JV

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Key Concepts

  • Accuracy
    • Definition: how close your data are to an accepted or true value.
    • Measured by how near the data/mean are to the true value.
    • Example metric: Absolute error \text{Absolute Error} = |x - x{\text{true}}| or for a set of measurements, \text{Accuracy of mean} = |\bar{x} - x{\text{true}}|.
  • Precision
    • Definition: how close your data are to each other (repeatability).
    • Measured by the spread of the measurements, not how close to the true value.
    • Example metrics: Range \text{Range} = \max(xi) - \min(xi), standard deviation s = \sqrt{\frac{1}{n-1}\sum (x_i - \bar{x})^2}.
  • Relationship between accuracy and precision
    • Accuracy concerns closeness to the accepted value, regardless of internal agreement.
    • Precision concerns consistency among measurements, regardless of closeness to the true value.
    • You can be accurate but not precise, precise but not accurate, both, or neither.

Worked Example: Density of Aluminum (accepted value = x_{\text{true}} = 2.7\ \mathrm{g/mL})

  • Data sets (all in \mathrm{g/mL})

    • John: initial remark "2.9 is not very close to 2.7"; detailed trials include around 2.9, specifically 2.924, \ 2.923.
    • Sally: 2.3,\ 2.5,\ 2.9.
    • Megan: 2.64,\ 2.73,\ 2.69.
    • Mike: 2.71,\ 2.699.

  • Accuracy assessment (closeness to 2.7)

    • John
    • Mean of displayed values (based on trials given): not explicitly averaged in the transcript, but the values are around 2.9 (e.g., 2.924,\ 2.923).
    • Conclusion from transcript: John is not accurate (data not close to 2.7).
    • Sally
    • Values include 2.3, 2.5, 2.9; not close to 2.7 and vary widely.
    • Conclusion: Sally is not accurate.
    • Megan
    • Values around 2.64–2.73; close to 2.7 overall.
    • Conclusion: Megan is fairly accurate (close to 2.7) compared to John and Sally.
    • Mike
    • Values 2.71 and 2.699; very close to 2.7.
    • Conclusion: Mike is accurate.

  • Precision assessment (consistency of values)

    • John
    • Trials near 2.9 and very close to each other: 2.924,\ 2.923\;\Rightarrow\; \text{high precision}.
    • Sally
    • Values span from 2.3 to 2.9; not close to each other: \text{Range} = 2.9 - 2.3 = 0.6\;\text{g/mL}. → Not precise.
    • Megan
    • Values: 2.64, 2.73, 2.69; spread is \text{Range} = 2.73 - 2.64 = 0.09\;\mathrm{g/mL}. → Some precision, but not as tight as Mike.
    • Mike
    • Values 2.71 and 2.699; \text{Range} = 2.71 - 2.699 = 0.011\;\mathrm{g/mL}. → Very precise.

  • Numerical calculations (to illustrate concepts)

    • John's data (2 measurements): \bar{x}{John} = \frac{2.924 + 2.923}{2} = 2.9235. |\bar{x}{John} - x_{\text{true}}| = |2.9235 - 2.7| = 0.2235. (Absolute error)
    • Sally's data: \bar{x}{Sally} = \frac{2.3 + 2.5 + 2.9}{3} = 2.566\overline{6}. |\bar{x}{Sally} - x_{\text{true}}| \approx |2.5667 - 2.7| = 0.1333.
    • Megan's data: \bar{x}{Megan} = \frac{2.64 + 2.73 + 2.69}{3} = 2.686\overline{6}. |\bar{x}{Megan} - x_{\text{true}}| \approx |2.6867 - 2.7| = 0.0133.
    • Mike's data: \bar{x}{Mike} = \frac{2.71 + 2.699}{2} = 2.7045. |\bar{x}{Mike} - x_{\text{true}}| = |2.7045 - 2.7| = 0.0045.
    • Mike's precision (range): \text{Range}_{Mike} = 2.71 - 2.699 = 0.011.

  • Verdict for the prompt question

    • Data that is accurate but not precise: Megan.
    • Why: Megan's mean is very close to 2.7 (accurate), but her individual measurements have a noticeable spread (not highly precise).

Summary of Lessons

  • Accuracy vs Precision recap
    • Accuracy measures closeness to the true value: lower absolute error means higher accuracy.
    • Precision measures consistency among measurements: smaller spread means higher precision.
  • In real experiments, aim to maximize both when possible, but recognize they are separate qualities.
    • If you need a reliable value, improve accuracy (calibrate instrument, use accurate standards).
    • If you need reproducible results, improve precision (control variables, repeat measurements, reduce random errors).
  • Real-world relevance
    • Errors can bias outcomes if only accuracy is considered (you might hit the true value on average but with high variability).
    • High precision with low accuracy can mislead if systematic bias exists in the equipment or method.

Practical Improvement Tips

  • For better accuracy
    • Calibrate instruments regularly.
    • Use standardized reference materials.
    • Correct for known biases in the method.
  • For better precision
    • Increase number of trials to average out random errors.
    • Control environmental factors (temperature, pressure, workflow).
    • Standardize procedures to reduce variability in technique.

Connections to Foundational Principles

  • Measurement uncertainty and error analysis underpin accuracy and precision.
  • Reproducibility and repeatability are core to experimental reliability.
  • Ethical reporting: disclose limitations and uncertainties to avoid overstatement of results.

Quick Takeaway

  • Accuracy = closeness to true value.
  • Precision = closeness among repeated measurements.
  • Megan in the example is accurate but not precise, illustrating the separation of the two concepts.