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:

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 $0.5\,\text{mm}$.

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.