Random Error, Systematic Error: Which One Repeating Actually Fixes

A measurement is never just a number. It is a number plus a claim about how far that number might be from the truth, and the second part is the part people skip.

Start with the awkward part. Measurement error is the difference between a measured value and its true value, and the true value is by definition unknown. You cannot point at your own error, because pointing requires knowing the answer.

That sounds like a dead end, and in the strict sense it is. What you can do is estimate it, and then report the estimate alongside the measurement. A length written as 32.3 plus or minus 0.5 centimeters is making two claims at once. Here is the number, and here is how far I think it might be wrong.

Some Error Is Built Into the Instrument

Error is built into the act of measuring. It is not a sign that somebody did something wrong. A ruler marked in whole centimeters gives lengths that are several millimeters out, every time, however careful the person using it is.

A micrometer resolves to about 0.01 mm, and that resolution is a property of the tool rather than of the person using it — Image: Glenn McKechnie, CC BY-SA 2.0, via Wikimedia Commons

This is worth separating from the idea of a faulty instrument. There are instruments that are broken, and there are instruments that work perfectly and still cannot tell you more than their graduations allow. The second case is the ordinary one, and it sets a floor under what any amount of care can achieve.

The scatter is the random error, and the line through it is an estimate of a value nobody can see — Image: M. W. Toews, CC BY 2.5, via Wikimedia Commons

The picture above is what almost every set of measurements looks like. Individual readings sit above and below some central value. The spread is the error you can see. The central value is the error you cannot, and the gap between them is the part you are really trying to estimate.

The Two Errors Need Opposite Treatment

Observations are marred by two distinct kinds of error, and the distinction is not a matter of degree. It decides what to do next.

	Random error	Systematic error
Where it comes from	the measurement process itself: where you place a ruler, how you read a meniscus, when you time something	a cause that acts the same way every time: a miscalibrated instrument, a zero offset, a technique that always overshoots
What repeating does	reduces it, and you can calculate by how much	nothing. Repeated identical measurements do not reduce it at all
What it damages	precision, the agreement between readings	accuracy, the closeness to the true value, while leaving precision intact
How you find it	measure several times and look at the spread	compare against a standard of known accuracy
How you fix it	take more readings, and improve technique	find the cause and remove it, or calibrate against a reference

The two kinds of error side by side. Only one of them responds to effort

Why averaging cannot rescue a bad instrument

This is the single most useful thing to know about the distinction. If an instrument reads 25 grams heavy every time, then a hundred readings average out to a number that is still 25 grams heavy. The average has improved. The answer has not moved.

Random error is mitigated by repeated measurement. Systematic error must instead be actively avoided, because it arises from causes that constantly act in the same way and always alter the result in the same direction. More readings from a biased instrument give you a more confident wrong answer, which is worse than an uncertain one because nothing in the data warns you.

Calibration is comparison, not adjustment

Since systematic error can only be found by comparison, that process needs a name, and the name has a strict meaning that is widely ignored.

Calibration means checking what a device reads against something whose accuracy you already know. That reference might be another measuring device. It might be a device that makes the thing being measured, such as a voltage or a sound tone. Or it might be a plain object, such as a metre rule.

The comparison can end three ways. No significant error is found. Or a significant error is found and left alone. Or the device is adjusted to bring the error to an acceptable level.

Strictly speaking, calibration is only the first step, the comparison itself, and does not include the adjustment that might follow. The distinction is not pedantic. An instrument adjusted without a record of its state beforehand has had its calibration destroyed, because there is now no way to check the adjustment later.

What Repetition Is Actually Worth

If repeating a measurement reduces random error, the obvious next question is by how much, and the answer is one of the most useful numbers in practical science.

A wider spread is a larger standard deviation, and the standard deviation is in the same unit as the data — Image: JRBrown, public domain, via Wikimedia Commons

The standard deviation measures how far the values of a set sit from their own average. A low one means the values cluster near the average, and a high one means they spread wider. It is written as the Greek letter sigma. Unlike the variance, it is in the same unit as the data itself, which is why it is the form anyone reports.

For a normal distribution, about 68 percent of values fall within one standard deviation, 95 percent within two, and 99.7 within three — Image: Inductiveload, public domain, via Wikimedia Commons

Now the part that answers the original question. The standard error of a statistic is the standard deviation of its sampling distribution. Put plainly, it is how much repeated samples of the same size would disagree with each other.

For a mean, it reduces to a simple rule. The standard error of the mean equals the standard deviation divided by the square root of the sample size. This holds because the variance of the sampling mean distribution equals the variance of the population divided by the sample size.

So the gain is real but it collapses fast. Nine readings give one third of the standard error of a single reading. To halve the standard error you need four times as many measurements. To get it down by a factor of ten you need a hundred. Repetition is the cheapest tool available for random error and it is still worth more thought than most people give it.

The instrument in the original example: repeating on a balance this good narrows the spread, and cannot move a zero offset — Image: US DEA, public domain, via Wikimedia Commons

Carrying the Error Into the Answer

Most reported quantities are not measured directly. They are calculated from several measurements, and their uncertainty has to be carried through the same calculation.

This is propagation of uncertainty: the effect that one variable's uncertainty has on a function built from it. The inputs are usually experimental measurements, and their uncertainties come from limits such as instrument precision. They spread into the answer because the function combines them.

The uncertainty can be written several ways. It may be given as an absolute error, or as a relative error, which is usually expressed as a percentage. Most commonly it is quantified by the standard deviation, and the value and its error are written as an interval, x plus or minus u.

The general method is to give the probability distribution of the uncertainty rather than a single number. For a normal distribution, the 68 percent confidence limits sit at about one standard deviation either side of the central value. The interval covers the true value in about 68 percent of cases.

When the inputs are not independent

The simple rules assume the inputs do not drag each other around. When they do, covariance has to be taken into account, and there are two separate reasons it can arise.

The first is that the measurement errors themselves are linked. The second is that the underlying values are linked across the group. That makes the uncertainties in group averages linked too. Ignore either one and you understate the real uncertainty.

For anything nonlinear, the standard approach is sampling by the Monte Carlo method. Interval arithmetic is the alternative when only bounds are wanted. Instead of a single number, it works with a range. A function applied to that range gives back a range holding every possible result. It is widely used in scientific software to track rounding errors and the uncertainty in physical parameters.

Digits that carry no information the instrument did not supply should never reach the reader — Image: Swpb, CC BY-SA 4.0, via Wikimedia Commons

This is also why a result should never claim more digits than the instrument resolved. Significant figures are the digits that are determined by the resolution of the instrument. Digits that are uncertain but still meaningful are counted, and so are spurious digits produced by a calculation more precise than the data going into it.

Accuracy, Trueness, and Two Meanings of Precision

Accuracy is where the words get slippery. The trouble is that the term really does have two meanings in use, and both of them are correct.

The International Organization for Standardization defines a related measure, trueness. It is the closeness of agreement between the mean of many test results and the accepted reference value.

Word	What it actually measures	What it depends on
Precision	a description of random error: how close the measurements are to one another	statistical variability, and nothing else
Trueness	a description of systematic error: how close the mean of many results is to the reference value	statistical bias of the central tendency
Accuracy, common use	the same thing as trueness. Independent of precision, so a set of data can be accurate, precise, both, or neither	systematic error only
Accuracy, ISO use	the combination of trueness and precision together, so that high accuracy requires both	both kinds of error at once

Why "accurate but not precise" is not a contradiction: the word is being used in the first sense, where it ignores precision entirely

A confidence interval does not mean what it sounds like

A confidence interval gives bounds that are likely to hold the true value of something you do not know. It is built with a set confidence level, usually 95 percent.

It does not follow that there is a 95 percent chance the true value falls inside this particular interval. That reading belongs to the Bayesian credible interval instead. The confidence level describes the long-run behavior of the method. If the same sampling procedure were repeated a hundred times on the same population, about 95 of the intervals would be expected to contain the true mean.

The difference is that the true population mean is treated as a fixed but unknown constant, while the interval endpoints are random variables that move around from sample to sample.

Repeatability is not reproducibility

Precision itself splits in two, and the split is where laboratory disagreements live.

Repeatability is the closeness of agreement between successive measurements of the same quantity under the same conditions. In practice that means one person, one instrument, on the same item, in a short period of time. This is the kind of precision a repeated reading in a single session measures, and it is the only kind that averaging improves.

Reproducibility is a different and harder thing. Other researchers, using the same method, should get the same answer. Only after one or more successful replications should a result count as scientific knowledge.

A result can be perfectly repeatable and completely irreproducible. That gap is why the same experiment gives different answers in different labs. No amount of repeated measuring in one place will reveal it.

A systematic error is only traceable if somebody has already checked the step below — Image: Imminent77, CC BY-SA 4.0, via Wikimedia Commons

Measuring a mass by balancing it against a local gravitational acceleration, at the far end of the chain — Image: Jennifer Lauren Lee, for NIST, public domain, via Wikimedia Commons

The chain once ended in a single physical cylinder kept in Paris — Image: Greg L, CC BY-SA 3.0, via Wikimedia Commons

Metrology is the science of measurement. It rests on three jobs that overlap: defining units, making those units in practice, and keeping measurements tied to reference standards. That last one is traceability, and it is the answer to the systematic error problem. It is a chain, and someone has to check every link in it.

That chain is also why the old kilogram is a warning. For most of the twentieth century, the kilogram was defined as one solid cylinder of platinum-iridium kept in Paris. Physical standards like that can be lost, damaged, or changed, and they carry error that better science cannot shrink. That is exactly the failure the rest of the chain exists to catch.

Key Terms in Plain English

These are the words in this article that would send you to a dictionary. Each one is given here the way it is actually used above.

Term	What it means here
Measurement error	the difference between a measured value and its true value, which is unknown by definition. That is why error is always estimated, never read off directly
True value	the value a quantity actually has. It is the thing error is measured against, and nobody can observe it
Uncertainty	the quantified estimate that travels with a measurement, written as x plus or minus u
Random error	error that changes from reading to reading. Repetition reduces it, and the amount is calculable
Systematic error	error from a cause acting the same way every time. Repetition does not reduce it at all
Precision	how close repeated measurements are to each other. It describes random error and nothing else
Accuracy	how close a result is to the true value. In common use it means the same as trueness; in ISO use it means trueness and precision together
Trueness	the ISO term for the closeness of the mean of many results to the reference value, which is systematic error
Statistical bias	the systematic pulling of a central tendency away from the true value, in one consistent direction
Standard deviation	how far a set of values sits from its own average, written sigma. In the same unit as the data, unlike the variance
Standard error	the standard deviation of a sampling distribution. For a mean it equals the standard deviation divided by the square root of the sample size
Confidence interval	a range built to contain the true value at a stated long-run rate. It does not put a probability on that one interval
Repeatability	precision under identical conditions: same operator, same equipment, same item, short interval
Reproducibility	getting the same result again when different researchers repeat the study. Repeatability does not guarantee it
Calibration	comparing a device against a standard of known accuracy. Strictly only the comparison, not the adjustment that follows
Traceability	the documented chain linking a measurement to reference standards
Metrology	the science of measurement: defining units, realising them, and keeping them traceable
Significant figures	the digits a measurement actually determines, set by instrument resolution. Claiming more is claiming knowledge you do not have

These are the terms that carry the argument above. Every one of them is used here in the sense given here, and nowhere else.

Key Takeaways

• Measurement error is the gap between a measured value and a true value nobody can observe, so it is always estimated and always reported.
• Random and systematic error are not degrees of the same problem. One shrinks when you measure again and the other does not move.
• Averaging cannot rescue a miscalibrated instrument. It produces a more confident wrong answer instead.
• Calibration means comparison, strictly. The adjustment that follows is a separate act, and skipping the record destroys the calibration.
• The standard error of the mean is the standard deviation over the square root of n. Nine readings give a third of the error of one.
• Precision is about random error only. Accuracy, in its common sense, is about systematic error, which is why the two words are independent.
• A 95 percent confidence level describes the long-run success rate of the method, not the odds for one particular interval.
• Repeatability and reproducibility are different. A result can be perfectly repeatable and fail to replicate anywhere else.
• Traceability is the answer to systematic error, and the old prototype kilogram shows what happens when the chain ends in a physical object.

Frequently Asked Questions

How many times should I repeat a measurement?

There is no fixed number, which is why the square-root rule matters more than any rule of thumb. Halving the standard error takes four times as many readings. If your spread is already small next to the effect you are trying to detect, repeating again buys very little, and the effort is better spent on systematic error.

Can a measurement be accurate but not precise?

Yes, and it is not a contradiction. Precision describes random error, so a tightly clustered set of readings is precise whatever its position. If every reading is 2 grams heavy, the cluster is tight and the mean is wrong.

Does averaging remove systematic error?

No. Systematic error comes from causes that act the same way every time, so repeating the measurement leaves it exactly where it was. The only remedies are to find the cause, or to compare the instrument against a reference of known accuracy.

Is precision something I can buy with a better instrument?

Often, but resolution and calibration are separate properties. A very sensitive instrument with a zero offset will produce tightly grouped readings that are all wrong. Precision is necessary and not enough, and no amount of it makes an uncalibrated instrument correct.

What does traceability actually give me?

A documented chain from your measurement up to a reference standard, with each link checked. It is what makes a systematic error findable. Without a chain to a known standard, there is nothing to compare against.

Why not just report more decimal places?

Because digits beyond what the instrument resolved carry no information. They came from the arithmetic, not from the measurement, and publishing them makes a result look more certain than the experiment was.

Sources: Wikipedia articles on measurement error, accuracy and precision, errors and residuals, calibration, standard deviation, standard error, sampling error, confidence interval, propagation of uncertainty, interval arithmetic, repeatability, reproducibility, metrology, significant figures and least squares. Images: Wikimedia Commons, with authors and licenses noted in each caption.