Systematic Error Calculator

What Is the Systematic Error Calculator?

How to use the systematic error calculator

1. Choose the unknown. The Solve for menu opens on Total error (rms), and the other three choices are Systematic error (bias), Random scatter of one reading and Number of repeat readings. Switching the menu returns every box to its opening value, so set the menu first and type second.
2. Pick one unit and use it in all three error boxes. Micrometres, millimetres, grams, seconds, volts — whatever your bench uses. What they cannot be is three different units, and that is the one mistake this page cannot detect for you.
3. Type the systematic error. The part that is the same in every reading, entered as a magnitude with no sign, opening on 0.020. The box accepts 0 to 1e9, and a zero here is meaningful rather than empty.
4. Type the random scatter of one reading. The standard deviation of a single measurement, not of the mean — the page divides by sqrt(n) for you. It opens on 0.004 and accepts 0 to 1e9.
5. Type the number of repeat readings. A whole number of 1 or more, opening on 9, with presets at 1, 9, 25 and 100. One reading is a legitimate entry and is where the whole argument starts.
6. Read the answer. The headline carries the answer to six significant figures, trailing zeros dropped, in the unit your three error boxes were in. Six rather than ten is deliberate: two of the four modes subtract the dominant term and are known to about seven figures.
7. Read the two terms and the bias share. The first reading gives sigma / sqrt(n) and the share of the mean square error the bias owns. Above about 90 per cent the instrument is the limit; below about 10 per cent your hand is.
8. Read the repeat ceiling. The second reading is (sigma / b)² rounded up — the readings at which the random error of the mean has fallen to the bias — and beside it is 4 (sigma / b)² rounded up in its own right, which is what it takes to see the fault at the two-sigma level. Two separate roundings, so the second count is at most four times the first, and both read a single reading once the bias is at least twice the scatter.
9. Read what a hundred readings would leave. The third reading runs your own state at n = 100 and prints the percentage it cuts. On the opening state that is a fraction of one per cent; with no bias at all it is the full tenfold gain.
10. Read the sensitivity chip before you quote the answer. The fourth reading prints (Erms - b) / Erms. Below about 1e-6 the scatter is far smaller than the bias, the two subtracting modes are working on your last digits, and the chip says so.
11. Open Show working. The steps restate the relation as a triangle, quote the boxes back, substitute, put the two terms beside the head-to-head sum you should not use, work the ceiling and the detection count beside it, price a hundred readings, print the conditioning ratio and close on what the equation does not model.

This page exists for the one step that turns a bias and a scatter into a single number, and then tells you whether more readings would move it. The systematic error guide is the prose half of it, where the two kinds of fault are separated, the null length is derived and the problems are worked. The n in the third box is a count of readings, not a figure count: how many digits you are entitled to write down is the significant figures calculator’s question and not this one.

Three mistakes account for most wrong answers here. The first is typing the scatter of the mean into the scatter box, which divides by the square root of the readings twice. The second is mixing units across the three error boxes, which the page cannot see. The third is adding the two errors head to head instead of in quadrature, which the working prints beside the right answer so you can see the size of the difference.

Worked example: change one thing at a time

The table starts at the defaults, runs the unknown through all four modes on one state, then changes exactly one thing at a time: the readings, the bias, the scatter. It ends on the boundary cases and the entries the calculator declines. Every answer and reading cell was taken from the running widget rather than worked out by hand, and where a cell and the tool ever part company, believe the tool.

Rows 1 to 4 are one state — a bias of 0.020, a scatter of 0.004 and nine readings — with the Solve for menu in each of its four positions. All three reading columns are identical down all four rows, because the readings describe the state rather than the mode. That is the round trip in one block: the headline prints the total error as 0.0200444, and the twelve-digit 0.0200443951712 the box carries returns 0.02, 0.004 and 9 as printed.

Rows 5 to 9 change exactly one thing at a time, and row 5 is the whole argument. A hundred readings instead of nine move the total error from 0.0200444 to 0.020004 — eleven times the work for a cut the third reading prices at 0.20 %. Row 7 cuts the bias to a twenty-fifth instead and the total error falls from 0.0200444 to 0.00155492, which is the same page making the opposite point.

Rows 8 to 12 are the cases worth keeping. At the crossover the bias share is exactly 50.00 % and the ceiling is 25 readings, and row 9 reaches that same count from the other direction by asking the readings mode for it. Rows 10 and 11 are the no-bias case in both directions: the total error is the random error of the mean, and solving back for the bias clamps to 0 rather than returning nothing at all.

Rows 13 and 14 are where the page stops trusting its own digits, and row 13 carries a rounding worth naming: the bias share prints 100.00 % because the chip is held to two decimals, while the share is really 99.9998 % — it is never exactly all of it while any scatter remains. Row 16 is the cancellation case: two faults that cancel hand the box a bias of 8.67e-19, the page snaps it to zero and reports no ceiling instead of 3.32e+29 readings. Rows 18 to 22 are the refusals, declined on the input rather than tidied up afterwards.

Formula and symbol reference

One relation does all the work: Erms² = b² + sigma² / n, which rearranges to b = (Erms² - sigma²/n)^(1/2), to sigma = (n (Erms² - b²))^(1/2) and to n = sigma² / (Erms² - b²). It comes from one fact about the mean: the systematic part is a constant and survives averaging intact, while the random part has standard deviation sigma / sqrt(n). Independent errors combine in quadrature, so the total is the hypotenuse of those two.

Only n carries a unit on this page, and it carries none: it is a count. The three error boxes share whichever unit you choose, which is why no SI unit is named in the table below — the SI units guide is where the base units and their prefixes are set out, and the significant figures calculator is where the question of how many digits a written number claims belongs.

Four symbols on this page are not boxes. sigma / sqrt(n) is the random error of the mean, nStar is the repeat ceiling, the bias share is a fraction of the mean square error rather than of the error itself, and (Erms - b) / Erms is the headroom the two subtracting modes need. All four appear in the readings beside the answer; none of them appears in the widget.

The physics: why the two errors add in quadrature, and what that costs you

A systematic error is defined by what repeating a measurement does not do to it. The random part of a reading differs from reading to reading, so the mean of n readings carries only sigma / sqrt(n) of it; the systematic part occurs with the same value every time the instrument is used the same way, so the mean carries all of it. That asymmetry is the whole subject, and it is why the two terms can never be traded against each other.

Independent errors add in quadrature because their squares add, not their sizes. So the total error is the hypotenuse of a right-angled triangle whose legs are the bias and the random error of the mean, and shrinking the shorter leg barely moves the hypotenuse. That one geometric fact is the reason “take more readings” stops being advice at a particular count.

That count is the repeat ceiling, nStar = (sigma / b)², and this page reaches it from the other direction: set the total error to the bias times the square root of two and the readings mode returns exactly (sigma / b)². At that target the two terms contribute equally and the bias owns exactly half the mean square error. Because the bias is squared in the denominator, halving it quadruples the ceiling — an inverse square, so the better the instrument, the sooner averaging stops paying.

Four figures are worth carrying away, and all four are this page’s own output rather than rules of thumb. A bias equal to the scatter of one reading reaches its ceiling at the first reading. A bias one fifth of the scatter reaches it at twenty-five, and seeing that fault at the two-sigma level takes a hundred readings. Going from twenty-five readings to a hundred halves the random error of the mean exactly and still cuts the total error by only about a fifth.

Three different things get called uncertainty and only one of them is on this page. A figure count is a statement about how a number was written, which is the significant figures guide’s subject and which that guide itself calls a coarse stand-in for precision rather than the precision itself. The Heisenberg uncertainty principle guide is a statement about nature, and a bias is a statement about an instrument. All three limit a measurement, and care about one buys nothing for the other two.

Where this arithmetic breaks down

The arithmetic here is exact to the last digit a double can hold in two of its four modes, and good to about seven figures in the other two. What goes wrong is physical, and the first item on this list is the one metrologists would raise first.

The random-and-systematic split is not how metrology books the arithmetic

This is the one to read. NIST’s summary of its Technical Note 1297 classifies components of uncertainty by the method used to evaluate them — Type A from the statistical analysis of a series of observations, Type B from anything else — and not by whether the cause wobbles. The same physical fault is Type A if you estimated it by measuring a standard twenty times and Type B if you read it off a certificate. The split this page uses is the right one to learn, because it answers the question of whether more readings help; it is not the one the paperwork uses.

A correction has its own uncertainty, so a corrected reading is not exact

NIST is explicit that the measurement equation should express a measurement process rather than simply a physical law, and that the quantities in it include the corrections. Subtracting a measured bias does not delete it; it replaces a bias with the smaller uncertainty of the correction. This page takes b as a given number and models neither where it came from nor how well you know it.

Repeats with the same instrument and the same observer cannot see the fault

The source is blunt about this: a fixed systematic error can only be found either by measuring a known quantity or by comparing the readings with readings made using a different apparatus known to be more accurate. Nothing in sigma / sqrt(n) warns you that a bias is there. What this page can give you is the largest bias your figures are consistent with, and the count it would take to see it.

Where the bias came from is not in this equation

A zero error that is the same at every length and a gain error proportional to the reading both enter here as one number b, and they behave completely differently. An instrument with one of each is correct at exactly one length — calibrate it there and it certifies itself as perfect, while reading low on a foil and high on a block. Separating the two takes two standards of known length rather than any amount of arithmetic, and the companion simulator is where that split has its own sliders.

A gain error is a percentage of a physical quantity, not of a scale reading

A proportional fault multiplies the quantity, and the quantity has to be measured from its own zero. A two per cent gain on a thermometer is not two per cent of the Celsius reading, because the Celsius scale’s zero is the ice point and not absolute zero — the absolute zero guide is where that distinction is set out. Say which one your percentage is of before you turn it into a bias for this page.

Drift breaks the constant-bias assumption, and partly averages out

A fault that walks with time is not the same value in every reading, so the mean of a symmetric ramp is its mid-value and half the drift does average away. Drift is also the easier fault to catch, because the readings trend with time instead of varying randomly about a mean, and it inflates the apparent scatter while a constant offset adds exactly none. That asymmetry is why a constant offset is the dangerous one.

Correlated readings make the standard error optimistic

sigma / sqrt(n) assumes the readings are independent. Nine readings taken by the same operator in the same five minutes are not, and nothing in the formula warns you — the printed random error of the mean is then too small and the bias share too low. The fix is a measurement design that breaks the correlation, not a different sum.

Quantisation makes the scatter uniform, not symmetric about the truth

A dial read to its last division rounds rather than wobbles, so the scatter is uniform on the resolution rather than bell-shaped, and below one resolution step it can read as exactly zero while the rounding bias is entirely real. Rounding to whole centimetres has a standard deviation of ten over the square root of twelve, about 2.9 mm, which is why the source’s claim that a whole-centimetre ruler gives an error of several millimetres is exactly right.

Sigma is usually estimated, not known

The relation treats the scatter as a known number. In practice it is estimated from the same handful of readings with n - 1 degrees of freedom, and that estimate is itself uncertain by roughly 1 / sqrt(2(n-1)) — about 25 per cent at nine readings. So a repeat ceiling computed from nine readings is itself only good to a factor, and the right reading of “25 readings” is “a few tens”.

The two subtracting modes lose digits, and the page says when

The scatter mode and the readings mode both subtract the dominant term when the bias is the bigger of the two. Measured over 32768 states of the companion lab their worst relative errors are 2.19e-7 and 4.38e-7 against 1.71e-10 for the bias mode — a factor of about a thousand. That is why the headline stops at six significant figures and why the fourth reading prints (Erms - b) / Erms.

Three different things get called uncertainty

A figure count describes how a number was written. The Heisenberg principle describes a limit in nature. A bias describes an instrument. The uncertainty principle calculator handles the second of those, and it is worth knowing that no calibration touches it — unlike a bias, which is exactly what calibration is for.

The calculator measures nothing — you supply the numbers

One bias, one scatter and one count go in, one relation is used, and the answer describes the situation you described. The case names on the buttons are shorthand for the figures behind them and are not claims about any particular micrometer, balance, stopwatch or meter.

Where the systematic error calculator is actually used

Deciding whether another forty readings are worth taking

Load A zero error, nine readings and the answer is on screen in one line: the ceiling was passed at the first reading, and the third chip prices a hundred readings at a fraction of one per cent of the error. That is the question this page was built for, and the honest answer is usually no.

Finding the largest bias your figures are consistent with

Point Solve for at the systematic error and feed it a total error from a comparison against a standard plus a scatter from a long run. The answer is an upper bound dressed as a number, and it is the one mode of the four that never subtracts the dominant term, so it is also the one you can quote to six figures without a caveat.

Setting a reading count from a target total error

Point Solve for at the readings and type the total error you need. If the target is at or below your bias the page declines, and that refusal is the useful answer: no number of readings reaches it, and the next move is a calibration. If it answers, the count beside the headline is the whole number to take.

Checking a published error budget adds up

A budget that quotes a bias, a scatter and a total error is testable in one line: square the first two terms, divide the second by the readings, add, take the root. If the published total matches, the budget combined its terms in quadrature. If it is larger by roughly the smaller term, the terms were added head to head.

Separating a timing fault from a timing wobble

A stopwatch that starts with a second already on it is the classic constant offset, and averaging twenty attempts leaves it entirely intact. Timing twenty swings once and dividing by twenty divides that offset by twenty, which is a change of measurement design rather than of statistics — the pendulum period calculator is where the period itself comes from, and this page is where you price the two procedures against each other.

Reading a calibration certificate against your own repeatability

A certificate gives a bias; a run of readings gives a scatter. Put both in, with the count you actually take, and the bias share tells you which of the two is worth spending money on. Above about 90 per cent the money goes on the instrument; below about 10 per cent it goes on the procedure.

Teaching the difference between accuracy and precision with numbers

Load No bias at all: the mean finds the truth and then A bias 250 times the scatter. The first has no ceiling and a tenfold gain from a hundred readings; the second is hopeless at one reading and hopeless at a thousand. Two clicks, two readings beside each other, and the words stop being interchangeable.

Where to go next

For the argument this arithmetic supports — the two kinds of fault, the single length at which a two-fault instrument reads correctly, the twenty-readings-against-twenty-swings case and the one in which a bias really can be beaten — read the systematic error guide, which this tool is the calculating half of.

Five posts pick up the pieces this page uses but does not explain. The significant figures guide and the Heisenberg uncertainty principle guide are the other two things called uncertainty, the SI units guide and the unit conversions guide are where the unit you choose comes from, and the physics constants reference is where every quoted value arrives with an uncertainty attached.

Four tools sit next to this one. The significant figures calculator handles precision as written, the uncertainty principle calculator handles the limit nature imposes, the pendulum period calculator gives the quantity the stopwatch case is about, and the unit conversions simulator is where a mixed-unit entry gets sorted out before it reaches this page.

The full physics lab library and the calculator index are open too.