1.1.5 · D5Measurement, Vectors & Kinematics
Question bank — Errors — absolute, relative, percentage; systematic vs random
Two words we will lean on constantly — pin them down first so no line below uses them unearned:
Before any line uses a symbol, here are the exact quantities we will trap you on. Nothing below is allowed to appear unless it is on this list.
True or false — justify
Every statement below is either true or false. Say which, then give the reason — a bare "true/false" scores nothing.
Taking more readings can eventually remove a systematic error.
False. Systematic error is the same nudge on every reading (e.g. always cm), so summing and dividing leaves that same cm behind. Only random scatter shrinks with averaging.
A precise measurement is automatically an accurate one.
False. Precise readings are tightly clustered but the whole cluster can sit far from the true value (a systematic bias). You can be "precisely wrong."
Relative error has the same units as the quantity measured.
False. Relative error is — units on top and bottom cancel, so it is dimensionless. That is exactly why it lets you compare a length error with a mass error.
An absolute error can be negative.
False. It is defined with a modulus, , so it is a magnitude — the size of the miss, never its direction.
A zero error on an instrument is a random error.
False. A zero error means the instrument reads a fixed non-zero amount at zero input, so it offsets every reading identically — that is the textbook signature of a systematic error.
Percentage error and relative error carry the same information.
True. Percentage error is just relative error ; multiplying by a constant reshapes the number for human eyes but adds no new information.
The mean absolute error and the standard deviation of a data set are always equal.
False. Both measure spread but by different recipes — MAE averages , while root-mean-squares them, so is generally the larger. They answer the same question with different sensitivity to outliers.
Doubling the number of readings halves the uncertainty of the mean.
False. The standard error falls as , so doubling divides the uncertainty by only ; you need four times the readings to halve it — and even then only random error shrinks.
If two students measure the same rod and get the same mean, they must have the same uncertainty.
False. Equal means can hide very different scatter; the one with widely spread readings has a larger mean absolute error, hence a larger range.
A measurement with error is always more trustworthy than one with error.
False for accuracy. A small percentage error only reflects tight, consistent readings; an unnoticed systematic bias can make the "" result confidently wrong. Small relative error means precise, not necessarily accurate.
Spot the error
Each line contains a specific mistake in reasoning or reporting. Name it and correct it.
"I measured a wire as mm — look how precise my error is."
The error is stated to too many significant figures; an uncertainty is itself uncertain, so round it to 1 (occasionally 2) sig figs and match the value's decimals: mm.
"My readings are g, so my result is perfectly accurate."
Tight readings prove precision, not accuracy. If the balance had a g zero error, all three are equally offset and the true mass is g — you cannot judge accuracy from spread alone.
"The error is mm and the length is m, so this is a bad measurement."
Judging quality by absolute error alone is the mistake. Relative error is — excellent. The same mm on a mm wire would be , terrible. Size of the miss must be weighed against the size of the thing.
"I averaged my shaky-hand readings, so I've fixed both my random and systematic errors."
Averaging only cancels the random scatter. Any systematic bias (parallax always leaning one way, a mis-calibrated scale) rides straight through the average untouched.
"Percentage error is always , full stop."
Careful — the denominator matters. Strictly, percentage error divides by whichever value you take as reference: the measured/mean value (fractional uncertainty, what school work uses) or the true/accepted value (when comparing against a known standard). The two agree only when the measurement is close to truth; state which you mean.
"My percentage error is because the digits I read were tiny like ."
The percentage depends on the fraction , not on how few digits the number has. Small numbers do not automatically mean large percentage error — compute the ratio before guessing.
"I found a systematic error of g, so I'll take more readings to remove it."
Wrong tool. Once a systematic error is identified and quantified, you correct it by subtraction, not by repetition — subtract g from every reading.
Why questions
These probe the mechanism. A correct final phrase without the reasoning is incomplete.
Why does averaging reduce random error but not systematic error?
Random errors are like coin-flip nudges, sometimes and sometimes , so they partly cancel in a sum; systematic error is the identical nudge every time, so summing copies and dividing by returns the same nudge. The two clouds in figure s02 show this directly.
Why does the uncertainty of the mean fall as and not as ?
Because random deviations partly cancel rather than fully cancel — they add up like a random walk, whose typical size grows as , so dividing the summed error by leaves a shrinkage. See Mean and Standard Deviation.
Why do we take the mean as the best estimate of the true value?
Because random errors scatter symmetrically about the truth, the value that minimises the total squared deviation is the arithmetic mean — it is the single number that "the highs and lows cancel toward."
Why do we divide by the value to get relative error instead of just quoting the absolute error?
A fixed absolute error means wildly different things at different scales; dividing by the value expresses the miss as a fraction of the thing itself, making errors of different quantities directly comparable and unit-free.
Why do we use the absolute value (modulus) when computing each reading's error?
We want the magnitude of the deviation. Without the modulus, positive and negative deviations would cancel and the mean deviation would collapse toward zero, hiding the real spread.
Why is a known systematic error actually the easier one to deal with, despite averaging failing?
Because it is constant and predictable — once you measure the offset (e.g. the zero reading), you subtract it exactly. Random error has no fixed value to subtract, only a spread to shrink statistically.
Why should reported uncertainty match the value's last decimal place?
Because claiming a value to more decimals than your uncertainty allows is claiming knowledge you don't have; the value is only meaningful down to the digit where the error bites. See Significant Figures and Rounding.
Edge cases
Boundary and degenerate situations the definitions still must handle.
If all readings are identical, what is the mean absolute error?
Exactly zero — every deviation from the mean is zero. But zero reported error does not mean zero true error; a hidden systematic bias or an instrument's least count still limits real accuracy.
You take only a single reading. Can you compute a mean absolute error?
No — with one reading the mean equals that reading and its deviation is zero, giving a meaningless "zero" uncertainty. The honest uncertainty must then come from the instrument's least count (see Least Count and Vernier Calipers), not from scatter.
You want to halve your current uncertainty. Roughly how many more readings do you need?
About four times as many in total, because the standard error scales as — and this only works if the dominant error is random; a systematic bias will not budge no matter how many you take.
Your mean absolute error comes out larger than the mean value itself (relative error ). What does that signal?
The measurement is essentially useless — the scatter swamps the signal. This typically means a blunder, wrong instrument, or a quantity so near zero that the fraction blows up.
The measured mean is exactly zero (e.g. a net force reading). What happens to the relative error?
It is undefined — you cannot divide by zero. Near a true value of zero, relative and percentage error lose meaning, so you must report the absolute error instead.
A reading that is a wild outlier (a clear mistake, like slipping the ruler) sits in your data set. Is it random error?
No — a genuine blunder or mistake is neither random nor systematic in the statistical sense; it should be discarded before analysis, not folded into the mean, where it would distort both the mean and the mean absolute error.
Both students hit the true value on average, but one used a biased instrument that happened to cancel out. Is that instrument accurate?
No — accidental cancellation is luck, not accuracy. A single reading from the biased instrument would still be off; only this particular data set happened to balance. Reliability requires the bias itself be absent.
Recall One-line self-test before you leave
Averaging fixes random error, subtraction fixes a known systematic error, the uncertainty of the mean shrinks like == (standard error), and small relative== (percentage) error means precise but not necessarily accurate. If you can say why for each, this page did its job. For how errors propagate when you combine measurements, continue to Combination of Errors.