Experimental Errors: Random, Systematic, and Zero Errors

Experimental errors are the differences between measured values and the true value; random errors cause readings to scatter unpredictably about the true value and can be reduced by averaging many readings, while systematic errors shift all readings in the same direction and cannot be removed by repeating; a zero error is a systematic error in which an instrument reads a nonzero value when it should read zero. In brief: random errors reduce precision and show as scatter; systematic errors reduce accuracy and show as a consistent offset (for example, a graph line that does not pass through the origin when it should). A zero error is corrected by subtracting it from every reading. Systematic errors are found by comparing with a different method or a standard, or by calibration.

Every measurement has errors, and a good experimenter knows their types, tries to reduce them, and takes them into account. This guide describes random and systematic errors and zero errors with examples, explains how to detect and reduce each type, compares them in a table, works through numerical examples (including a micrometer with a zero error) and corrects common mistakes.

What are random errors?

Random errors are unpredictable variations in readings, which are equally likely to be above or below the true value. Causes: the limited ability of the experimenter to read an instrument (judging a pointer between scale divisions), the reaction time in starting and stopping a stopwatch (varying from one attempt to the next), small changes in conditions (drafts, vibrations, temperature fluctuations), electrical noise in sensors, and the random nature of radioactive decay. Properties: they give scatter in repeated readings; averaging many readings reduces them (the uncertainty of the mean decreases as 1/√N for N readings); they limit the precision of the result. Reduction: take many repeated measurements and use the mean; use an instrument with a finer resolution; measure a larger quantity (the time for 20 swings instead of one); and use automatic data logging (light gates instead of a stopwatch). On a graph, random errors cause points to be scattered about the line of best fit.

What are systematic errors?

Systematic errors are consistent: all readings are shifted by the same amount (an offset) or the same proportion (a scale factor) in the same direction. Causes: a badly calibrated instrument (a clock that runs fast, a ruler with a worn end, a thermometer reading 2 °C too high); a flaw in the method (heat losses to the surroundings in calorimetry, always leading to a lower measured specific heat capacity or temperature rise); parallax when always reading the scale from the same side; neglecting the effect of the weight of a hanger or air resistance; and the experimenter’s consistent bias. Properties: they do not show up in the scatter, so repeating the measurement does not reveal or reduce them; they affect the accuracy (closeness to the true value). Detection and reduction: calibrate the instruments against a standard; compare the results with those from a different method; test the apparatus with a known value; check for zero errors; identify the physical causes (insulation to reduce heat losses); and consider the sign of the likely error. On a graph, a systematic offset appears as an intercept where the line should pass through the origin; a scale-factor error changes the gradient.

What are zero errors?

A zero error occurs when an instrument does not read zero when the true value is zero. Examples: a micrometer or vernier caliper that reads 0.02 mm with the jaws closed (a positive zero error); an electronic balance that has not been tared and reads 0.3 g with nothing on it; a voltmeter or analog ammeter with the pointer not at zero; a meter ruler with a worn end; a stopwatch that was not reset. The correction is made by subtracting the zero error from all readings: true reading = observed reading − zero error (with the sign of the zero error: positive if it reads above zero, negative if below). A zero error is a type of systematic error that is constant for all readings (an offset). Where possible, the instrument is adjusted to zero before use (“taring” a balance, resetting the pointer); otherwise, the zero reading is recorded before and after the measurements and used to correct them.

Error types compared

Worked example: identifying and correcting errors

1. Micrometer. With the jaws closed, a micrometer reads +0.02 mm. A wire measured with it reads 1.35 mm. The true diameter = 1.35 − 0.02 = 1.33 mm. If the closed reading had been −0.03 mm (below zero), the true diameter = 1.35 − (−0.03) = 1.38 mm.
2. Balance. An electronic balance reads 0.3 g with nothing on it. A sample reads 25.8 g. The true mass = 25.8 − 0.3 = 25.5 g. The percentage error of not correcting is 0.3 ÷ 25.5 = 1.2 percent.
3. Random scatter. Five readings of the time for 10 oscillations of a pendulum: 20.1, 20.4, 19.9, 20.3, 20.0 s. The mean is 20.14 s, the scatter (half-range) is (20.4 − 19.9) ÷ 2 = 0.25 s. The random error is reduced by taking the mean: the uncertainty of the mean is about 0.25 ÷ √5 = 0.11 s (an estimate).
4. Systematic offset in a graph. In an experiment on Ohm’s law, the graph of V against I for a resistor has a positive intercept on the V axis of 0.15 V instead of passing through the origin. This points to a systematic error (a zero error of the voltmeter of +0.15 V, or a contact potential). Correcting it by subtracting 0.15 V from all voltages gives a line through the origin. The gradient (resistance) is unaffected by an offset, but an error in the scale of the meter would change it.
5. Calorimetry. A student heats 0.200 kg of water with a 50 W heater for 120 s. The expected energy: 50 × 120 = 6,000 J, and ΔT = 6,000 ÷ (0.200 × 4,180) = 7.2 K. The measured rise is 6.0 K, always lower in repeated trials. The consistent shortfall of 1.2 K (17 percent) points to a systematic error (heat lost to the surroundings and the container), not random scatter. Lagging the container reduces it.
6. Distinguishing the cases. A set of readings is 9.8, 9.9, 9.8, 9.9 and 9.8 against a known value of 10.5: the results are precise (small scatter of 0.1) but inaccurate (offset of −0.7): a systematic error. Readings 9.2, 10.9, 10.1, 9.7 and 10.6 average 10.1, scattered widely: random errors dominate, and the mean is close to the true value.

How were errors treated historically?

Carl Friedrich Gauss developed the theory of random errors and the method of least squares in the early 1800s, introducing the normal (Gaussian) distribution of errors, from the analysis of astronomical observations. Friedrich Bessel recognized in 1823 the “personal equation”, the systematic differences between observers timing the transits of stars. Today, the Guide to the Expression of Uncertainty in Measurement (GUM, 1993) distinguishes between uncertainties evaluated statistically (Type A) and by other means (Type B), and the metrological institutes calibrate instruments against national standards.

Which mistakes are common?

Mistake 1: Calling any mistake or human blunder a “random error”

Blunders (reading the wrong scale, miscalculation) are not errors in the technical sense; they should be avoided or the result discarded.

Mistake 2: Thinking that repeating readings removes systematic errors

Repeated readings with the same bias give the same offset. Use calibration or another method.

Mistake 3: Adding the zero error instead of subtracting it

Corrected reading = observed reading − zero error (with sign).

Mistake 4: Blaming “human error” without being specific

Explain the specific source (for example, the reaction time of about 0.2 s in starting a stopwatch) and its effect (random scatter in the times), and the improvement (light gates or a longer time).