Introduction to Combining Uncertainties

In Physics, we rarely just take one measurement and stop there. Usually, we take several different measurements and plug them into a formula to find a final answer. For example, to find the speed of a trolley, you need to measure distance and time.

But here is the catch: every single measurement has its own uncertainty. When you use these measurements in a calculation, those uncertainties don't just disappear—they "pile up" or combine. In this chapter, we will learn the simple rules for combining these uncertainties so you can be confident in your final scientific results.

Note: If you need a refresher on what uncertainty actually is, check out the chapter "Accuracy, precision and uncertainty".

The Golden Rule: Identifying the "Worst Offender"

Before we dive into the math, remember this key concept from the OCR syllabus: always identify the largest source of uncertainty.

In many experiments, one measurement will be much "fuzzier" than the others. For example, if you are measuring the width of a wire with a high-precision micrometer (very low uncertainty) but measuring the time it takes to fall with a manual stopwatch (high uncertainty due to reaction time), the stopwatch is your "worst offender." Improving the precision of the micrometer won't help much if the timing is still wildly uncertain!

Quick Tip: In exam questions, if you are asked how to improve an experiment, look for the measurement with the largest percentage uncertainty and suggest a better way to measure that specific thing.

Rule 1: Adding or Subtracting Measurements

When you add or subtract two measurements, you add the absolute uncertainties together.

Imagine you are measuring the total length of two pieces of wood placed end-to-end:

  • Piece A: \(10.0 \pm 0.1 \text{ cm}\)
  • Piece B: \(5.0 \pm 0.2 \text{ cm}\)

To find the total length: \(10.0 + 5.0 = 15.0 \text{ cm}\).
To find the total uncertainty: \(0.1 + 0.2 = 0.3 \text{ cm}\).
Final answer: \(15.0 \pm 0.3 \text{ cm}\).

Important! Even if you are subtracting the numbers, you still add the uncertainties. Uncertainty never cancels out; it always grows!

Key Takeaway:

For \(y = a + b\) or \(y = a - b\), the uncertainty in \(y\) is \(\Delta y = \Delta a + \Delta b\).

Rule 2: Multiplying or Dividing Measurements

When you multiply or divide measurements, you cannot just add the absolute uncertainties (like \(0.1 \text{ cm}\)). Instead, you must add the percentage uncertainties.

Step-by-step process:
1. Calculate the percentage uncertainty for each measurement.
2. Add these percentages together to get the total percentage uncertainty.
3. (Optional) Convert that total percentage back into an absolute number for your final answer.

Example: Finding the area of a rectangle
Length = \(2.0 \pm 0.1 \text{ m}\) (This is a \(5\%\) uncertainty)
Width = \(5.0 \pm 0.5 \text{ m}\) (This is a \(10\%\) uncertainty)

Area = \(2.0 \times 5.0 = 10.0 \text{ m}^2\).
Total percentage uncertainty = \(5\% + 10\% = 15\%\).
Final answer: \(10.0 \text{ m}^2 \pm 15\%\).

Key Takeaway:

For \(y = a \times b\) or \(y = a / b\), the \(\% \text{ uncertainty in } y = \% \text{ uncertainty in } a + \% \text{ uncertainty in } b\).

Rule 3: Raising to a Power

When a measurement is squared, cubed, or raised to any power \(n\), you simply multiply the percentage uncertainty by that power.

Think of it this way: if you calculate the area of a square (\(\text{Side} \times \text{Side}\)), you are multiplying the same measurement twice. So, you add the percentage uncertainty to itself.

Example: Kinetic Energy
The formula for kinetic energy is \(E_k = \frac{1}{2} m v^2\).
If the uncertainty in velocity \(v\) is \(3\%\), the uncertainty contributed by \(v^2\) is \(3\% \times 2 = 6\%\).

Don't worry! Constants (like the \(\frac{1}{2}\) in the formula above) have no uncertainty, so you just ignore them when combining uncertainties.

The "Extreme Value" Method

For your OCR Physics B course, you don't always need complex formulas. The syllabus suggests a practical "cheat code" called the extreme value method. This is especially useful if you find the rules above confusing.

To find the uncertainty of a calculated value, you can:

  1. Calculate the "best" value (using the means).
  2. Calculate the "maximum" possible value (using the extremes of your measurements).
  3. The difference between the "best" and the "maximum" is your uncertainty.

Example:
You measure a voltage \(V = 5.0 \pm 0.2 \text{ V}\) and a current \(I = 2.0 \pm 0.1 \text{ A}\).
You want to find Resistance \(R = V / I\).

Best value: \(5.0 / 2.0 = 2.5 \text{ } \Omega\).
Max value: To get the biggest \(R\), use the biggest \(V\) and the smallest \(I\):
\(R_{max} = 5.2 / 1.9 = 2.74 \text{ } \Omega\).
Uncertainty: \(2.74 - 2.5 = 0.24 \text{ } \Omega\).
Final Result: \(2.5 \pm 0.2 \text{ } \Omega\) (rounded to appropriate sig figs).

Common Pitfalls to Avoid

  • Forgetting to convert: Don't add an absolute uncertainty (like \(0.2 \text{ cm}\)) to a percentage uncertainty (like \(5\%\)). They are different "languages"—convert them so they match!
  • Subtracting uncertainties: Remember, when you subtract measurements, the error still gets bigger. Always add the uncertainties.
  • Ignoring the power: If a value is squared in a formula, its uncertainty is twice as important!

Quick Review Summary

Adding/Subtracting: Add the absolute uncertainties (\(\Delta y = \Delta a + \Delta b\)).
Multiplying/Dividing: Add the percentage uncertainties (\(\% \Delta y = \% \Delta a + \% \Delta b\)).
Powers: Multiply the percentage uncertainty by the power (\(\% \Delta y = n \times \% \Delta a\)).
Extreme Values: If in doubt, calculate the "worst-case scenario" and find the difference from the mean.