JEE Mains Physics · Units and Measurements

Propagation of Errors

For a product of powers, the maximum relative error is each relative error times the size of its power, all added; for a sum or a difference, the absolute errors add.

All Units and Measurements notes JEE Mains Physics strategy

Why this matters

Twenty-seven PYQs, twenty-two of them multiple choice, and five from 2026. Fifteen give percentage errors and a formula built from powers; seven give each measurement as value ± error, so the relative errors must be worked out first; five add errors in a sum, a parallel combination or a mean of readings. Two rules cover all of them, and every error in them is added.

Concept 1 of 3: Relative error of a product of powers

If a quantity is a product of powers, a small relative error in one factor is multiplied by its power. The question asks for the MAXIMUM error, the worst case, so every contribution is added with a plus sign, even for a quantity in the denominator and even if one error is stated as negative.

Definition

$Q = \dfrac{a^{p}b^{q}}{c^{r}}$ gives $\dfrac{\Delta Q}{Q} = p\dfrac{\Delta a}{a} + q\dfrac{\Delta b}{b} + r\dfrac{\Delta c}{c}$ .
Use the size of each power: a square root counts ½, a cube root ⅓.
Multiply by 100 for a percentage error. Constants such as 4π² carry no error.
Signs of individual errors do not reduce the maximum error: a 2% positive error in T and a 1% negative error in m still give $1 + 2 \times 2 = 5\%$ in $k = 4\pi^{2}m/T^{2}$ .
A diameter used for an area or a volume enters with power 2 or 3; d and r have the same relative error.
Exponential factor $e^{-\beta t}$ : it adds the ABSOLUTE amount $\beta\,\Delta t$ to the relative error, not a percentage of t.

Maximum relative error

Q = \frac{a^{p}b^{q}}{c^{r}} \Rightarrow \frac{\Delta Q}{Q} = p\frac{\Delta a}{a} + q\frac{\Delta b}{b} + r\frac{\Delta c}{c}

Worked example

Q = \dfrac{A^{3}B^{2}}{\sqrt{C}}

. The percentage errors in A, B and C are 0.5%, 1.5% and 2%. Find the maximum percentage error in Q.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2023 · 10 April 2023 · Q1Moderate

Example 1 · Units and Measurements · Propagation of Errors

A physical quantity

P

is given as

P =\frac{a^{2}b^{3}}{c\sqrt{d}}

The percentage error in the measurement of a, b, c and

d

are

1\%,2\%,3\%

and

4\%

respectively. The percentage error in the measurement of quantity

P

will be

An error in the denominator is added, not subtracted

Dividing by c does not cancel c's error. The maximum relative error of a/c is Δa/a + Δc/c. Subtracting gives a smaller number, which is usually one of the options.

A stated negative error still adds

For the maximum error, a '1% negative error' counts as 1%. Signs matter only for the most likely error, which JEE questions do not ask for.

An exponential adds βΔt, not a percentage

In E = α³e^(−βt), the exponential contributes β·Δt to ΔE/E. With β = 0.2 s⁻¹ and Δt = 0.5 s, that is 0.1, or 10%, whatever the value of t.

Drill 14 more on relative error of a product of powers

Concept 2 of 3: Percentage error from values given with their absolute errors

Often the question gives each measurement as value ± absolute error. Turn each into a relative error, Δx/x, then apply the power rule. If the answer is wanted as value ± error, multiply the relative error back by the value.

Definition

Relative error $= \dfrac{\Delta x}{x}$ ; percentage error $= \dfrac{\Delta x}{x} \times 100$ .
Apply the power rule to the relative errors.
Absolute error of the result: $\Delta Q = Q \times \dfrac{\Delta Q}{Q}$ .
Density of a cylinder $\rho = \dfrac{4m}{\pi d^{2}l}$ : $\dfrac{\Delta m}{m} + 2\dfrac{\Delta d}{d} + \dfrac{\Delta l}{l}$ . Of a sphere $\rho = \dfrac{3m}{4\pi r^{3}}$ : $\dfrac{\Delta m}{m} + 3\dfrac{\Delta r}{r}$ .
Keep units consistent within each ratio; the ratio itself has no unit.

Absolute error from relative error

\Delta Q = Q\left(p\frac{\Delta a}{a} + q\frac{\Delta b}{b} + r\frac{\Delta c}{c}\right)

Worked example

A sphere has mass

(25.0 \pm 0.1)

g and radius

(2.00 \pm 0.01)

cm. Find the maximum percentage error in its density.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2026 · 6 Apr 2026 Shift 1 · Q1Moderate

Example 2 · Units and Measurements · Propagation of Errors

The density

\rho

of a uniform cylinder is determined by measuring its mass m , length

l

and diameter d . The measured values of

m,l

and d are

97.42 \pm 0.02\text{ }g,8.35 \pm 0.05\text{ }mm

and

20.20 \pm 0.02\text{ }mm

, respectively. Calculated percentage fractional error in

\rho

\_\_\_\_

A diameter's error counts twice in an area

Area is πd²/4, so its relative error is 2Δd/d. Using Δd/d once, as if the area grew in proportion to d, halves that contribution.

Drill 6 more on percentage error from values given with their absolute errors

Concept 3 of 3: Errors in sums, parallel combinations and means

When quantities are added or subtracted, their absolute errors add: the worst case is every reading off in the same direction. Only after the sum is formed do you divide by it to get a percentage. A parallel combination is a sum of reciprocals, so it needs one more step.

Definition

$Z = A + B$ or $A - B$ : $\Delta Z = \Delta A + \Delta B$ .
Springs in parallel or resistors in series: $k = k_1 + k_2$ , so $\Delta k = \Delta k_1 + \Delta k_2$ .
Resistors in parallel, $\dfrac{1}{R} = \dfrac{1}{R_1} + \dfrac{1}{R_2}$ : $\dfrac{\Delta R}{R^{2}} = \dfrac{\Delta R_1}{R_1^{2}} + \dfrac{\Delta R_2}{R_2^{2}}$ .
Mean absolute error: $\overline{\Delta x} = \dfrac{1}{n}\sum|x_i - \bar x|$ ; relative error $= \overline{\Delta x}/\bar x$ .

Sums and reciprocal sums

\Delta(A \pm B) = \Delta A + \Delta B \qquad \frac{\Delta R}{R^{2}} = \frac{\Delta R_1}{R_1^{2}} + \frac{\Delta R_2}{R_2^{2}}

Worked example

Two resistors are

R_1 = (6.0 \pm 0.2)\ \Omega

and

R_2 = (12.0 \pm 0.3)\ \Omega

. Find the equivalent resistance with its error (a) in series and (b) in parallel.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2026 · 21 Jan 2026 Shift 1 · Q5Moderate

Example 3 · Units and Measurements · Propagation of Errors

In an experiment the values of two spring constants were measured as

k_{1}= (10 \pm 0.2)N/m

and

k_{2}= (20 \pm 0.3)N/m

. If these springs are connected in parallel, then the percentage error in equivalent spring constant is :

Errors add in a difference too

For Z = A − B, ΔZ = ΔA + ΔB. Subtracting the errors assumes they cancel, but the maximum error comes when they do not.

Do not add percentage errors in a sum

For k = k₁ + k₂, add the ABSOLUTE errors and then divide by k. Adding the two percentage errors overstates the answer: (10 ± 0.3) + (30 ± 0.3) is 40 ± 0.6, which is 1.5%, not 3% + 1% = 4%.

Drill 4 more on errors in sums, parallel combinations and means

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

Relative error of a product of powers
Maximum relative error
$Q = \frac{a^{p}b^{q}}{c^{r}} \Rightarrow \frac{\Delta Q}{Q} = p\frac{\Delta a}{a} + q\frac{\Delta b}{b} + r\frac{\Delta c}{c}$
Percentage error from values given with their absolute errors
Absolute error from relative error
$\Delta Q = Q\left(p\frac{\Delta a}{a} + q\frac{\Delta b}{b} + r\frac{\Delta c}{c}\right)$
Errors in sums, parallel combinations and means
Sums and reciprocal sums
$\Delta(A \pm B) = \Delta A + \Delta B \qquad \frac{\Delta R}{R^{2}} = \frac{\Delta R_1}{R_1^{2}} + \frac{\Delta R_2}{R_2^{2}}$

Watch out for (6)

An error in the denominator is added, not subtracted
→ Relative error of a product of powers
A stated negative error still adds
→ Relative error of a product of powers
An exponential adds βΔt, not a percentage
→ Relative error of a product of powers
A diameter's error counts twice in an area
→ Percentage error from values given with their absolute errors
Errors add in a difference too
→ Errors in sums, parallel combinations and means
Do not add percentage errors in a sum
→ Errors in sums, parallel combinations and means

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