Limitations of Physical Measurements: Mastering Errors and Uncertainty (9630)
Welcome to one of the most important practical topics in Physics! You might think measurement is simple, but in the real world, no measurement is ever truly perfect. Understanding the limitations of physical measurements—how good they are and how we account for their imperfections—is crucial for all scientists.
This chapter teaches you how to identify where things go wrong (errors), how to judge the quality of your data (precision and accuracy), and how to communicate exactly how certain (or uncertain) you are about your final results. Let's get started!
1. Identifying and Managing Errors in Experiments
Every time you take a reading, there is a chance of error. We classify these imperfections into two main types: Random Errors and Systematic Errors.
1.1 Random Errors
Random errors cause readings to spread randomly around the true value. They are unpredictable, meaning sometimes the measurement is too high, and sometimes it is too low.
- Effect: They reduce the precision of your measurement.
- Causes:
Fluctuations in reading position (parallax error).
Environmental changes (e.g., small draughts, temperature variation).
The limit of human judgment when timing or estimating a reading. - Suggestion for Removal/Reduction:
The best way to reduce the impact of random errors is to repeat the measurement several times and calculate the mean (average).
1.2 Systematic Errors
Systematic errors cause all readings to be shifted by the same amount in the same direction (either consistently too high or consistently too low).
- Effect: They reduce the accuracy of your measurement.
- Causes:
Zero error (The instrument doesn't read zero when it should, e.g., a measuring cylinder with the bottom of the meniscus above the zero mark).
Incorrect calibration of the instrument (e.g., a ruler that shrank slightly).
A faulty experimental technique (e.g., consistently measuring time late). - Suggestion for Removal/Correction:
Systematic errors cannot be reduced by averaging. You must identify the cause (check the equipment, check the method) and correct the offset (e.g., measure the zero error and subtract it from all readings).
Analogy: Shooting at a Target
Imagine aiming at the bullseye.
If your shots are spread all over the target but centred generally near the bullseye, you have random errors.
If all your shots are tightly clustered together, but 3 cm to the left of the bullseye, you have good precision but poor accuracy due to a systematic error (perhaps your sight is misaligned).
Key Takeaway: Random errors affect precision and are fixed by averaging. Systematic errors affect accuracy and are fixed by calibration and method correction.
2. Defining the Quality of Data: Precision, Accuracy, and Resolution
These terms are often confused! Let’s clarify exactly what they mean in the context of your practical work.
2.1 Resolution
Resolution refers to the smallest reading or quantity an instrument can measure.
Example: A standard meter ruler has a resolution of 1 mm (or 0.001 m). A stopwatch might have a resolution of 0.01 s.
2.2 Precision and Accuracy
- Accuracy: How close a measurement is to the true or accepted value. (High accuracy means low systematic error).
- Precision: How close repeated measurements are to each other. (High precision means low random error).
2.3 Repeatability and Reproducibility
These terms relate to the reliability of your procedure:
- Repeatability: The variation in measurements when the same person repeats the experiment using the same equipment and method over a short period.
- Reproducibility: The variation in measurements when the experiment is repeated by different people, using different equipment, or in different locations.
Did you know? If your experiment has high repeatability but low reproducibility, it suggests the equipment or method is highly sensitive to small external factors or specific set-up details that need careful standardisation.
Key Takeaway: Resolution is the limit of the device. Precision is about consistency (closeness of readings). Accuracy is about correctness (closeness to the true value).
3. Quantifying Uncertainty (\(\Delta X\))
Since no measurement is perfect, we must quote a measure of doubt, known as uncertainty, alongside our measured value.
3.1 Determining Absolute Uncertainty
The absolute uncertainty (\(\Delta X\)) has the same units as the measured quantity \(X\).
Rule 1: Uncertainty in a Single Reading (using instrument resolution)
When using a digital instrument, the uncertainty is usually \(\pm\) the smallest scale division (resolution).
When using an analogue instrument (like a ruler or thermometer), the uncertainty is often taken as \(\pm\) half the resolution.
Example: If a ruler has 1 mm resolution, the reading uncertainty is \(\pm 0.5 \text{ mm}\).
Rule 2: Uncertainty from Repeated Readings (using range)
If you take multiple readings, the best estimate for the absolute uncertainty is often taken as half the range (Maximum value - Minimum value).
\[\Delta X = \frac{(X_{\text{max}} - X_{\text{min}})}{2}\]
3.2 Fractional and Percentage Uncertainty
These are used to compare the significance of the uncertainty relative to the measurement itself.
- Fractional Uncertainty: This is the ratio of the absolute uncertainty to the measured value. \[\text{Fractional Uncertainty} = \frac{\Delta X}{X}\]
- Percentage Uncertainty: This is simply the fractional uncertainty multiplied by 100%. \[\text{Percentage Uncertainty} = \frac{\Delta X}{X} \times 100\%\]
Example: If a measured length \(L = 10.0 \text{ cm}\) has an absolute uncertainty \(\Delta L = 0.5 \text{ cm}\).
Fractional uncertainty is \(\frac{0.5}{10.0} = 0.05\).
Percentage uncertainty is \(0.05 \times 100\% = 5\%\).
Quick Review: Absolute is measured in units (e.g., m), Fractional has no units, Percentage is a ratio (\(\times 100\)).
4. Combining Uncertainties (Error Propagation)
When you calculate a final result (\(R\)) using several measured quantities (\(A, B, C\)), you must combine the individual uncertainties into a final uncertainty (\(\Delta R\)).
4.1 Combination Rule 1: Addition and Subtraction
If the final quantity \(R\) is found by adding or subtracting measured quantities (e.g., \(R = A + B\) or \(R = A - B\)), you add the absolute uncertainties.
\[\Delta R = \Delta A + \Delta B\]
Example: Measuring the length of an object by subtracting the start reading \(x_1\) from the end reading \(x_2\). If \(x_1\) and \(x_2\) both have an uncertainty of \(\pm 0.5 \text{ mm}\), the uncertainty in the final length \(\Delta L\) is \(0.5 \text{ mm} + 0.5 \text{ mm} = 1.0 \text{ mm}\).
4.2 Combination Rule 2: Multiplication and Division
If the final quantity \(R\) is found by multiplying or dividing measured quantities (e.g., \(R = A \times B\) or \(R = A / B\)), you add the fractional or percentage uncertainties.
\[\frac{\Delta R}{R} = \frac{\Delta A}{A} + \frac{\Delta B}{B}\]
Example: Calculating speed, \(v = \frac{d}{t}\). If distance \(d\) has 2% uncertainty and time \(t\) has 3% uncertainty, the uncertainty in speed \(v\) is \(2\% + 3\% = 5\%\).
4.3 Combination Rule 3: Powers
If the final quantity \(R\) involves a quantity raised to a power \(n\) (e.g., \(R = A^n\)), you multiply the fractional or percentage uncertainty by the power \(n\).
\[\frac{\Delta R}{R} = |n| \times \frac{\Delta A}{A}\]
Example: Calculating the area of a square, \(A = L^2\). If the length \(L\) has an uncertainty of 4%, the uncertainty in the area \(A\) is \(2 \times 4\% = 8\%\).
Important Note: In this AS/A Level syllabus, you do not need to combine uncertainties involving trigonometric or logarithmic functions.
Key Takeaway: Absolute uncertainties add for sums/differences. Percentage uncertainties add for products/quotients.
5. Uncertainty in Graphs and Error Bars
Graphs are powerful tools, but they must represent the uncertainty of the collected data points.
5.1 Representing Uncertainty: Error Bars
An error bar is a line drawn through a data point on a graph that shows the range within which the true value of the measurement is expected to lie.
If a data point is plotted at \((x, y)\) and the uncertainties are \(\Delta x\) and \(\Delta y\), the error bars extend:
- Vertically from \(y - \Delta y\) to \(y + \Delta y\).
- Horizontally from \(x - \Delta x\) to \(x + \Delta x\).
5.2 Determining Uncertainty in Gradient and Intercept
When you determine the gradient (\(m\)) or intercept (\(c\)) of a straight-line graph, these values also have associated uncertainties.
Step-by-Step Process:
- Draw the Line of Best Fit: This line should pass as close as possible to all data points.
- Draw the Maximum and Minimum Gradient Lines: These are the steepest (\(m_{\text{max}}\)) and shallowest (\(m_{\text{min}}\)) possible straight lines that still pass through all the error bars.
- Calculate the Gradients: Find \(m_{\text{max}}\) and \(m_{\text{min}}\) using points taken directly from the lines (not the data points).
- Calculate the Uncertainty in Gradient (\(\Delta m\)): \[\Delta m = \frac{|m_{\text{max}} - m_{\text{min}}|}{2}\]
- Calculate the Uncertainty in Intercept (\(\Delta c\)):
This is found by determining where the \(m_{\text{max}}\) and \(m_{\text{min}}\) lines cross the y-axis (\(c_{\text{max}}\) and \(c_{\text{min}}\)). \[\Delta c = \frac{|c_{\text{max}} - c_{\text{min}}|}{2}\]
Don't worry if this seems tricky at first—it requires practice. The key is ensuring your maximum and minimum lines respect the limits defined by your error bars!
Key Takeaway: Error bars show the absolute uncertainty for each point. Graph uncertainty is found by calculating half the difference between the steepest and shallowest valid lines.
6. Significant Figures and Associated Uncertainty
There is a critical link between the way you quote your uncertainty and the number of significant figures (SF) or decimal places (dp) in your final answer.
6.1 The Golden Rule for Reporting Data
The final value of a quantity should be reported such that its precision matches the precision of its absolute uncertainty.
- Uncertainty (\(\Delta X\)): The absolute uncertainty should generally be quoted to one significant figure (1 SF). (However, 2 SF is acceptable if the first digit is a '1', e.g., \(\pm 0.14\)).
- Value (\(X\)): The measured or calculated value must be rounded so that its last significant digit is in the same decimal place as the single significant figure of the absolute uncertainty.
Example Set 1: Getting the Format Right
- If calculation gives: \(T = 1.6457 \text{ s}\) and \(\Delta T = 0.083 \text{ s}\)
- Step 1 (Round Uncertainty): \(\Delta T = 0.08 \text{ s}\) (1 SF, hundredths place)
- Step 2 (Round Value): Round \(T\) to the hundredths place. \(T = 1.65 \text{ s}\)
- Final Answer: \(T = (1.65 \pm 0.08) \text{ s}\)
Example Set 2: The Decimal Place Must Match
-
If the value is \(L = 4.76 \text{ m}\) and \(\Delta L = 0.2 \text{ m}\) (tenths place).
Correct: \(L = (4.8 \pm 0.2) \text{ m}\) (The value 4.76 is rounded to 4.8, matching the tenths place). -
If the value is \(R = 12.02 \Omega\) and \(\Delta R = 1.4 \Omega\) (units place).
Correct: \(R = (12 \pm 1) \Omega\) (The value 12.02 is rounded to 12, matching the units place after rounding the uncertainty 1.4 to 1).
Common Mistake to Avoid:
Do not quote your calculated value with many decimal places when your uncertainty is large. A reading of \((15.423 \pm 5) \text{ kg}\) suggests you know the mass to three decimal places, but your uncertainty says you only know it to the nearest 5 kg! Correctly, this would be \((15 \pm 5) \text{ kg}\).
Key Takeaway: The number of significant figures in the value is determined by the decimal place of the uncertainty. Uncertainty is usually quoted to 1 SF.