Introduction to Uncertainty, Error and Accuracy

In Physics, we love to measure things—from the tiny diameter of a wire to the speed of a falling ball. However, no measurement is ever "perfect." Even with the most expensive equipment, there is always a limit to how sure we can be of our result. This chapter is all about understanding those limits. We will learn how to describe how "good" our measurements are and how to calculate the uncertainty that comes with every experiment. Don't worry if this seems a bit abstract at first; once you learn the specific rules Edexcel expects, it becomes much easier!

Note: This chapter focuses on the definitions and calculations of uncertainty. For details on how to choose specific tools like vernier calipers or micrometers, see the chapter on "Instruments, resolution and measuring technique."

1. Key Vocabulary: What do these words actually mean?

In everyday life, we use "error," "mistake," and "uncertainty" to mean the same thing. In Physics, they have very specific meanings. It is vital to use these correctly in your Unit 3 exam.

  • True Value: This is the value that would be obtained by a perfect measurement. In reality, the true value is unknowable, unless we are talking about a defined fundamental constant.
  • Error: The difference between your individual measurement and the true value. Important: An error is not a mistake (like misreading a scale). It is a natural part of measuring.
  • Uncertainty: The interval or range within which the true value is considered to lie. We usually write this as \( \text{Value} \pm \text{Uncertainty} \).
  • Resolution: The smallest change in the quantity being measured that gives a perceptible change in the reading of the measuring instrument. For example, a standard ruler has a resolution of \( 1 \text{ mm} \).

Quick Tip: If you accidentally write down the wrong number from a screen, that is a mistake or a misreading, not a physical error!

2. Accuracy vs. Precision

These two terms are often confused, but they describe different things about your data.

Accuracy

Accuracy is how close your measurement is to the true value. According to the syllabus, accuracy itself cannot be quantified (you can't give it a single number), but we can judge it. We say a result is accurate if the "accepted value" (from a textbook) falls within our uncertainty range, or if the percentage difference is below 5%.

Precision

Precision is the agreement between repeated measurements. It has nothing to do with the true value. If you measure the length of a string five times and get exactly \( 15.2 \text{ cm} \) every time, your measurements are very precise, even if the string was actually \( 16.0 \text{ cm} \) long!

Precision is affected only by random effects.

Summary: The Dartboard Analogy
  • High Accuracy, High Precision: All darts are in the bullseye.
  • Low Accuracy, High Precision: All darts are bunched tightly together, but far from the bullseye.
  • High Accuracy, Low Precision: Darts are spread out, but their average position is the bullseye.

3. Repeatability and Reproducibility

When you do an experiment, you want to know if you can trust the results. We use two terms to describe this:

Repeatability: Can you (the same operator) get the same results using the same method and same apparatus over a short period of time?

Reproducibility: Can someone else (different operator) get the same results using different apparatus or a different method?

Key Takeaway: A percentage uncertainty below 5% suggests that the measurement is repeatable.

4. How to Calculate Uncertainty

In the Unit 3 exam, you will often be asked to calculate the uncertainty of a single reading or a set of repeat readings. Here are the rules:

For a Single Reading

If you only take one measurement (or if all your repeats are identical), the uncertainty is half the resolution of the instrument.

Example: A thermometer has a resolution of \( 1^\circ\text{C} \). You record a temperature of \( 22^\circ\text{C} \).
The uncertainty is \( \pm 0.5^\circ\text{C} \).
The measurement is recorded as \( 22.0 \pm 0.5^\circ\text{C} \).

For Repeat Readings

When you have several different values for the same measurement, use the half-range rule:

\( \text{Uncertainty} = \frac{\text{Maximum value} - \text{Minimum value}}{2} \)

Alternatively, you can use the distance from the mean (average) to the furthest reading.

Common Mistake to Avoid: When calculating the mean or range, always check for anomalies (inconsistent readings). If one value is wildly different from the others, ignore it before doing your calculations!

5. Percentage Uncertainty

This is a way of expressing the uncertainty as a fraction of the total measurement. It helps us see how significant the uncertainty actually is.

The Formula:
\( \text{Percentage Uncertainty} = \left( \frac{\text{Uncertainty}}{\text{Measurement}} \right) \times 100\% \)

Rules for Significant Figures:
  • Percentage Uncertainty: Usually quoted to one or two significant figures.
  • Processed Data: Usually quoted to three significant figures.

Example:
You measure a voltage as \( 4.50 \pm 0.05 \text{ V} \).
\( \text{Percentage Uncertainty} = \left( \frac{0.05}{4.50} \right) \times 100\% = 1.1\% \)

Note: In Unit 3 (IAS), you are not expected to "compound" (add) percentage uncertainties for different variables. That is only required for the full A-level (Unit 6).

6. Systematic Errors and Graphs

A systematic error is an error that follows a set pattern. For example, if your weighing scales show \( 0.1 \text{ g} \) when nothing is on them, every reading you take will be \( 0.1 \text{ g} \) too high.

How to spot systematic error on a graph:
If you plot your data and find a perfect straight line (a line of best fit), but it misses the origin (the point \( 0,0 \)) when it was expected to pass through it, this indicates a systematic error.

Reducing Errors:
- Random errors: Reduced by taking repeat readings and calculating a mean.
- Systematic errors: Reduced by re-calibrating equipment or performing a zero check.

Summary Checklist

Before moving on, make sure you can:

  • Define Accuracy, Precision, Resolution, and Uncertainty using the exam board's wording.
  • Distinguish between Repeatability and Reproducibility.
  • Calculate uncertainty for single readings (\( \frac{1}{2} \text{ resolution} \)) and repeats (\( \frac{1}{2} \text{ range} \)).
  • Calculate Percentage Uncertainty.
  • Identify Systematic Error from a graph that misses the origin.
  • Remember the 5% rule for judging if a result is "accurate" or "repeatable."