What a weight on a label means: balances, calibration and the honest size of a small number

A weight looks like the simplest measurement there is. Something is placed on a pan, a number appears, and the number is written down. Nothing about the display suggests doubt. It shows the same confident digits whether the instrument was checked this morning or five years ago, whether the room is still or a door has just been opened, and whether the object on the pan is a brass cylinder or a few specks of powder at the very bottom of the instrument's range.

Yet every stated quantity on a container, and every "measured amount" on a report, rests on exactly this act. If the weighing was sound, the figure is a fact about the contents. If it was not, the figure is a fact about the display. The two look the same on paper. Telling them apart means knowing what the instrument can do, how that was established, and how small a quantity it was being asked to see.

What follows is a reference guide to that reasoning: what a balance senses, the four separate properties that get folded into the word "accurate", how readings are tied back to a reference, the handful of ordinary influences that disturb a small weighing, the idea of a minimum weight, how a net amount is obtained from two readings, and what a stated amount on a label is actually promising. It closes with eight errors that recur and a ten-point check that can be applied to any reported weight.

It is written for small settings, where there is one balance on one bench and nobody whose whole job is metrology. It is deliberately general. It does not describe, recommend, compare or make any claim about any material, item, treatment or substance, and it never tells a reader what to do with one.

Mass, weight and what the instrument senses

Mass is the amount of matter in an object. Weight, strictly, is the force that gravity exerts on it. A balance cannot sense mass directly. It senses a force and reports it in units of mass, on the assumption that gravity at the bench is the same as it was when the instrument was last adjusted.

That assumption is good, and it is not perfect. Gravity varies by roughly half a percent between the equator and the poles, and it falls slightly with height. A balance adjusted in one city and carried to another without readjustment can be wrong by an amount far larger than its last displayed digit. This is the first reason an instrument's history matters more than its display: the number on the screen is a force that was converted using a factor somebody set, at some place, on some day.

Older balances compared the object against known weights on a second pan, which cancels gravity out entirely. Almost every modern laboratory balance instead uses electromagnetic force compensation, shown in the figure above. The pan hangs from one end of a lever. A coil sitting in a permanent magnet hangs from the other. When a load is placed on the pan the lever tips, an optical sensor notices, and a controller increases the current in the coil until the lever is level again. The current is proportional to the force. The display is, in effect, an ammeter with a conversion factor.

Two consequences follow. The magnet's strength changes with temperature, so the instrument needs time to warm up and dislikes a room that drifts. And the conversion factor has to come from somewhere, which is the subject of calibration.

Four numbers that are not the same number

In ordinary speech a balance is "accurate to" its last digit. In a specification sheet that one idea is split into at least four, and a reported weight can only be judged by the right one.

A fifth property, eccentricity, describes how much the reading changes when the load sits off the center of the pan. It matters most with wide containers.

For small quantities, repeatability dominates everything else. A balance that reads to a hundredth of a milligram but repeats only to within three hundredths is, for practical purposes, a three-hundredths instrument. The extra digit is decoration. This is why a careful report states the repeatability that was measured on the bench, in that room, rather than quoting the figure from the brochure.

Calibration, adjustment and the chain upwards

Two words are used loosely and mean different things. Calibration is a comparison: a reference weight of known value is placed on the pan, the reading is recorded, and the difference is written down together with its uncertainty. Nothing is changed. Adjustment is an intervention: the conversion factor inside the instrument is altered so that the reading agrees with the reference.

The distinction matters because many balances adjust themselves. An internal weight is lowered onto the mechanism by a motor, on a timer or when the temperature moves, and the factor is reset. This is useful and it is not calibration. It tells nobody how wrong the instrument was before the adjustment, and the internal weight is itself never seen or checked by the user. An instrument that has only ever adjusted itself has no documented relationship with anything outside its own case.

That outside relationship is called traceability. A working weight in a laboratory was compared with a better weight at a calibration provider. That weight was compared with a better one again, and the chain continues upwards to a national institute and finally to the definition of the kilogram, which since 2019 has rested on a fixed value of the Planck constant rather than on a metal cylinder in a vault. Each link adds a little uncertainty, and each link has a certificate that says how much.

Reference weights are sold in accuracy classes defined by the international recommendation OIML R 111. The class sets how far the true value of a weight may differ from the value stamped on it.

A summary of the scheme, not the standard itself. The permitted error for each class and each nominal value is tabulated in the recommendation.

A fingerprint weighs something. So does a film of dust or a patch of corrosion. A reference weight that has been handled carelessly is no longer the weight its certificate describes, and nothing on its surface will say so.

What disturbs a small weighing

At the scale of grams, most disturbances are too small to notice. At the scale of milligrams they are the main event. Six of them account for nearly all the trouble.

Air movement. A draft that cannot be felt on the skin exerts a measurable force on a pan. This is why fine balances sit inside glass draft shields, and why the doors of the shield are closed before a reading is taken, every time.

Temperature difference. An object warmer than the air around it sets up a rising current along its sides, which lifts it slightly, and the reading comes out low. An object colder than the air reads high, and it may also collect a film of condensation that makes it heavier in fact. A container taken from cold storage needs to reach room temperature, closed, before it goes on the pan.

Static charge. Dry powders and plastic or glass containers pick up charge from handling, especially in air below about 40 percent relative humidity. A charged object is attracted to or repelled by the grounded metal around the pan, and the reading drifts slowly in one direction instead of settling. Drift of this kind is the signature of static, and an ionizer or simply a higher room humidity removes it.

Moisture exchange. Many dry, porous or freeze-dried materials take up water from the air within seconds of being exposed, and some lose it. The reading climbs or falls steadily while the operator watches. There is no correct moment to stop the reading; the only sound approach is to weigh in a closed container and work quickly.

Vibration and level. A balance on a flexible bench, near a door, a centrifuge or a refrigerator compressor will scatter. One that is not level will read wrong in a consistent direction, because part of the load is pushing sideways.

Air buoyancy. Everything on the pan is pushed upwards slightly by the air it displaces. Balances are adjusted with dense steel weights, so a light, bulky object of the same mass reads a little low, by about a tenth of a percent for something with the density of water. For most work this is ignored. In the most careful work it is corrected.

The floor under a balance: minimum weight

The scatter of a balance is roughly the same size whatever is on the pan. A spread of two hundredths of a milligram is nothing when weighing a gram and is a great deal when weighing a single milligram. So the relative uncertainty of a weighing grows as the quantity shrinks, slowly at first and then very steeply.

The minimum weight is the point on that curve where the relative uncertainty reaches the largest value the task can tolerate. A widely used rule takes twice the measured repeatability and divides it by the required tolerance. With a repeatability of 0.02 milligrams and a tolerance of one tenth of a percent, the floor is 0.04 divided by 0.001, which is 40 milligrams. Anything lighter can still be placed on the pan and will still produce a tidy reading. That reading is simply not good to a tenth of a percent.

Three practical points follow. The floor belongs to the instrument as installed: the same model on a worse bench has a higher floor. The floor applies to the net amount, so a heavy container does not rescue a tiny sample. And a sensible laboratory works with a margin above the floor, because repeatability on a bad day is worse than on the day it was measured.

Gross, tare and net

The contents of a container are almost never weighed on their own. The container is weighed full, which gives the gross value. It is weighed empty, which gives the tare. The difference is the net amount. Each of the two readings carries its own scatter, so the net figure is less certain than either reading alone, by a factor of about 1.4 when the two are similar.

The order of the two readings depends on the situation. When filling, the empty container is weighed first and the tare button sets the display to zero. When checking something already filled and sealed, the full container is weighed first, the contents are removed completely, and the cleaned, dried container and its closure are weighed afterwards. This is called weighing by difference, and it is the only way to learn the net contents of a single sealed container without assuming what the container weighs.

The assumption is the weak point. Containers and closures from one production lot differ from each other, sometimes by more than the contents weigh. A net figure obtained by subtracting an average tare from an individual gross reading inherits all of that variation. For small contents, a result based on an average tare should be read as an estimate.

What a stated amount promises

A quantity printed on a label is a claim, and the same printed figure can stand for three different claims. Which one applies is set by the rules of the trade the item belongs to, and it is rarely spelled out on the label.

Under a minimum or an average rule, a filler deliberately aims above the stated figure. The extra is called overage or overfill, and its size is set by how much the filling process scatters: the wider the scatter, the higher the aim has to be to keep the short tail above the line. A measured amount that is several percent above the stated one is therefore not, on its own, a sign of a mistake. It is what a process with a floor looks like from outside.

The reverse is less forgiving. A measured amount below the stated one deserves a second look at two things before any conclusion is drawn: the uncertainty of the measurement, and the kind of claim the label is making. A result that is two percent low with an uncertainty of three percent has not shown a shortfall. It has shown that the method could not tell.

Reading a reported weight

A report that gives a measured amount for one unit from a batch is making a narrow statement. It says what one laboratory found for the unit it received, on the day it weighed it, using its own instrument and procedure. Four questions turn that statement into something usable.

How many units? One unit says where that unit fell. It says little about the spread of the batch. Three or more begin to describe the process.

By what route? A direct weighing by difference measures everything in the container, including any water the contents have taken up. An indirect route, in which the contents are dissolved and the amount of one component is determined by an instrument, measures only that component. The two can legitimately differ, and a report should say which was used.

With what uncertainty? A figure quoted to two decimal places with no stated uncertainty invites the reader to assume the last digit is exact. It almost never is.

Which batch? A result belongs to the lot named on the report. It transfers to another container only if that container carries the same lot identifier.

Routine checks and the logbook

Between formal calibrations, a balance is kept honest by short checks that take a minute and are written down. The usual pattern has three layers.

Before each session, the operator confirms that the instrument is level, has been powered for long enough to be warm, and reads zero with the shield closed. Daily or weekly, one working reference weight near the loads normally weighed is placed on the pan and the reading is compared with warning and action limits set in advance. At longer intervals, a fuller test measures repeatability with ten placements of the same weight and checks the corners of the pan for eccentricity.

The record of these checks is as important as the checks themselves. A log that shows the check weight reading the same value, within limits, on every working day for a year is the evidence that any single weighing during that year can be trusted. A log with a gap is a period in which nobody knows. And a reading that creeps steadily in one direction over months is an early warning that no single check would raise.

Eight errors that recur

Most poor weighings come from a short list of habits, each of which seems reasonable at the bench.

1. Trusting the last digit. Reading the display's resolution as the instrument's uncertainty.
2. Weighing below the floor. Using a balance for quantities under its minimum weight because it "still gives a number".
3. Weighing cold or warm objects. Putting a container on the pan before it has reached room temperature.
4. Accepting a drifting reading. Picking a moment to stop rather than asking why the value is moving.
5. Leaving the shield open. Reading with a door ajar because it is quicker.
6. Assuming the tare. Subtracting a typical container weight instead of the weight of the actual container.
7. Treating self-adjustment as calibration. Having no external reference weight and no record of what the instrument read before it corrected itself.
8. Handling reference weights by hand. Adding skin oils to the one object whose value is supposed to be known.

A ten-point check of a reported weight

Whenever a weight is offered as evidence, on a label, a report or a specification, these ten questions decide how much it can carry.

1. Is the instrument identified, with its readability and its measured repeatability?
2. Was it calibrated against external reference weights, and when?
3. Do those reference weights have certificates that trace upwards without a gap?
4. Was the quantity weighed above the instrument's minimum weight for the tolerance claimed?
5. Is the result a net value, and was the tare measured on the same container?
6. Was the object at room temperature, and was the reading stable before it was taken?
7. Does the result state an uncertainty, or at least enough to estimate one?
8. How many units were weighed, and is the spread between them reported?
9. Is the route stated: direct weighing, or an indirect determination of one component?
10. Does the batch or lot identifier on the result match the item it is being applied to?

A weight that answers all ten is a measurement. One that answers three or four may still be correct, and nobody, including the person who produced it, is in a position to know. The difference between the two is never visible in the number itself. It lives entirely in the record around it.