JV

Accuracy, Precision, and Significant Figures

Accuracy

  • Accuracy = how close a measurement is to the actual or true value.
  • It is the degree of closeness to the actual number. The drawback: you may not know the actual number ahead of time, so you must measure and compare to a known/accepted value.
  • In science, accuracy is the gold standard you want to maximize, but it relies on knowing or agreeing on the true value for comparison.
  • The term accuracy is sometimes confused with precision in everyday language; precision is not the same as accuracy, though in practice both are important.

Precision

  • Precision = how close repeated measurements are to each other.
  • It reflects the repeatability of the measurement process: if you measure the same thing many times, do you get very similar results?
  • Precision is about the spread of values (the range) and how tightly clustered the measurements are.
  • A measurement process can be precise but not accurate if the readings are consistently close to each other but far from the true value (systematic error).
  • A measurement process can be accurate but not precise if readings happen to be close to the true value but show a large spread due to random errors.
  • Conceptual example (noise): When using a balance with a dial to weigh an item, the needle lands between marks due to reading uncertainty or noise; you have to guess between marks, which reflects measurement uncertainty.

Noise, Uncertainty, and Significant Figures

  • Noise in measurement refers to the uncertainty introduced by the instrument, reading, and human judgment when interpreting where the value lies between marks.
  • Significant figures are used to convey the uncertainty and the precision of reported measurements; they indicate how many meaningful digits reflect the measurement's precision.
  • In practice, you report numbers with the correct number of significant figures to reflect the instrument’s precision and the uncertainty of the measurement.

Measurement in Practice: Empirical vs Theoretical

  • Science is empirical: measurements and observations form the basis of data and conclusions.
  • Chemistry is described as an empirical science because its conclusions are grounded in observations and experiments.
  • Some theoretical sciences are not based on direct observation, but an observation (measurement) still underpins most scientific conclusions.

A Concrete Measurement Scenario: Reading a Balance

  • Old-fashioned pan balances with a dial illustrate reading uncertainty: you must interpolate between marks to estimate a value (e.g., three pounds vs. two-and-a-half pounds).
  • This illustrates how uncertainty (noise) translates into the need to report values with appropriate precision and to consider significant figures.

Data Example: Two Students Measuring the Same String

  • Setup: Two students measure the same string four times each.
  • Objective: Assess accuracy and precision by comparing to a known true value.

Experiment 1 data (student 1)

  • Measurements (example):
    • x1 = 19.3, x2 = 20.1, x3 = 19.4, x4 = 19.2
  • Average (mean):
    • \bar{x}_1 = \frac{1}{4}(19.3 + 20.1 + 19.4 + 19.2) = 19.5
  • True (actual) value: x_{true} = 19.2
  • Accuracy assessment (percent error):
    • Experiment 1:
    • \% ext{Error}1 = \left|\frac{\bar{x}1 - x{true}}{x{true}}\right| \times 100\% = \left|\frac{19.5 - 19.2}{19.2}\right| \times 100\% \approx 1.6\%
  • Precision assessment (range and standard deviation):
    • Range:
    • R_1 = \max(x) - \min(x) = 20.1 - 19.2 = 0.9
    • Standard deviation (sample):
    • Given as s_1 = 0.41 (these values reflect scatter of the four measurements around the mean)

Experiment 2 data (student 2)

  • Measurements (example):
    • x1 = 19.5, x2 = 19.0, x3 = 19.3, x4 = 19.4
  • Average (mean):
    • \bar{x}_2 = \frac{1}{4}(19.5 + 19.0 + 19.3 + 19.4) = 19.3
  • True (actual) value: x_{true} = 19.2
  • Accuracy assessment (percent error):
    • Experiment 2:
    • \% ext{Error}2 = \left|\frac{\bar{x}2 - x{true}}{x{true}}\right| \times 100\% = \left|\frac{19.3 - 19.2}{19.2}\right| \times 100\% \approx 0.52\%
  • Precision assessment (range and standard deviation):
    • Range:
    • R_2 = \max(x) - \min(x) = 19.5 - 19.0 = 0.5
    • Standard deviation (sample):
    • Given as s_2 = 0.22

Comparison of experiments

  • Which set is more precise? Experiment 2 (smaller range: 0.5 vs 0.9; smaller standard deviation: 0.22 vs 0.41).
  • Which set is more accurate? Experiment 2 has the smaller percent error (0.52% vs 1.6%).
  • Important distinction: high accuracy does not guarantee high precision, and high precision does not guarantee high accuracy.

How to Quantify Accuracy and Precision (Summary of Methods)

  • Step 1: Compute the average of repeated measurements:
    • \bar{x} = \frac{1}{n}\sum{i=1}^{n} xi
  • Step 2: Assess accuracy using percent error:
    • \% ext{Error} = \left|\frac{\bar{x} - x{true}}{x{true}}\right| \times 100\%
  • Step 3: Assess precision using the range:
    • \text{Range} = \maxi xi - \mini xi
  • Step 4: Assess precision using standard deviation (to quantify spread):
    • s = \sqrt{\frac{1}{n-1}\sum{i=1}^{n} (xi - \bar{x})^2}
  • Step 5: Interpret standard deviation (statistical meaning):
    • Approximately 68\% of measurements fall within \bar{x} \pm s
    • Approximately 95\% of measurements fall within \bar{x} \pm 2s
  • Step 6: Use these tools to identify outliers and assess consistency with similar experiments or populations:
    • Values outside the mean ± 2s may indicate a different population, measurement error, or an outlier.

Real-World Context and Significance

  • Example: Counting molecules inside a vesicle requires measurement and comparison to literature values; results were checked for consistency against other data, illustrating how accuracy and precision feed into scientific conclusions.
  • The standard deviation provides a statistical basis for judging whether new measurements belong to the same population as prior measurements or represent something different.
  • Practical takeaways:
    • Aim for high accuracy (closeness to true value) and high precision (consistency among repetitions).
    • Report measurements with appropriate significant figures to reflect measurement uncertainty.
    • Use averages and statistical descriptors (range, standard deviation) to summarize measurement data.

Quick Takeaways

  • Accuracy = closeness to true value; depends on knowing the true value.
  • Precision = closeness of repeated measurements to each other; reflects repeatability.
  • Noise/uncertainty is inherent in measurement; significant figures communicate this uncertainty.
  • Percent error compares measured average to true value to quantify accuracy.
  • Range and standard deviation quantify precision; standard deviation also provides a probabilistic interpretation (68% within ±s, 95% within ±2s).
  • Real-world measurements often require verification across multiple data sets and literature to establish reliability and relevance.