Physics (9702) | Errors and uncertainties

Welcome to the World of Errors and Uncertainties!

Hey there! In Physics, getting an exact, perfect answer is almost impossible. Every measurement you take in the lab, from timing a pendulum swing to measuring a voltage, will have some inherent imperfection.

This chapter is absolutely vital because it teaches you how to recognize those imperfections (called errors) and how to quantify how good your measurements are (using uncertainties). Mastering this is crucial for Paper 3 (Practical Skills) and Paper 5 (Planning, Analysis and Evaluation).

What you will learn:

• The difference between types of errors (random and systematic).
• How to distinguish between precision and accuracy.
• The essential rules for calculating the total uncertainty in a complex experiment.

1. The Two Main Types of Errors in Measurement

When you take a reading, the deviation of your reading from the true value is called an error. Errors are generally categorized into two main types: Random and Systematic. Understanding the difference is key to improving your experimental work.

1.1 Random Errors

Definition: Random errors are unpredictable variations in measured data that lead to readings being scattered randomly around the true value. They occur due to factors that change unpredictably during the experiment.

Causes of Random Errors:

• Parallax error: Reading a scale from a different angle each time.
• Fluctuations: Small, unpredictable changes in the environment (e.g., air currents, temperature changes).
• Human reaction time: Inconsistencies when starting or stopping a timer.

How to Minimise Random Errors:

Since random errors scatter the results evenly (some too high, some too low), they can be significantly reduced by:

1. Taking multiple measurements of the same quantity.
2. Calculating the mean (average) of these measurements. (The more readings you take, the closer your mean is likely to be to the true value.)

Memory Aid: R for Random, R for Repeat!

1.2 Systematic Errors

Definition: Systematic errors cause all readings to be shifted consistently in the same direction (either all too high or all too low). They are usually associated with the equipment or the design of the experiment.

Causes of Systematic Errors:

• Faulty calibration: Using a balance that always reads 0.5 g too high.
• Flawed technique: Always viewing a liquid meniscus from above, resulting in consistently low volume readings.
• Zero error: This is a common and important type of systematic error.

What is a Zero Error?

A zero error occurs when the measuring instrument does not read exactly zero when it should.

Example: A micrometer screw gauge reads \(+0.02 \text{ mm}\) when the jaws are closed. Every reading taken with that micrometer will be \(0.02 \text{ mm}\) too high. To correct this, you must subtract the zero error from your final reading.

How to Minimise Systematic Errors:

Repeating the measurement (like you do for random errors) won't help here, because every reading will still be shifted by the same amount. To eliminate systematic errors, you must:

1. Identify and remove the fault (e.g., recalibrate the instrument).
2. If the error cannot be fixed (like a non-zero start point), apply a correction to all recorded data.

Key Takeaway: Random errors affect precision and are averaged out. Systematic errors affect accuracy and must be identified and corrected.

2. Precision and Accuracy

Don't worry if these two terms seem tricky—they are often confused in everyday language, but in Physics, they have distinct meanings.

2.1 Accuracy

Definition: Accuracy refers to how close a measurement is to the true value of the quantity being measured.

High accuracy means low systematic error.

2.2 Precision

Definition: Precision refers to how close successive measurements are to each other. It also relates to the smallest division on the measuring instrument (the resolution).

High precision means low random error.

Analogy: The Dartboard

• Accurate & Precise: All darts hit very close together, right in the bullseye. (Good equipment, good technique).
• Precise but Inaccurate: All darts hit very close together, but far away from the bullseye (e.g., in the corner). (This indicates low random error but high systematic error—perhaps the target was shifted, like a zero error).
• Inaccurate & Imprecise: Darts are scattered all over the board. (High random error).

Key Takeaway: You can be precise without being accurate (if you have systematic errors), but generally, you cannot be accurate without being precise.

3. Assessing Uncertainty in a Single Measurement

Uncertainty is the range of values within which the true value of the measurement is expected to lie. We express a measurement \(Q\) as \(Q \pm \Delta Q\), where \(\Delta Q\) is the uncertainty.

3.1 Absolute Uncertainty (\(\Delta Q\))

The simplest way to estimate the absolute uncertainty for a reading taken using an analog device (like a ruler or thermometer) is usually taken as:

• Half the smallest scale division (e.g., if a ruler has a smallest division of 1 mm, the uncertainty is \(\pm 0.5 \text{ mm}\)).

For digital devices, the uncertainty is usually:

• The smallest digit shown (e.g., if a voltmeter reads 5.23 V, the uncertainty is \(\pm 0.01 \text{ V}\)).

Note: When measuring a length using a ruler where you read both the start and end points, the uncertainty is often taken as the resolution (smallest division) because you introduce uncertainty at both ends of the measurement.

3.2 Fractional and Percentage Uncertainty

To compare the quality of different measurements, we use relative uncertainties:

• Fractional Uncertainty: \(\frac{\text{Absolute Uncertainty}}{\text{Measured Value}} = \frac{\Delta Q}{Q}\)
• Percentage Uncertainty: Fractional Uncertainty \(\times 100\%\)
  $$\text{Percentage Uncertainty} = \frac{\Delta Q}{Q} \times 100\%$$

Example: If a mass \(M = 50 \text{ g}\) has an absolute uncertainty of \(\pm 1 \text{ g}\):
Fractional uncertainty is \(\frac{1}{50} = 0.02\)
Percentage uncertainty is \(0.02 \times 100\% = 2\%\)

Did you know? In Physics, we strive to keep percentage uncertainties below 5%. If your uncertainty is much higher, you probably need a better method or better equipment!

4. Combining Uncertainties in Derived Quantities (Propagation)

In an experiment, you rarely measure the final quantity directly. Instead, you measure several quantities (A, B, C...) and use a formula to calculate the final result (Q). The total uncertainty in Q must account for the uncertainties in A, B, and C.

The syllabus requires you to use the technique of simple addition of absolute or percentage uncertainties (Syllabus 1.3.3).

Rule 1: Addition and Subtraction

When calculating a quantity \(Q\) by adding or subtracting measured values, we add the absolute uncertainties.

If \(Q = A + B\) or \(Q = A - B\), then:

$$ \Delta Q = \Delta A + \Delta B $$

Example: You measure the extension of a spring by reading the initial length (\(L_1\)) and final length (\(L_2\)).
\(L_1 = (10.0 \pm 0.1) \text{ cm}\)
\(L_2 = (15.5 \pm 0.1) \text{ cm}\)
Extension \(E = L_2 - L_1 = 5.5 \text{ cm}\)
Absolute uncertainty in \(E\): \(\Delta E = \Delta L_1 + \Delta L_2 = 0.1 \text{ cm} + 0.1 \text{ cm} = 0.2 \text{ cm}\)
Result: \(E = (5.5 \pm 0.2) \text{ cm}\)

Think of it this way: when you subtract two values, you are still introducing two potential sources of error, so the uncertainties always add up!

Rule 2: Multiplication and Division

When calculating a quantity \(Q\) by multiplying or dividing measured values, we add the percentage (or fractional) uncertainties.

If \(Q = A \times B\) or \(Q = \frac{A}{B}\) or \(Q = \frac{A \times B}{C}\), then:

$$ \frac{\Delta Q}{Q} = \frac{\Delta A}{A} + \frac{\Delta B}{B} + \frac{\Delta C}{C} $$

(The same rule applies if you use percentage uncertainties: %Uncertainty Q = %Uncertainty A + %Uncertainty B + ...)

Rule 3: Powers (or Indices)

If a quantity \(A\) is raised to a power \(n\) to find \(Q\):

If \(Q = A^n\), we multiply the percentage (or fractional) uncertainty in A by the power \(n\).

$$ \frac{\Delta Q}{Q} = n \times \frac{\Delta A}{A} $$

Example: If the radius of a circle \(r = (2.0 \pm 0.1) \text{ m}\). What is the percentage uncertainty in the Area \(A = \pi r^2\)?
Percentage uncertainty in \(r\): \(\frac{0.1}{2.0} \times 100\% = 5\%\)
Since Area depends on \(r^2\) (\(n=2\)), the percentage uncertainty in Area is:
\(\text{% Unc in } A = 2 \times (\text{% Unc in } r) = 2 \times 5\% = 10\%\)

Note: Constants like \(\pi\) (pi) or the number 2 in the formula \(E_k = \frac{1}{2}mv^2\) have zero uncertainty, so they are ignored in the uncertainty calculation.

Step-by-Step Example of Combining Uncertainties

Let's find the uncertainty in Density (\(\rho\)), where \(\rho = \frac{M}{V}\).
Suppose Mass \(M = (20.0 \pm 0.5) \text{ g}\) and Volume \(V = (5.0 \pm 0.1) \text{ cm}^3\).

1. Calculate the actual value: $$\rho = \frac{20.0}{5.0} = 4.0 \text{ g/cm}^3$$
2. Calculate Percentage Uncertainties for M and V: $$\text{% Unc } M = \frac{0.5}{20.0} \times 100\% = 2.5\%$$ $$\text{% Unc } V = \frac{0.1}{5.0} \times 100\% = 2.0\%$$
3. Add Percentage Uncertainties (Rule 2: Division): $$\text{% Unc } \rho = \text{% Unc } M + \text{% Unc } V = 2.5\% + 2.0\% = 4.5\%$$
4. Convert back to Absolute Uncertainty: $$\Delta \rho = \text{% Unc } \rho \times \rho = 0.045 \times 4.0 \text{ g/cm}^3 = 0.18 \text{ g/cm}^3$$
5. State the final result (rounded to one significant figure for uncertainty): $$\Delta \rho \approx 0.2 \text{ g/cm}^3$$ $$\rho = (4.0 \pm 0.2) \text{ g/cm}^3$$

Key Takeaway: For multiplication/division/powers, ADD the percentage uncertainties. For addition/subtraction, ADD the absolute uncertainties.