Errors, Significant Figures, and Precision Measurement

Errors: Nature & Classification

• All measurements are limited by human and instrument precision; no reading is absolutely exact.

• Four canonical categories of experimental error:

  • Personal (Observer) Errors

  • Stem from bias, carelessness, or improper observation techniques.

  • Common manifestations:

    • Trusting a first trial and forcing later trials to match it.

    • Poor adjustment of instruments.

    • Faulty estimation of fractional divisions.

    • Parallax error: apparent displacement of the measured point when the eye is not perpendicular to the scale.

    • Example (Fig. 1-1):

      • Eye along $AD$ → read $4.4$ cm.

      • Eye along $CD$ → read $4.6$ cm.

      • Correct eye line $BD$ → $4.5$ cm.

    • Mitigation: keep scale as close as possible to the object; place meter sticks edge-wise on the surface.

  • Accidental (Random) Errors

  • Arise from uncontrollable environmental fluctuations: building vibration, wind, temperature drift, etc.

  • Instrumental Errors

  • Occur when an instrument is used outside its calibration range, is worn, or lacks required precision.

  • Example: measuring a wire diameter (≈ 1 mm) with a meter stick whose smallest division is 1 mm.

  • Systematic Errors

  • Consistent, uni-directional deviations due to faulty calibration or set-up.

  • Example (Fig. 1-2): Using the worn end of a meter stick; every reading is too large.

Quantifying Error

• Actual (Absolute) Error: $|x<em>{\text{exp}}-x</em>{\text{true}}|$.

• Relative (Fractional) Error: $\dfrac{|x<em>{\text{exp}}-x</em>{\text{true}}|}{x_{\text{true}}}$.

• Percentage Error: $\text{Relative error}\times100\%$.

  • Street example: $\dfrac{10\,\text{ft}}{400\,\text{ft}}=0.025\,(2.5\%)$.

  • Box example: $\dfrac{0.2\,\text{ft}}{2.0\,\text{ft}}=0.10\,(10\%)$ .

• Percentage Difference (when no accepted value exists): $\dfrac{|x<em>1-x</em>2|}{\frac{x<em>1+x</em>2}{2}}\times100\%$.

  • Example: $\dfrac{4.2-4.0}{4.1}\times100=4.9\%$.

Significant Figures (Sig-Figs)

• Definition: Digits required to report a quantity to the same accuracy as the measurement.

• Leading zeros only locate the decimal; they are not significant.

  • $0.00002064\,\text{km}$ has 4 sig-figs (2-0-6-4).

• Estimated (Doubtful) Digit: last recorded digit; always include it, even when zero.

  • “Exactly 20 cm” measured with a meter stick → write 20.00 cm (precision ≈ $0.01$ cm).

• Error bound for a reading ending in the doubtful digit is ± half the smallest resolvable division.

  • 20.64 cm → possible error ± 0.005 cm.

Operations & Sig-Fig Rules

• Multiplication / Division

  • Final result keeps sig-figs so that its % error matches the least accurate factor.

  • Carry one extra sig-fig in intermediate steps.

  • Cylinder example:

  • Diameter $d=2.25\,\text{cm}$ (3 sig-figs) → radius $r=1.125\,\text{cm}$ (4 kept).

  • $r^2=1.27\,\text{cm}^2$ (one doubtful digit retained).

  • Volume $V=\pi r^2h$; when $h=26.2$ cm, choose π such that extra digits do not out-class the least accurate term.

    • Guideline proposed: match π’s sig-figs to those of the largest factor containing only one doubtful digit.

• Addition / Subtraction

  • Align decimal points; answer is cut off at the first doubtful place.

  • Task example: add $81.372 + 710 + 0.03 + 24.3$.

Experiment 1: Errors & Significant Figures

Special Apparatus

• Meter stick; aluminum rectangular block; aluminum cylindrical block.

Part I – Rectangular Block

• Measure length, width, thickness (3 trials each, in cm).

• Measure length again (3 trials) in inches.

• Compute average conversion $1\,\text{in}=\ldots\,\text{cm}$.

• Determine % error vs. standard $2.54\,\text{cm/in}$.

• Compute volume $V=\ell w t$.

• Measure mass → density $\rho = m/V$.

• Compare to accepted $\rho_{\text{Al}}=2.7\,\text{g/cm}^3$; compute % error.

Part II – Hollow Cylinder

• Measure inner diameter $d<em>1$, outer diameter $d</em>2$, and length $L$ (3 trials each).

• Cross-sectional area $A = \dfrac{\pi}{4}\,(d<em>2^2-d</em>1^2)$.

• Volume $V = AL = \dfrac{\pi}{4}(d<em>2^2-d</em>1^2)L$.

• Determine mass and density; evaluate % error.

Measurement Instruments (Experiment 2)

Vernier Caliper

• Two scales: main (fixed, mm) & vernier (movable, 20 divisions = 1 mm).

• Reading procedure:

  1. Main scale gives whole & first decimal mm.

  2. Vernier mark that aligns gives additional $\frac{n}{20}$ mm.

• Example (Fig. 2-2): zero lies between 11 mm & 12 mm; 13th vernier mark aligns → $11 + \frac{13}{20}=11.65$ mm.

Micrometer Caliper

• Components: anvil, spindle (jaw B), screw, sleeve scale (mm), thimble scale (50 divisions), ratchet.

• Pitch usually $0.5\,\text{mm/rev}$; thus each thimble division = $0.01$ mm.

• Example reading: sleeve shows >13 mm; thimble at 43 → $13.43$ mm.

• Zero error (systematic) must be corrected every reading.

Laboratory Balance

• Compares gravitational force of unknown mass (left pan) to standard masses (right pan).

• Rider on the graduated beam provides fine adjustment (adds effective mass to right pan).

• Steps:

  1. Zero rider; pointer should oscillate symmetrically.

  2. Place unknown on left pan.

  3. Add a single close weight to right pan; bracket with next heavier/lighter.

  4. Use smaller weights and finally the rider for exact balance.

Density Concept

• $\rho = \dfrac{m}{V}$ with common units g cm$^{-3}$ (solid & liquid tables provided).

• Utilised for material identification & %-error comparison.

Experiment 2: Measurement of Length & Mass

• Apparatus: vernier caliper, micrometer, hollow metal cylinder, small steel sphere, laboratory balance, meter stick, weights.

• Sequential Tasks:

  1. Measure $D,d,h$ of cylinder with vernier.

  2. Determine micrometer zero error.

  3. Measure sphere diameter with micrometer.

  4. Weigh both objects.

  5. Calculations (sig-fig compliant):

  • Cylinder volume: $V=\dfrac{\pi}{4}(D^2-d^2)h$.

  • Sphere volume: $V=\dfrac{\pi}{6}d^3$.

  • Densities $\rho = m/V$.

  1. Identify materials; compute %-error vs. accepted densities (Appendix table: Al 2.70, Brass 8.56, Copper 8.92, Iron 7.86, Steel 7.8).

  2. Show detailed sample calculations wherever * indicated.

Physical Constants (Appendix Excerpts)

• Solid Densities $\times10^3\,\text{kg/m}^3$: Aluminum 2.70; Brass 8.56; Copper 8.92; Iron 7.86; Steel 7.8; Silver 10.57.

• Liquid Density: Glycerine 1.26.

• Velocity of sound at 20 °C (m/s): Al 5104; Brass 3500; Copper 3560; Iron 5130; Steel 5000.

• Young’s Modulus $\times10^{11}\,\text{N/m}^2$: Al 0.7; Brass 0.92; Copper 1.1; Iron 1.98; Steel 2.2.

Practical & Philosophical Take-Aways

• A small absolute error can signify poor work if the measured quantity is itself small; always evaluate relative error.

• Record the doubtful digit; omit it only sacrifices hard-earned precision.

• Ethical lab practice: discard blunders, but never massage data to match a preconceived result.

• Instrument choice should match required precision; otherwise systematic errors are inevitable.

• Consistency of sig-figs across calculations ensures reported answers truthfully reflect underlying uncertainty.