Lesson 1.3 – Uncertainties and Deviations in Measurement

Context & Motivation

  • Everyday technologies illustrate measurement uncertainty; e.g., Global Positioning System (GPS) often places a user several meters from their real position.
  • Scientific conclusions are not absolute; they evolve with improvements in instruments, techniques, and theory.

Learning Targets

  • Differentiate accuracy from precision.
  • Describe & compare the precision of common measuring devices (tape, ruler, vernier caliper, analog/digital meters).
  • Compute the arithmetic mean and estimated standard deviation for small data sets.
  • List practical tips to minimize measurement uncertainties.

Measurement Fundamentals

  • Measurement = assigning a numerical value to a property (length, mass, voltage, pH, etc.).
  • Choice of unit and instrument is essential (e.g., thickness of a book → millimeters; preferable instrument → vernier caliper over ruler or tape for higher precision).

Selecting Instruments

  • Ask 3 guiding questions:
    • What quantity & unit do I need?
    • Which device offers sufficient resolution/least uncertainty?
    • Can the device be read & handled properly for this task?
  • Comparison for book-thickness example:
    • Tape measure: large scale divisions → low precision.
    • Ruler: medium precision (mm divisions; practical limit ≈ half a mm).
    • Vernier caliper: smallest scale/divisional reading (0.02 mm or 0.01 mm) → best choice.

Error & Uncertainty

  • Error = true value − measured value.
  • Uncertainty = quantitative expression of doubt in the reported value; arises from cumulative errors.
  • Reporting convention: value ± uncertainty (e.g., 21.7\,\text{mm} \pm 0.2\,\text{mm}).

Accuracy

  • Closeness of a single (or mean) measurement to the true value.
  • Can be conveyed by:
    • Explicit uncertainty (range: 21.7\,\text{mm}\,\pm 0.2 gives possible true values 21.5!\text{–}!21.9\,\text{mm}).
    • Fractional/percent error: \text{Percent error} = \frac{|\text{true} - \text{measured}|}{\text{true}}\times100\%.
    • Correct use of significant figures (sig. figs. imply unspoken uncertainty of ±1 in last digit).

Precision

  • Describes scatter/variability among a set of measurements.
  • Example: 21.7, 21.6, 21.5\,\text{mm} are closely grouped → high precision regardless of closeness to the true value.

Accuracy vs. Precision Matrix

  • Four possible cases illustrated by target/bull’s-eye analogy:
    • Accurate & Precise (cluster at center).
    • Accurate but Not Precise (widely scattered around center).
    • Precise but Not Accurate (tight cluster offset from center).
    • Neither Accurate nor Precise (scattered & off-center).
  • Key questions posed:
    • Does high accuracy guarantee high precision? (No.)
    • Does high precision guarantee high accuracy? (No.)

Precision of Measuring Devices

Analog Devices

  • Rule of thumb: best achievable precision ≈ half of the smallest scale division.
  • Illustration:
    • Dial/analog gauges A vs. B: Device A has finer gradations → more precise reading; B lacks enough gradations for the same resolution.

Digital Devices

  • Readout shows a fixed number of digits; uncertainty usually lies in the last displayed digit.
  • Manufacturer may quote explicit specification, e.g., 5.61\,\text{V} \pm 0.005\,\text{V}.

Statistical Tools for Repeated Measurements

Arithmetic Mean (Average)

  • Gives the best estimate of the true value when random errors dominate.
  • Formula: \bar{x}=\frac{\sum x}{n}.
  • Practical notes:
    • 4–10 trials are typically adequate.
    • Improves accuracy compared to a single reading.

Estimated Standard Deviation (Sample SD)

  • Measures spread of data; informs reliability.
  • Formula for small sample: s=\sqrt{\frac{\sum (x_i-\bar{x})^2}{n-1}}.
  • Look at both mean and SD to judge data quality.

Worked Examples & Practice Problems

  • Mass of water (5 readings): 2.5, 2.6, 2.5, 2.4, 2.6\,\text{kg} → \bar{x}=2.52\,\text{kg}\,(\approx 2.5\,\text{kg}).
  • Missing-value problems given for car speed & steel mass illustrate algebraic rearrangement of mean formula.
  • Hall-length data in mixed units (m, km, cm) normalized to meters gave s = 2.51\,\text{m}.
  • Challenge problem: convert °F and K to °C first, then compute SD for hot-water temperatures.

Practical Tips to Reduce Uncertainty

  • Follow manufacturer instructions & proper maintenance.
  • Calibrate instruments and apply corrections.
  • Use trained personnel; validate software.
  • Choose instrument with smallest available uncertainty & appropriate range.
  • Repeat measurements; use peer repetition.
  • Round only at final steps; check copied calculations; record data immediately.

Key Formulas Recap

  • Mean: \bar{x}=\frac{\sum x}{n}.
  • Estimated SD: s=\sqrt{\frac{\sum (x_i-\bar{x})^2}{n-1}}.
  • Percent error: \frac{|\text{true} - \text{measured}|}{\text{true}} \times 100\%.

Concept Checks

  • True/False statements:
    1. High precision always implies high accuracy → False.
    2. Values close to each other imply accuracy → False (they imply precision).
    3. Values close to true value imply precision → False (they imply accuracy).
  • Challenge: Is a pH of 5.76 more accurate & precise than 5.8? → Depends on the true pH & device resolution; absolute difference alone cannot answer without uncertainty data.

Broader Significance

  • Understanding & quantifying uncertainty underpins trustworthy science and engineering.
  • Ethical obligation: report uncertainties so others can judge confidence.
  • Real-world: navigation systems, medical dosages, industrial manufacturing tolerances, climate modeling.

Further Reading (Sources)

  • Bauer & Westfall, "University Physics with Modern Physics" (2013)
  • Serway & Serway, "College Physics" 7th ed. (2006)
  • Knight, "Physics for Scientists & Engineers" (2017)
  • Serway & Jewett, "Physics for Scientists & Engineers" 9th ed. (2014)
  • Young, Freedman & Ford, "University Physics" 13th ed. (2012)