PhysicsISC Class 11 · Units and Measurements

Combination of Errors: Absolute, Relative and Percentage Error in Results

Updated 17 October 2026 · Fact-checked

When quantities with errors are added or subtracted, their absolute errors add. When they are multiplied or divided, their relative (fractional) errors add. For a power Aⁿ, the relative error is n times the relative error in A. The result is the maximum possible error.

Orange strips are the errors: for A ± B absolute errors add, for A × B relative errors add, and for Aⁿ the relative error is multiplied by n.

Illustration drawn for QuestionBuddy.

Understand Combination of Errors

Types of error. The absolute error in a reading is |a − mean|. The mean absolute error Δa is the average of the absolute errors of all readings. The relative (fractional) error is Δa/a, and the percentage error is (Δa/a) × 100.

Sum and difference. If Z = A ± B, then ΔZ = ΔA + ΔB. The errors add even for a difference, because in the worst case the two errors act in opposite senses on the quantities. This is why subtracting two nearly equal measurements gives a result with a very large relative error.

Product and quotient. If Z = A × B or Z = A/B, then ΔZ/Z = ΔA/A + ΔB/B. Think of it as adding percentage errors.

Powers. If Z = Aⁿ × Bᵐ / Cᵖ, then ΔZ/Z = n(ΔA/A) + m(ΔB/B) + p(ΔC/C). The index multiplies the fractional error, and every term is added, whatever the sign of the index.

Reporting. The final answer is written as (mean ± Δ) with units, and Δ is kept to one significant figure (or two at most), with the mean rounded to the same decimal place. Never report an error smaller than the least count of the instrument.

Which measurement matters most? In a formula with powers, the quantity with the largest (index × percentage error) contributes most. Improving that measurement is the best way to improve the result.

Combination of Errors: formulas and key facts

Mean absolute error: $Δ a = (∣Δ a_{1} ∣ + ∣Δ a_{2} ∣ + ... + ∣Δ a_{n} ∣) / n ∣ w h er e Δ a_{i} = a_{i} - m e an$; Mean of the readings is used as the true value.
Sum or difference: $Z = A \pm B ∣Δ Z = Δ A + Δ B$; Absolute errors add.
Product or quotient: $Z = A B or A / B ∣Δ Z / Z = Δ A / A + Δ B / B$; Relative errors add.
Power: $Z = A^{n} ∣Δ Z / Z = n \times Δ A / A$; Applies to every factor in a product of powers.

Combination of Errors in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Error in the density of a metal cylinder

Suppose a lab technician weighs a metal cylinder as (62.0 ± 0.1) g and measures its radius as (1.00 ± 0.01) cm and length as (10.0 ± 0.1) cm.

Find the density and the percentage error.

Try it first, then show the solutionSolution

Density ρ = m/(πr²l) = 62.0/(π × 1.00² × 10.0) = 62.0/31.416 = 1.97 g cm⁻³.
Relative errors: 0.1/62.0 = 0.0016; 2 × 0.01/1.00 = 0.020; 0.1/10.0 = 0.010.
Total = 0.0016 + 0.020 + 0.010 = 0.0316, that is 3.2%.
Δρ = 1.97 × 0.0316 = 0.06 g cm⁻³.

Answer: ρ = (1.97 ± 0.06) g cm⁻³, percentage error 3.2%.

Memory hook: The radius is squared, so its error counts twice and it dominates.

Tricks to remember Combination of Errors

Short memory aids. Say each one aloud once, then use it on a question the same day.

Add for + and −, add percentages for × and ÷
Errors never cancel in the maximum-error method: always add them.
Use it when: Choosing the right rule at the start of any error numerical.
Power times percentage
Multiply the index by the percentage error of that quantity before adding.
Use it when: Quantities such as r² or v², where the index is not 1.

Combination of Errors: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Error in an area · 1 mark · MCQBoard level

The length and breadth of a rectangle are (5.0 ± 0.1) cm and (3.0 ± 0.1) cm. The absolute error in its area is:
(a) 0.2 cm²
(b) 0.4 cm²
(c) 0.8 cm²
(d) 1.6 cm²

Try it first, then show the solutionSolution

A = 5.0 × 3.0 = 15 cm².
ΔA/A = 0.1/5.0 + 0.1/3.0 = 0.02 + 0.0333 = 0.0533.
ΔA = 15 × 0.0533 = 0.8 cm².

Answer: (c) 0.8 cm²

Example 2: Mean and mean absolute error · 3 marks · Short answerBoard level

Five measurements of the length of a rod are 2.63, 2.56, 2.42, 2.71 and 2.80 cm. Find the mean value, the mean absolute error and the percentage error.

Try it first, then show the solutionSolution

Sum = 13.12, mean = 13.12/5 = 2.624 cm ≈ 2.62 cm. (1 mark)
Absolute errors: 0.006, 0.064, 0.204, 0.086, 0.176; sum = 0.536. Mean absolute error = 0.536/5 = 0.1072 ≈ 0.11 cm. (1 mark)
Percentage error = (0.1072/2.624) × 100 = 4.1%. (1 mark)

Answer: Length = (2.62 ± 0.11) cm; percentage error about 4.1%.

Example 3: Density of a cylinder · 3 marks · Short answerHarder board level

A cylinder has mass (62.0 ± 0.1) g, radius (1.00 ± 0.01) cm and length (10.0 ± 0.1) cm. Find its density with the absolute error.

Try it first, then show the solutionSolution

ρ = m/(πr²l) = 62.0/(π × 1 × 10.0) = 1.9735 g cm⁻³. (1 mark)
Δρ/ρ = 0.1/62.0 + 2(0.01/1.00) + 0.1/10.0 = 0.0016 + 0.02 + 0.01 = 0.0316. (1 mark)
Δρ = 1.9735 × 0.0316 = 0.062 ≈ 0.06 g cm⁻³. (1 mark)

Answer: ρ = (1.97 ± 0.06) g cm⁻³ (3.2% error).

Example 4: g from a simple pendulum · 4 marks · Short answerHarder board level

In an experiment on a simple pendulum, L = (1.000 ± 0.001) m, and 20 oscillations take (40.0 ± 0.2) s. Find g and its percentage error using g = 4π²L/T².

Try it first, then show the solutionSolution

T = 40.0/20 = 2.00 s; ΔT = 0.2/20 = 0.01 s. (1 mark)
g = 4π² × 1.000/(2.00)² = 39.478/4.00 = 9.87 m s⁻². (1 mark)
Δg/g = ΔL/L + 2ΔT/T = 0.001 + 2(0.005) = 0.011, that is 1.1%. (1 mark)
Δg = 9.87 × 0.011 = 0.11 m s⁻², so g = (9.87 ± 0.11) m s⁻². (1 mark)

Answer: g = (9.87 ± 0.11) m s⁻²; percentage error 1.1%.

Example 5: Which measurement to improve · 5 marks · Long answerHOTS / competency

A quantity Z = A²B³/√C is calculated from A = (2.00 ± 0.02), B = (3.00 ± 0.06) and C = (4.00 ± 0.04), all in suitable units. Find Z, its maximum percentage error and absolute error. Which of the three measurements contributes most to the error?

Try it first, then show the solutionSolution

Z = (2.00)² × (3.00)³ / √4.00 = 4 × 27/2 = 54. (1 mark)
Percentage errors: A: 0.02/2.00 = 1%; B: 0.06/3.00 = 2%; C: 0.04/4.00 = 1%. (1 mark)
Contributions: A: 2 × 1 = 2%; B: 3 × 2 = 6%; C: ½ × 1 = 0.5%. (1 mark)
Maximum percentage error = 2 + 6 + 0.5 = 8.5%; ΔZ = 54 × 0.085 = 4.59 ≈ 4.6. (1 mark)
B contributes most (6 of the 8.5%), so improving B is the best use of effort. (1 mark)

Answer: Z = 54 ± 4.6 (8.5%); the error in B dominates.

Where students lose marks in Combination of Errors

The slips examiners see most often, and what to write instead.

Mistake: Subtracting errors in a difference
Students write ΔZ = ΔA − ΔB for Z = A − B.
Do this: Errors always add: ΔZ = ΔA + ΔB.
Usually costs: 1 to 2 marks.
Mistake: Ignoring the index
In ρ = m/(πr²l) the error in r is written once instead of twice.
Do this: Multiply the relative error of r by 2 before adding.
Usually costs: 1 mark.
Mistake: Reporting too many digits
An answer such as 9.8696 ± 0.1086 suggests more accuracy than the data have.
Do this: Round the error to one significant figure and match the decimal place of the mean.
Usually costs: 1 mark.

Exam tips

Write the rule you are using (add absolute or add relative) before substituting.
Show each relative error separately, then add them.
State the result as (value ± error) with unit, and give the percentage error when asked.
For a difference of nearly equal numbers, comment that the percentage error becomes large.

Practice questions on Combination of Errors

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
The percentage error in measuring the side of a cube is 2%. The percentage error in its volume is:
(a) 2%
(b) 4%
(c) 6%
(d) 8%
Hint
V = a³, so the index is 3.
Answer
(c) 6%
Q2 · 2 marks · Short answerBoard level
A resistance is found from R = V/I. V = (100 ± 5) V and I = (10 ± 0.2) A. Find R with the absolute error.
Hint
Add the relative errors.
Answer
R = 10 Ω. ΔR/R = 5/100 + 0.2/10 = 0.05 + 0.02 = 0.07, so ΔR = 0.7 Ω. R = (10.0 ± 0.7) Ω.
Q3 · 3 marks · Short answerHarder board level
Two lengths are (10.5 ± 0.1) cm and (26.6 ± 0.1) cm. Find their sum and difference with errors, and say which has the larger percentage error.
Hint
Absolute errors add in both cases.
Answer
Sum = 37.1 ± 0.2 cm (0.54%). Difference = 16.1 ± 0.2 cm (1.2%). The difference has the larger percentage error.
Q4 · 3 marks · Short answerHarder board level
The mass of a body is (2.0 ± 0.1) kg and its speed is (10 ± 0.2) m s⁻¹. Find the kinetic energy and its percentage error.
Hint
K = ½mv²: the speed is squared.
Answer
K = ½ × 2.0 × 100 = 100 J. Percentage error = 0.1/2.0 × 100 + 2 × (0.2/10 × 100) = 5% + 4% = 9%. K = (100 ± 9) J.
Q5 · 4 marks · Short answerHarder board level
For a convex lens, u = (30.0 ± 0.5) cm and v = (60.0 ± 1.0) cm (magnitudes). Using 1/f = 1/u + 1/v, find f and its absolute error.
Hint
Find Δ(1/f) = Δu/u² + Δv/v², then Δf = f² Δ(1/f).
Answer
f = uv/(u + v) = 1800/90 = 20.0 cm. Δ(1/f) = 0.5/900 + 1.0/3600 = 0.000556 + 0.000278 = 0.000833. Δf = 400 × 0.000833 = 0.33 cm. f = (20.0 ± 0.3) cm.
Q6 · 4 marks · Case-basedHOTS / competency
Case study: Suppose a student measures the diameter of a wire with a screw gauge (least count 0.01 mm) and records 1.20, 1.22, 1.21, 1.19 and 1.23 mm.
(i) Find the mean diameter. (1 mark)
(ii) Find the mean absolute error. (1 mark)
(iii) Find the percentage error in the cross-sectional area πd²/4. (1 mark)
(iv) Write the diameter in the correct form with the error rounded sensibly. (1 mark)
Hint
Area depends on d², so double the relative error.
Answer
(i) 6.05/5 = 1.21 mm. (ii) Deviations 0.01, 0.01, 0, 0.02, 0.02; mean = 0.06/5 = 0.012 mm. (iii) 2 × 0.012/1.21 = 0.0198, about 2.0%. (iv) d = (1.21 ± 0.01) mm.

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.Error rule for Z = A + B?Show answer
ΔZ = ΔA + ΔB.
2.Error rule for Z = A/B?Show answer
ΔZ/Z = ΔA/A + ΔB/B.
3.Percentage error in Z = A³?Show answer
Three times the percentage error in A.
4.Define relative error.Show answer
Mean absolute error divided by the mean value.

Combination of Errors: frequently asked questions

In Z = A − B, why is ΔZ not ΔA − ΔB?

The maximum-error method takes the worst case, where the two errors push the result the same way. The error in a difference is therefore the sum of the two absolute errors, never their difference.

What is the difference between random and systematic error?

Random errors vary unpredictably from reading to reading and are reduced by averaging many readings. Systematic errors push every reading the same way, as with a zero error, and are removed by correcting the instrument or the method.

How many digits should the error have?

Usually one significant figure. The mean is then rounded to the same decimal place as the error.