understand the distinction between precision and accuracy

📏 Errors and Uncertainties in Physics

What are Errors?

When we measure something, the result we get is never exactly the true value.

The difference between the measured value and the true value is called an error.

Errors can be grouped into two main types:

  • Random errors – caused by unpredictable fluctuations (e.g. the hand tremor when reading a scale).

  • Systematic errors – caused by a bias in the measuring instrument or method (e.g. a ruler that is 1 mm too long).

Uncertainty

The uncertainty is our best estimate of how far the true value could be from the measured value.

It is usually expressed as a plus‑minus range:

Measured value = \$x \pm \Delta x\$

where \$\Delta x\$ is the uncertainty.

Precision vs Accuracy

These two terms are often mixed up. Think of them as two different aspects of a measurement:

AspectWhat it MeansAnalogy
PrecisionHow close repeated measurements are to each other.All arrows hit the same spot on a target, but far from the bullseye.
AccuracyHow close a measurement is to the true value.Arrows hit the bullseye, but spread out widely.

In practice, a good experiment aims for both: many measurements that are close together (high precision) and close to the true value (high accuracy).

Example: Measuring the Length of a Book

  1. Use a ruler marked in millimetres. Measure the book 10 times.

  2. Record each reading: 210 mm, 211 mm, 209 mm, 210 mm, 210 mm, 211 mm, 209 mm, 210 mm, 210 mm, 211 mm.

  3. Average value: \$x = \frac{210+211+209+210+210+211+209+210+210+211}{10} = 210.0\,\text{mm}\$.

  4. Standard deviation (a measure of precision): \$\sigma \approx 0.7\,\text{mm}\$.

  5. If the book’s true length is 210.5 mm, the measurement is accurate to within \$0.5\,\text{mm}\$.

📌 Examination Tip: When asked to calculate uncertainty, always:

  1. Identify the type of error (random or systematic).

  2. Use the appropriate formula (e.g. \$\Delta x = \sqrt{\sum (\Delta x_i)^2}\$ for combined uncertainties).

  3. State the final result as \$x \pm \Delta x\$ with the correct significant figures.

🎯 Quick Check: If your repeated measurements are all 100 cm but the true value is 101 cm, you have:

  • High precision (all 100 cm are the same).

  • Low accuracy (far from 101 cm).

Common Mistakes to Avoid

  • Reporting only the mean value without its uncertainty.

  • Ignoring systematic errors (e.g. a miscalibrated scale).

  • Using too few measurements to claim high precision.

📝 Final Exam Reminder:

Always write the uncertainty in the same units as the measured quantity.

Use significant figures: if the uncertainty is \$0.03\$, report the measurement as \$x = 5.12 \pm 0.03\$ (not \$5.120 \pm 0.030\$).