Systematic and Random Errors

Every measurement in physics is imperfect. No matter how careful you are, no matter how expensive your apparatus, the value you record is never exactly the "true" value of the quantity being measured. Physicists are professional realists about this: they divide the discrepancy between "measured" and "true" into two distinct categories — systematic errors and random errors — and treat each very differently.

OCR's A-Level Physics A specification requires you to: define systematic and random errors; distinguish them; give examples; understand how each can be detected and reduced; and carry these ideas forward into uncertainty propagation (Lessons 6–8). This lesson lays the conceptual groundwork.

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The Three Key Words: Error, Uncertainty, and Accuracy

Before we classify errors, it is essential to get the vocabulary right. OCR is strict about these definitions.

• Error — the difference between a measured value and the true value of a quantity. An error is a specific numerical discrepancy that (in principle) exists but may not be known.
• Uncertainty — the range within which you believe the true value lies, based on your measurement procedure. It is a quantitative statement about how much confidence to place in a result.
• Accuracy — the degree to which a measurement is close to the true value. High accuracy means small error.

Note that "error" in physics does not mean "mistake". A measurement subject to a systematic error is not a blunder; it is a measurement carrying an unavoidable bias that needs to be identified and corrected.

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Random Errors

Random errors are fluctuations that cause repeated measurements to scatter around a central value. They are inherent to the measurement process and arise from:

• Limitations of the observer (timing with a stopwatch, reading a scale)
• Fluctuations in environmental conditions (draughts, vibrations, temperature changes)
• Random noise in electronic instruments
• The quantum nature of the measurement in extreme cases

Because random errors scatter both above and below the true value, they can be reduced by averaging. If you take N repeated measurements, the standard error in the mean is approximately 1/√N of the standard deviation of the individual measurements. Doubling the number of measurements roughly halves the random uncertainty — but only approximately.

Example: Timing a Pendulum

Imagine timing 10 oscillations of a pendulum with a hand-operated stopwatch. You might get:

14.5, 14.8, 14.3, 14.7, 14.6, 14.4, 14.8, 14.5, 14.6, 14.5 seconds

The values scatter around a mean of 14.57 s. No single value equals the "true" period; but the mean is a much better estimate than any individual reading. This is the classic signature of random error — the scatter averages away with repetition.

Reducing Random Errors

• Repeat measurements. Take several readings and calculate the mean.
• Use automated timing. Light gates are far more precise than stopwatches because they eliminate human reaction time.
• Use instruments with finer resolution. A micrometer (0.01 mm) is better than a ruler (1 mm) for measuring a wire diameter.
• Minimise environmental influences. Close doors and windows to reduce air currents, for example.
• Take many samples and plot them. A graph averages out individual random deviations.

Exam Tip: When asked how to reduce random error, the magic words are "repeat and take the mean" and "use an instrument of higher resolution" or "use automated timing such as light gates". These phrases appear verbatim in many mark schemes.

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Systematic Errors

Systematic errors are biases that shift every measurement in the same direction by the same (or a similar) amount. They arise from problems with the apparatus or method, not from the random limits of reading. Typical sources:

• Calibration errors — a ruler whose "zero" is not at the end; a balance that reads 0.05 g when empty.
• Zero errors — an ammeter that reads 0.02 A with no current flowing.
• Parallax errors — consistently reading a scale from a non-perpendicular angle.
• Environmental biases — a thermometer calibrated at sea level being used at altitude.
• Poor technique — always starting the stopwatch slightly late.

Systematic errors cannot be reduced by averaging. Repeating a measurement ten times with a faulty ammeter simply gives ten identically-biased readings. The mean is no closer to the truth than a single reading.

Example: A Ruler Worn at the Zero

Suppose you measure the length of a pencil with a plastic ruler. The "0" mark on the ruler is 1 mm inside the physical edge of the ruler, due to wear. Every measurement you take will be 1 mm too long. Ten measurements averaged will still be 1 mm too long. No amount of repetition helps.

Example: Zero Error on an Ammeter

An analogue ammeter reads +0.02 A when disconnected. You measure a current and get 0.48 A. The true value is 0.48 − 0.02 = 0.46 A. The systematic error is +0.02 A, and subtracting it yields the correct answer.

Reducing Systematic Errors

• Check for zero errors before starting — and either adjust the instrument or subtract the zero reading from every measurement.
• Calibrate instruments against a known standard (e.g. check a thermometer against pure melting ice at 0 °C and boiling water at 100 °C).
• Change the method. If a systematic error is inherent to a technique (e.g. end-correction in a resonance tube), use a different method to measure the same quantity.
• Use instruments of known accuracy — e.g. traceable weights, calibrated multimeters.
• Identify and subtract the known bias from each reading.
• Use the "difference method" — for example, in a resonance tube, measure L₂ − L₁ so that the end-correction cancels.

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Visualising the Difference

Imagine shooting arrows at a target. You fire five arrows.

graph LR
    A[Small random, small systematic<br/>arrows clustered at centre<br/>ACCURATE and PRECISE]
    B[Small random, large systematic<br/>arrows clustered but off-centre<br/>PRECISE but NOT ACCURATE]
    C[Large random, small systematic<br/>arrows scattered around centre<br/>ACCURATE on average, NOT PRECISE]
    D[Large random, large systematic<br/>arrows scattered and off-centre<br/>NEITHER ACCURATE NOR PRECISE]