Welcome to Uncertainties and Evaluation!

In Physics, we love measuring things—the speed of light, the mass of an electron, or even just the length of a wire. But here is a secret: no measurement is ever perfect. Every time you use a ruler, a stopwatch, or a voltmeter, there is a little bit of doubt about the exact number. This isn't because you are doing it wrong; it is simply a part of science!

In this chapter, we will learn how to describe that "doubt" using uncertainties and how to evaluate our experiments to see if our results are actually reliable. Understanding this is the difference between just "doing a lab" and being a real physicist.

Note: This chapter focuses on the final evaluation of your data. For details on how to set up your equipment or plot your initial points, check out "Apparatus, Range and Resolution (IAS)" and "Processing Results and Graphs (IAS)".

1. The Vocabulary of Measurement

Before we crunch the numbers, we need to speak the language. The exam board is very specific about these definitions, so let's get them right!

True Value, Error, and Uncertainty
  • True Value: This is the value that would be obtained by a perfect measurement. In reality, the true value is unknowable (except for fixed fundamental constants).
  • Error: This is the difference between your measured value and the true value. Remember: an error is not a mistake! If you knock over your equipment, that is a mistake (or a "blunder"). If your ruler is slightly expanded by heat, that creates an error.
  • Uncertainty: This is the interval (the range) within which the true value is considered to lie. For example, if you say a length is \( 10.0\text{ cm} \pm 0.1\text{ cm} \), you are saying you are confident the true value is somewhere between \( 9.9\text{ cm} \) and \( 10.1\text{ cm} \).
Accuracy vs. Precision

These two words are often used interchangeably in daily life, but in Physics, they mean very different things!

  • Accuracy: How close your measurement is to the true value. We cannot "quantify" accuracy (give it a specific number), but we can judge it.
  • Precision: How close repeated measurements are to each other. Precision is only affected by random effects. If you measure a wire five times and get the exact same number every time, your measurements are precise.
Repeatability and Reproducibility
  • Repeatability: Can you get the same results again using the same method and equipment in a short amount of time?
  • Reproducibility: Can someone else (or you, using different equipment/methods) get the same results?

Quick Tip: Think of a dartboard. If all your darts hit the bullseye, you are accurate and precise. If all your darts land in a tight cluster at the very edge of the board, you are precise but not accurate!

2. Calculating Uncertainty

How do we actually decide what the \( \pm \) value should be? There are two main ways depending on your data.

Case A: A Single Reading (or identical repeats)

If you only take one reading, or if you repeat the measurement and get the exact same value every time, we use the resolution of the instrument.

Uncertainty = \( \frac{1}{2} \times \text{resolution} \)

Example: A standard ruler has a resolution of \( 1\text{ mm} \). The uncertainty of a single reading is \( \pm 0.5\text{ mm} \).

Case B: Repeated Readings (with variation)

If your repeated readings are different (which happens often due to random errors), we use the range of the results.

Uncertainty = \( \frac{1}{2} \times \text{range} \)

(Alternatively: The distance from the mean to the furthest reading.)

Example: You measure the time for a pendulum to swing: \( 1.2\text{ s}, 1.4\text{ s}, 1.3\text{ s} \).
1. Find the mean: \( \frac{1.2 + 1.4 + 1.3}{3} = 1.3\text{ s} \).
2. Find the range: \( 1.4 - 1.2 = 0.2\text{ s} \).
3. Uncertainty: \( \frac{0.2}{2} = 0.1\text{ s} \).
4. Final value: \( 1.3 \pm 0.1\text{ s} \).

3. Percentage Uncertainty

To understand how much an uncertainty actually matters, we turn it into a percentage. A \( 1\text{ cm} \) error matters a lot if you are measuring a finger, but not at all if you are measuring a football pitch!

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

How many significant figures?
  • Percentage uncertainty is usually written to 1 or 2 significant figures.
  • Processed data (your final calculated answer) is usually written to 3 significant figures.

Key Takeaway: If your percentage uncertainty is below \( 5\% \), it suggests that your measurement is repeatable and reliable.

4. Evaluating the Experiment

Once you have your results, you need to "criticise" them. This is where you earn the high marks in Unit 3.

Identifying Systematic Errors

A systematic error is one that follows a pattern—usually because something is wrong with the equipment or the set-up. All your readings will be too high or too low by the same amount.

  • The Origin Check: If you plot a graph of two quantities that should be directly proportional (like Force and Extension), the line should go through the origin \( (0,0) \). If the best-fit line misses the origin, it is a big clue that you have a systematic error (like a "zero error" on a scale).
Judging Accuracy

How do you know if your experiment was "successful"? We compare our value to an "accepted value" (like \( g = 9.81\text{ m s}^{-2} \)).

Your result is considered accurate if:
1. The accepted value lies within your uncertainty range.
2. OR, the percentage difference between your value and the accepted value is less than \( 5\% \).

Using Graphs for Evaluation

When determining a relationship or a constant (like the Young Modulus), always use a graph. To evaluate properly:
1. Draw a best-fit line.
2. Calculate the gradient using a large triangle (this reduces the percentage uncertainty in your gradient calculation).
3. Check for anomalous points—readings that don't fit the trend. If you see one, check your notes to see if it was an inconsistent reading.

Summary Checklist

- Resolution: Smallest interval on the device.
- Single Reading Uncertainty: \( \pm 0.5 \times \text{resolution} \).
- Repeat Reading Uncertainty: \( \pm 0.5 \times \text{range} \).
- Percentage Uncertainty: \( \frac{\text{Uncertainty}}{\text{Value}} \times 100 \).
- Accuracy Check: Is the difference \( < 5\% \) or within the uncertainty range?
- Systematic Error: Look for a graph that doesn't hit the origin when it should.

Don't worry if this feels like a lot of definitions! Just remember: Precision is about the "grouping" of your results, while Accuracy is about hitting the "target." In Unit 3, always show your workings for uncertainties to get those calculation marks!