Compound Uncertainties, Precision and Accuracy (IA2)

Q: What is the difference between accuracy and precision in physics?

Accuracy is how close a measurement is to the true value, while precision is the degree of agreement between repeated measurements.

Q: How are repeatability and reproducibility distinguished?

Repeatability refers to getting consistent results using the same operator and equipment, while reproducibility refers to obtaining similar results using different methods or equipment.

Q: How is the uncertainty of a single reading determined?

For a single reading, the uncertainty is taken as half the resolution of the measuring instrument.

Q: How do you calculate the uncertainty for a set of repeat readings ?

The uncertainty is calculated as half the range : \( \text{Uncertainty} = \frac{\text{Max} - \text{Min}}{2} \).

Q: What is the rule for compounding uncertainties during multiplication or division ?

When multiplying or dividing quantities, you must add the percentage uncertainties of each quantity.

Q: What is the rule for uncertainty when a measurement is raised to a power ?

Multiply the percentage uncertainty of the measurement by the power \(n\).

Q: What is the rule for compounding uncertainties during addition or subtraction ?

When adding or subtracting quantities, you add the absolute uncertainties .

Q: In Edexcel IA2, what is the threshold for a measurement to be considered repeatable ?

A measurement is generally considered repeatable if its percentage uncertainty is below 5% .

Q: How can you use the accepted value to determine if a result is accurate ?

A result is accurate if the accepted value falls within your uncertainty range , or if the percentage difference is less than 5%.

Physics (YPH11) · Pearson Edexcel International A Level · 5 min read · Updated: 7 Sept 2026

Welcome to Compound Uncertainties, Precision, and Accuracy

In Unit 3, you learned how to take measurements and calculate simple uncertainties. Now that you are in IA2 (Unit 6), we take those skills to the next level. In this chapter, we will learn how to "compound" uncertainties—which means figuring out how the tiny errors in individual measurements (like mass or length) combine to affect your final calculated result (like density). Understanding this is the difference between just doing an experiment and truly understanding your data.

Note: This chapter builds on the basic measurement techniques you've seen before. For details on how to use specific instruments like micrometer screw gauges, see the chapter on "Planning an Experiment (IA2)".

1. Mastering the Vocabulary

The examiners are very specific about the words they want you to use. In Physics, "error" doesn't mean you made a mistake; it has a very specific technical meaning.

The Core Definitions:

True Value: The value that would be obtained by a perfect measurement. In reality, this is unknowable (except for fundamental constants).
Accuracy: How close your measurement is to the true value. Important: You cannot "quantify" accuracy with a single number; you can only judge it.
Precision: How much agreement there is between your repeated measurements. This is only affected by random effects.
Error: The difference between your measurement and the true value.
Uncertainty: The interval within which the true value is considered to lie (e.g., \(20.0 \pm 0.1 \text{ cm}\)).
Resolution: The smallest change in the quantity being measured that gives a perceptible change in the reading (the smallest "tick mark" on your tool).

Repeatability vs. Reproducibility:

Repeatability: You get the same results using the same operator, same equipment, and same method over a short period.
Reproducibility: Someone else gets the same results using different equipment or a different method.

Quick Analogy: Imagine an archer. If all the arrows hit the bullseye, they are accurate and precise. If all the arrows land in a tight cluster far from the bullseye, they are precise but inaccurate. If the arrows are scattered everywhere, they are neither!

2. Calculating Basic Uncertainties

Before we combine them, we need to find the uncertainty of a single variable.

For a Single Reading

If you only have one reading (or if all your repeats are exactly the same), the uncertainty is half the resolution of the instrument.

Example: A ruler has a resolution of \(1 \text{ mm}\). A single measurement of \(15.0 \text{ cm}\) has an uncertainty of \(\pm 0.5 \text{ mm}\) (or \(\pm 0.05 \text{ cm}\)).

For Repeat Readings

If you have a set of different readings, the uncertainty is half the range. To find this:

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

Alternatively, you can use the distance from the mean to the furthest reading if that is larger.

Percentage Uncertainty

To compare uncertainties of different types, we turn them into percentages:

\( \% \text{ uncertainty} = \left( \frac{\text{uncertainty}}{\text{measurement}} \right) \times 100\% \)

Note: In Unit 6, percentage uncertainties are usually given to one or two significant figures, while your processed data should usually be to three.

3. Compounding Uncertainties (The Unit 6 Essential)

This is the most important part of Unit 6. When you use your measurements in a formula, the uncertainties "add up." Follow these three golden rules:

Rule 1: Multiplication or Division

If your formula involves multiplying or dividing quantities, add the percentage uncertainties together.

Example: To find speed (\(v = \frac{d}{t}\)), if distance \(d\) has a \(2\%\) uncertainty and time \(t\) has a \(3\%\) uncertainty, the speed \(v\) has a \(2\% + 3\% = 5\%\) uncertainty.

Rule 2: Raising to a Power

If a measurement is raised to a power \(n\), multiply the percentage uncertainty by that power.

Example: The area of a circle is \(A = \pi r^2\). If the radius \(r\) has a \(3\%\) uncertainty, the area \(A\) has a \(3\% \times 2 = 6\%\) uncertainty.

Rule 3: Addition or Subtraction

If you are adding or subtracting values, add the absolute uncertainties (the actual \(\pm\) values, not the percentages).

Example: If you measure a change in temperature \(\Delta \theta = \theta_2 - \theta_1\), and each thermometer reading has an uncertainty of \(\pm 0.5^{\circ}\text{C}\), the total uncertainty in \(\Delta \theta\) is \(0.5 + 0.5 = 1.0^{\circ}\text{C}\).

Key Takeaway:

Always convert to percentage uncertainties first for anything involving multiplication, division, or powers!

4. Judging Your Results

Once you have your calculated uncertainties, examiners will ask you to comment on the experiment's success.

Is the measurement repeatable?

In the Pearson Edexcel context, if the percentage uncertainty is below \(5\%\), it suggests the measurement is repeatable.

Is the result accurate?

You can judge accuracy in two ways:

The Range Check: Does the accepted value (from a textbook or data sheet) lie within your uncertainty range? If yes, your result is accurate.
The \(5\%\) Rule: Calculate the percentage difference between your result and the accepted value. If the percentage difference is below \(5\%\), the result is usually considered accurate.

\( \% \text{ difference} = \frac{|\text{your value} - \text{accepted value}|}{\text{accepted value}} \times 100\% \)

5. Common Pitfalls and Tips

Zero Errors: Don't forget that systematic errors (like a "zero error" on a micrometer) affect accuracy but not precision. A best-fit line that misses the origin when it should pass through it is a classic sign of a systematic error.
Gradients: When finding a constant from a graph, always use a large triangle to calculate the gradient. This reduces the percentage uncertainty in your gradient value.
Significant Figures: Don't over-quote! Your final answer should not have more significant figures than the least precise piece of data you used.
Math delimiters: In your exam, always show the substitution of values into the uncertainty formulas to gain "method marks."

Quick Review Box:
- Precision = consistency of repeats.
- Accuracy = closeness to truth.
- Compounding = Add % for \(\times\) and \(\div\); Multiply % by power for \(x^n\).
- Threshold = \(5\%\) is the magic number for judging repeatability/accuracy.

For more on how to handle data using logarithms, see the chapter "Log Graphs and Data Analysis (IA2)".

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What is the difference between accuracy and precision in physics?

Accuracy is how close a measurement is to the true value, while precision is the degree of agreement between repeated measurements.

How are repeatability and reproducibility distinguished?

Repeatability refers to getting consistent results using the same operator and equipment, while reproducibility refers to obtaining similar results using different methods or equipment.

How is the uncertainty of a single reading determined?

For a single reading, the uncertainty is taken as half the resolution of the measuring instrument.

How do you calculate the uncertainty for a set of repeat readings?

The uncertainty is calculated as half the range: \( \text{Uncertainty} = \frac{\text{Max} - \text{Min}}{2} \).

What is the formula for percentage uncertainty?

\( \% \text{ uncertainty} = \left( \frac{\text{uncertainty}}{\text{measurement}} \right) \times 100\% \)

What is the rule for compounding uncertainties during multiplication or division?

When multiplying or dividing quantities, you must add the percentage uncertainties of each quantity.

What is the rule for uncertainty when a measurement is raised to a power?

Multiply the percentage uncertainty of the measurement by the power \(n\).

What is the rule for compounding uncertainties during addition or subtraction?

When adding or subtracting quantities, you add the absolute uncertainties.

In Edexcel IA2, what is the threshold for a measurement to be considered repeatable?

A measurement is generally considered repeatable if its percentage uncertainty is below 5%.

How can you use the accepted value to determine if a result is accurate?

A result is accurate if the accepted value falls within your uncertainty range, or if the percentage difference is less than 5%.

How does a systematic error, like a zero error, appear on a graph?

A systematic error typically causes a non-zero intercept on a graph where the line of best fit should theoretically pass through the origin.

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