Units and Measurements • Topic 3 of 3

Errors & Significant Figures

No measurement is perfect. The reported value of any measured quantity carries an unavoidable uncertainty, and a trained physicist always states the result with its error. Two words are often confused here. Accuracy describes how close a measurement is to the true value; precision describes how closely repeated measurements agree with one another and is set by the resolution of the instrument. A clock that is fast by exactly the same amount every reading is precise but not accurate; a worn ruler giving scattered readings near the truth is accurate on average but not precise.

Errors are classified as systematic (consistent, one-directional — instrument zero-error, faulty calibration, personal bias) and random (unpredictable scatter from uncontrolled fluctuations). Random errors are reduced by repeating the measurement and averaging.

Given $n$ readings $a_1, a_2, \\dots, a_n$, the best estimate is the mean $a_{\\text{mean}}$. The error in each reading is its absolute error $|\\Delta a_i| = |a_{\\text{mean}} - a_i|$, and their average is the mean absolute error $\\Delta a_{\\text{mean}}$. Two derived measures matter most:

$$\\text{Relative error} = \\frac{\\Delta a_{\\text{mean}}}{a_{\\text{mean}}}, \\qquad \\text{Percentage error} = \\frac{\\Delta a_{\\text{mean}}}{a_{\\text{mean}}} \\times 100\\%$$

When measured quantities are combined, their errors propagate by simple rules:

  • Sums and differences: absolute errors add. If $Z = A \\pm B$ then $\\Delta Z = \\Delta A + \\Delta B$.
  • Products and quotients: relative errors add. If $Z = \\dfrac{AB}{C}$ then $\\dfrac{\\Delta Z}{Z} = \\dfrac{\\Delta A}{A} + \\dfrac{\\Delta B}{B} + \\dfrac{\\Delta C}{C}$.
  • Powers: the relative error is multiplied by the power. If $Z = A^p B^q / C^r$ then $\\dfrac{\\Delta Z}{Z} = p\\dfrac{\\Delta A}{A} + q\\dfrac{\\Delta B}{B} + r\\dfrac{\\Delta C}{C}$.

Significant figures are the digits in a measurement that are reliably known plus the first uncertain one. The rules: all non-zero digits count; zeros between non-zero digits count; leading zeros never count ($0.0025$ has two); trailing zeros count only when a decimal point is present ($2.500$ has four, but $2500$ is ambiguous). Scientific notation removes all ambiguity. In calculations, the result of a multiplication or division keeps as many significant figures as the least-precise factor, while addition or subtraction keeps as many decimal places as the term with the fewest.

Rounding off follows the standard rule: if the digit to be dropped is less than 5 round down, greater than 5 round up, and if it is exactly 5, round to make the preceding digit even (the "round-half-to-even" convention that avoids systematic bias).

Deeper Insight — your answer can never be more precise than your data: Students routinely copy down all eight digits a calculator displays. That is physically dishonest — if you measured a length to three significant figures, the fourth digit of any result computed from it is meaningless noise. The significant-figure rules are not arithmetic pedantry; they are how you keep your stated precision truthful. Pair them with the error rules and you can always answer the real question: not just "what is the value?" but "how much should I trust it?"

Accuracy versus precision shown as dartboardsAccuracy vs PrecisionNeitherPrecise, not accurateAccurate & preciseA measurement on a number line with its uncertainty bandReporting a value with its error(2.54 ± 0.02) cm2.522.56
Solved Examples
1
Worked Example
Four readings of a length give $2.63, 2.56, 2.42$ and $2.71\\,\\text{cm}$. Find the mean value, mean absolute error and percentage error.
Solution
  1. Mean $= \\dfrac{2.63 + 2.56 + 2.42 + 2.71}{4} = \\dfrac{10.32}{4} = 2.58\\,\\text{cm}$.
  2. Absolute errors: $|2.58-2.63|=0.05$, $|2.58-2.56|=0.02$, $|2.58-2.42|=0.16$, $|2.58-2.71|=0.13$.
  3. Mean absolute error $= \\dfrac{0.05+0.02+0.16+0.13}{4} = \\dfrac{0.36}{4} = 0.09\\,\\text{cm}$.
  4. Percentage error $= \\dfrac{0.09}{2.58}\\times 100 \\approx 3.5\\%$.

Answer: $(2.58 \\pm 0.09)\\,\\text{cm}$, percentage error $\\approx 3.5\\%$.

2
Worked Example
The sides of a rectangle are $(5.0 \\pm 0.1)\\,\\text{cm}$ and $(4.0 \\pm 0.1)\\,\\text{cm}$. Find the area and its error.
Solution
  1. Area $A = 5.0 \\times 4.0 = 20\\,\\text{cm}^2$.
  2. For products, relative errors add: $\\dfrac{\\Delta A}{A} = \\dfrac{0.1}{5.0} + \\dfrac{0.1}{4.0} = 0.02 + 0.025 = 0.045$.
  3. $\\Delta A = 0.045 \\times 20 = 0.9\\,\\text{cm}^2$.

Answer: Area $= (20 \\pm 0.9)\\,\\text{cm}^2$.

3
Worked Example
State the number of significant figures in: (a) $0.0025$, (b) $1.080$, (c) $2500$, (d) $6.02 \\times 10^{23}$.
Solution
  1. (a) Leading zeros don't count: digits $2,5$ → 2 sig. figs.
  2. (b) Zero between/after with decimal counts: $1,0,8,0$ → 4 sig. figs.
  3. (c) Trailing zeros without a decimal are ambiguous → conventionally 2 sig. figs.
  4. (d) Only the mantissa counts: $6,0,2$ → 3 sig. figs.

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

4
Worked Example
Add $3.14\\,\\text{m}$, $0.2\\,\\text{m}$ and $12.357\\,\\text{m}$ to the correct number of significant figures.
Solution
  1. Raw sum $= 3.14 + 0.2 + 12.357 = 15.697\\,\\text{m}$.
  2. For addition, keep the fewest decimal places — here $0.2$ has only one.
  3. Round $15.697$ to one decimal place.

Answer: $15.7\\,\\text{m}$.

5
Worked Example
A physical quantity is given by $P = \\dfrac{a^2 b^3}{c\\sqrt{d}}$. If the percentage errors in $a, b, c, d$ are $1\\%, 2\\%, 3\\%$ and $4\\%$, find the maximum percentage error in $P$.
Solution
  1. Apply the power rule: $\\dfrac{\\Delta P}{P} = 2\\dfrac{\\Delta a}{a} + 3\\dfrac{\\Delta b}{b} + \\dfrac{\\Delta c}{c} + \\tfrac{1}{2}\\dfrac{\\Delta d}{d}$.
  2. $= 2(1) + 3(2) + 1(3) + \\tfrac{1}{2}(4)$.
  3. $= 2 + 6 + 3 + 2 = 13\\%$.

Answer: Maximum percentage error in $P$ is $13\\%$.

6
Worked Example
Round each to three significant figures using the round-half-to-even rule where needed: (a) $4.567$, (b) $0.06245$, (c) $2.345$.
Solution
  1. (a) Dropped digit $7 > 5$ → round up: $4.57$.
  2. (b) Three sig. figs of $0.06245$ → look at the 4th figure ($5$ followed by nothing significant); preceding digit $4$ is even, keep it: $0.0624$.
  3. (c) Dropping the final $5$: preceding digit $4$ is even, keep it: $2.34$.

Answer: (a) $4.57$, (b) $0.0624$, (c) $2.34$.

Key Points

  • Accuracy = closeness to the true value; precision = agreement of repeated readings (set by instrument resolution).
  • Relative error $= \\dfrac{\\Delta a}{a}$; percentage error multiplies this by $100$.
  • Sums/differences: absolute errors add. Products/quotients: relative errors add. Powers: multiply the relative error by the power.
  • Significant figures = reliable digits plus one uncertain; leading zeros never count, trailing zeros count only with a decimal point.
  • Multiplication/division keep the least sig. figs; addition/subtraction keep the least decimal places; round half to even.
Quick Check — 4 MCQs
Tap an option to check your answer0 / 4
Q1.The number of significant figures in $0.00420$ is:
Explanation: Leading zeros don't count; $4$, $2$ and the trailing $0$ (after a decimal) do — three sig. figs.
Q2.For $Z = A + B$, the error $\\Delta Z$ is:
Explanation: For sums and differences, absolute errors add.
Q3.An instrument that gives the same wrong reading every time shows mainly:
Explanation: A consistent one-directional offset is a systematic error (e.g. zero error).
Q4.If $Z = A^2$, the relative error in $Z$ is:
Explanation: For a power, the relative error is multiplied by the exponent: $2\\,\\Delta A/A$.
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