PhysicsCBSE Class 11 · Units and Measurement

Significant Figures and Error Propagation Class 11

Updated 17 October 2026 · Fact-checked

Significant figures are the reliably known digits of a measurement plus one uncertain digit. In a sum or difference keep the fewest decimal places; in a product or quotient keep the fewest significant figures. For errors, absolute errors add in a sum or difference, and percentage errors add in a product or quotient, with each power multiplying the percentage error of its quantity.

Leading zeros never count, and errors add: absolute errors for A ± B, percentage errors for products, with powers multiplying the percentage.

Illustration drawn for QuestionBuddy.

Understand Significant Figures and Error Propagation

A measurement is never exact, so how it is written must show how well it is known.

Counting significant figures.

All non-zero digits count (2.47 has 3).
Zeros between non-zero digits count (5006 has 4).
Leading zeros do not count; they only fix the decimal point (0.00520 has 3).
Trailing zeros count only when there is a decimal point (100.0 has 4; 100 may have 1 to 3, so write $1.00 \times 1 0^{2}$ when you mean 3).
Changing units does not change the count: $4.700$ m $= 470.0$ cm $= 4.700 \times 1 0^{- 3}$ km.

Arithmetic. In addition and subtraction the answer has as many decimal places as the term with the fewest decimal places. In multiplication and division it has as many significant figures as the factor with the fewest. Carry one extra digit in intermediate steps and round only at the end. Exact numbers (a count of 5 oscillations, the 2 in $2 π$ ) have unlimited significant figures.

Rounding. If the digit to be dropped is below 5, leave the preceding digit; if above 5, raise it by one. For a dropped 5 exactly, a common school rule raises the preceding digit only if it is odd.

Error propagation. If a quantity is measured as $a \pm Δ a$ , the absolute error is $Δ a$ and the relative error is $Δ a / a$ (times 100 for percentage). For maximum error:

$Z = A \pm B$ : $Δ Z = Δ A + Δ B$ (errors add even for a difference).
$Z = A B$ or $A / B$ : $\frac{Δ Z}{Z} = \frac{Δ A}{A} + \frac{Δ B}{B}$ .
$Z = A^{n} B^{m}$ : $\frac{Δ Z}{Z} = n \frac{Δ A}{A} + m \frac{Δ B}{B}$ (the powers appear as positive multipliers).

For repeated readings take the mean as the best value; the mean absolute error is the mean of $∣ a_{i} - \overset{a}{ˉ} ∣$ .

Significant Figures and Error Propagation: formulas and key facts

Absolute error: $Δ a_{i} = ∣ \overset{a}{ˉ} - a_{i} ∣$; Mean of all readings is the best value.
Mean absolute error: $Δ \overset{a}{ˉ} = \frac{1}{n} \sum ∣Δ a_{i} ∣$; Reported as $\overset{a}{ˉ} \pm Δ \overset{a}{ˉ}$ .
Relative / percentage error: $\frac{Δ a}{a ˉ}, \frac{Δ a}{a ˉ} \times 100%$; Dimensionless.
Sum or difference: $Δ Z = Δ A + Δ B$; Absolute errors add.
Product or quotient: $\frac{Δ Z}{Z} = \frac{Δ A}{A} + \frac{Δ B}{B}$; Relative errors add.
Power law: $\frac{Δ Z}{Z} = n \frac{Δ A}{A} + m \frac{Δ B}{B}$; For $Z = A^{n} B^{m}$ .

How to solve Significant Figures and Error Propagation questions

Separate the two jobs: first get the numerical value, then decide how many digits and what error to quote.

1Write each reading with its uncertainty and count its significant figures.
2Decide the operation type: sum or difference (decimal places) or product or quotient (significant figures).
3For errors in a formula, find the relative error of every factor and multiply by its power.
4Add the relative errors (never subtract), then convert to an absolute error with $Δ Z = Z \times$ relative error.
5Round the error to one or two significant figures and round the value to the same decimal place.
6State the final answer as $Z \pm Δ Z$ with units.

Significant Figures and Error Propagation 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.

Weighing ingredients

Suppose a kitchen scale reads 250 g for flour and 18.4 g for yeast, and a student totals them as 268.4 g.

Is quoting 268.4 g sensible? Give the correct total.

Try it first, then show the solutionSolution

The flour reading 250 g has no decimal place, so its last known digit is the grams place.
The sum can be only as precise as the least precise term: round 268.4 to the grams place.

Answer: 268 g.

Memory hook: The coarsest instrument sets the precision of the total.

A plank cut to size

Suppose a carpenter measures a plank as $(120.0 \pm 0.2)$ cm long and $(15.0 \pm 0.1)$ cm wide.

Find the area and its percentage error.

Try it first, then show the solutionSolution

A = 120.0 × 15.0 = 1800 cm².
Relative error = 0.2/120.0 + 0.1/15.0 = 0.00167 + 0.00667 = 0.00833, i.e. 0.83%.
Absolute error = 1800 × 0.00833 = 15 cm².

Answer: $(1.80 \pm 0.015) \times 1 0^{3}$ cm², about 0.83%.

Memory hook: The narrow side contributes most of the error.

Tricks to remember Significant Figures and Error Propagation

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

Sum: decimals; product: figures
Add or subtract by decimal places, multiply or divide by significant figures.
Use it when: Choosing how to round.
Errors always add
Whether you add or subtract, or multiply or divide, the uncertainty only grows.
Use it when: Error questions.
Power multiplies
For $A^{3}$ the percentage error is 3 times that of $A$ .
Use it when: Volume, g from a pendulum.
Zeros: leading never, captive always
Leading zeros never count; zeros between digits always count; trailing zeros count only after a decimal point.
Use it when: Counting figures.

Significant Figures and Error Propagation: solved examples

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

Example 1: Counting significant figures · 2 marks · Short answerBoard level

State the number of significant figures in (a) 0.00520, (b) $4.300 \times 1 0^{3}$ , (c) 100.0, (d) 5006, (e) 0.0030.

Try it first, then show the solutionSolution

(a) leading zeros do not count: 5, 2, 0 give 3. (b) all of 4, 3, 0, 0 count: 4. (1)
(c) trailing zeros after a decimal point count: 4. (d) a zero between digits counts: 4. (e) leading zeros ignored, the trailing 0 counts: 2. (1)

Answer: (a) 3, (b) 4, (c) 4, (d) 4, (e) 2.

Example 2: Adding with different precisions · 2 marks · Very short answerBoard level

Add 14.26 g, 3.5 g and 0.087 g and give the result with the right number of significant figures.

Try it first, then show the solutionSolution

Sum = 14.26 + 3.5 + 0.087 = 17.847 g. (1)
The term 3.5 has one decimal place, so round to one decimal: 17.8 g. (1)

Answer: 17.8 g.

Example 3: Density from mass and volume · 2 marks · Very short answerBoard level

The mass of a solid is 6.452 g and its volume is 3.05 cm³. Find the density with the correct number of significant figures.

Try it first, then show the solutionSolution

ρ = 6.452/3.05 = 2.1154 g/cm³. (1)
3.05 has 3 significant figures (6.452 has 4), so ρ = 2.12 g/cm³. (1)

Answer: 2.12 g/cm³.

Example 4: Error in a resistance · 3 marks · Short answerHarder board level

A potential difference of $(100 \pm 5)$ V is applied across a conductor and the current is $(10 \pm 0.2)$ A. Find R with its maximum percentage error and absolute error.

Try it first, then show the solutionSolution

R = V/I = 100/10 = 10 Ω. (1)
Relative errors: ΔV/V = 5/100 = 5%, ΔI/I = 0.2/10 = 2%; they add: 7%. (1)
ΔR = 10 × 0.07 = 0.7 Ω, so R = (10 ± 0.7) Ω. (1)

Answer: R = (10 ± 0.7) Ω, maximum error 7%.

Example 5: g from a simple pendulum · 4 marks · Long answerHOTS / competency

In an experiment with a simple pendulum, L = $(0.980 \pm 0.005)$ m and the time period is T = $(2.00 \pm 0.02)$ s. Using $g = 4 π^{2} L / T^{2}$ , find g and its maximum percentage and absolute error.

Try it first, then show the solutionSolution

g = 4π² × 0.980/(2.00)² = 38.689/4.00 = 9.672 m/s². (1)
ΔL/L = 0.005/0.980 = 0.51%. ΔT/T = 0.02/2.00 = 1%. (1)
T appears squared, so its contribution doubles: Δg/g = 0.51% + 2 × 1% = 2.51%. (1)
Δg = 9.672 × 0.0251 = 0.24 m/s², so g = (9.67 ± 0.24) m/s². (1)

Answer: g = (9.67 ± 0.24) m/s², maximum error about 2.5%.

Where students lose marks in Significant Figures and Error Propagation

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

Mistake: Subtracting percentage errors for a quotient.
Thinking a quotient behaves like a difference.
Do this: Errors add for every operation: relative errors add for × and ÷.
Usually costs: Both error marks.
Mistake: Forgetting the power multiplier, e.g. for $T^{2}$ .
Treating T as a plain factor.
Do this: Multiply its relative error by the power.
Usually costs: 1 mark.
Mistake: Quoting 17.847 g after adding 3.5 g.
Keeping all digits from the calculator.
Do this: Round to the decimal places of the least precise term.
Usually costs: The final-answer mark.
Mistake: Counting leading zeros as significant.
Reading 0.0030 as 5 figures.
Do this: Leading zeros only locate the decimal point.
Usually costs: 1 mark.

Exam tips

Write the operation rule (decimal places or significant figures) in words; it is a separate mark.
Quote the value and the error to the same decimal place.
For maximum-error questions write the formula for Z first, then each relative error, then the sum.
Exact numbers such as the 2 in 2π never limit the significant figures.
Do not round in the middle of a calculation; round once at the end.

Practice questions on Significant Figures and Error Propagation

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 number of significant figures in 0.0450 is
(a) 2 (b) 3 (c) 4 (d) 5
Hint
Leading zeros do not count; the trailing 0 after the decimal point does.
Answer
(b) 3.
Q2 · 3 marks · Short answerHarder board level
A cube has edge (2.50 ± 0.02) cm. Find its volume with the maximum percentage error and write the result with the correct number of significant figures.
Hint
V = a³, so the error multiplies by 3.
Answer
V = 15.625 cm³; error = 3 × (0.02/2.50) = 3 × 0.8% = 2.4%; ΔV = 0.375 cm³, so V = (15.6 ± 0.4) cm³.
Q3 · 1 mark · Assertion-reasonHarder board level
Assertion (A): 4.700 m and 470.0 cm have the same number of significant figures.
Reason (R): The number of significant figures depends on the least count of the instrument and not on the unit chosen.
(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true but R is not the correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
Hint
Both have four figures.
Answer
(a).
Q4 · 3 marks · Short answerHarder board level
The length and breadth of a plate are $(12.5 \pm 0.1)$ cm and $(8.0 \pm 0.1)$ cm. Find its area with the maximum percentage error.
Hint
Add relative errors.
Answer
A = 12.5 × 8.0 = 100 cm²; error = 0.1/12.5 + 0.1/8.0 = 0.8% + 1.25% = 2.05%, so about 2.1%; ΔA ≈ 2 cm², A = (100 ± 2) cm².
Q5 · 4 marks · Long answerHOTS / competency
Five readings of a length are 5.23, 5.25, 5.21, 5.24 and 5.22 cm. Find the mean, the mean absolute error and the percentage error, and write the result.
Hint
Mean first, then |mean − reading|.
Answer
Mean = 26.15/5 = 5.23 cm. Absolute errors 0.00, 0.02, 0.02, 0.01, 0.01; mean = 0.06/5 = 0.012 cm. Percentage error = 0.012/5.23 × 100 = 0.23%. Result: (5.23 ± 0.01) cm.

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.Rule for sums?Show answer
Fewest decimal places.
2.Rule for products?Show answer
Fewest significant figures.
3.Error in A − B?Show answer
ΔA + ΔB.
4.Error in A²B?Show answer
2ΔA/A + ΔB/B.
5.Significant figures in 100.0?Show answer
Four.
6.Best value of repeated readings?Show answer
Their arithmetic mean.

Significant Figures and Error Propagation: frequently asked questions

Why do errors add even when we subtract?

Each reading could be off in the worst direction, so the maximum possible error in the difference is the sum of both errors.

Do zeros after a decimal point count?

Yes. Trailing zeros after a decimal point show that those digits were measured, so 2.50 has 3 significant figures.

Is the mean absolute error the same as the standard deviation?

No. At this level use the mean absolute error; the standard deviation is not required.

When should I round off?

Only at the end; keep one extra digit in intermediate steps.

What is the least count?

The smallest reading an instrument can measure; the error in a single reading is taken as the least count.