mistake and give the correct result.
Show answer
Model answer: Uncertainties are not multiplied. Percentages add: 0.1 ÷ 6.0 = 1.7 % and 0.1 ÷ 3.0 = 3.3 %, total 5 %. 5 % of 18 cm² = 0.9 cm², so area = (18.0 ± 0.9) cm².
Common mistakeMultiplying 0.1 × 0.1 makes the uncertainty tiny; combining measurements can only make the uncertainty larger, never smaller.
49Short answer
A learner records the length of the school football pitch as 100 000 mm, measured to the nearest metre. Rewrite it in metres in standard form and explain why the standard-form value is clearer.
Show answer
Model answer: 100 000 mm = 100 m = 1.00 × 10² m. Standard form shows exactly three significant figures (to the nearest metre), while 100 000 mm hides how many of the zeros are significant.
Common mistakeA common slip is to write 1 × 10⁵ m, forgetting to divide by 1000 when changing mm to m.
50Short answer · ★ Challenge
Five learners measure the width of a classroom door: 82.4 cm, 82.6 cm, 82.5 cm, 82.3 cm and 82.7 cm. Find the mean and the mean absolute deviation, and write the result as mean ± deviation.
Show answer
Model answer: Mean = 412.5 ÷ 5 = 82.5 cm. Deviations: 0.1, 0.1, 0.0, 0.2, 0.2 cm; mean deviation = 0.6 ÷ 5 = 0.12 cm ≈ 0.1 cm. Width = (82.5 ± 0.1) cm.
Common mistakeLearners often add the deviations with their signs (+ and −); they then cancel to zero. Use the size of each deviation.