480 chairs went into 24 classrooms as two seat heights, 240 at 380 mm and 240 at 430 mm. Every one of the 50 chairs the inspector measured was the right chair, exactly to tolerance, and the delivery was still 40 chairs short for the secondary block.

The order named two heights and one packing list

A school group in the north of England was replacing the seating across a primary and a secondary block: 480 student chairs, one consignment, one address, unloaded and sorted by the caretaker team who had done it every year.

The 2 sizes are the 2 school-chair sizes every European school contract names. BS EN 1729 sets them out. Size 4 is a 380 mm seat, a 640 mm table, a red size mark, and suits roughly 8 to 11 year olds. Size 5 is a 430 mm seat, a 710 mm table, a green size mark, and suits roughly 11 to 14. The chair itself is a plain steel-framed classroom chair, identical in both sizes apart from the leg length and the colour.

Each size pairs with its own table height, and the pairing is what fixes the clearance between seat and table: 260 mm for Size 4 and 280 mm for Size 5. Nothing on the order said so, because the buyer assumed the sizes could not be confused.

The order was clear enough. Two lines on the purchase order: 240 chairs at seat height 380 mm and 240 at 430 mm, colour-coded to EN 1729. Unit price $38, so $18,240 for the consignment. 30 per cent, $5,472, on order. 70 per cent, $12,768, against loading.

What the order did not say, and what turned out to matter more than anything it did say, was how the 2 sizes would be told apart at the container door.

Forty Size 4 chairs where Size 5 should have been

The legs are the whole difference. A Size 5 chair is the same frame as a Size 4 chair with a longer leg set and a different bracket. The factory held Size 4 legs in stock and had ordered Size 5 legs against a delivery date that slipped.

The buyer was told about the slip. What came back was not a new date but a proposal to make up the number: build 40 more Size 4 chairs and ship them in place of the 40 Size 5 chairs that could not be built in time. Same frame, same finish, same red size mark, correct to the Size 4 specification. The count would be right. The consignment would be 480.

The factory's packing list said 480 chairs in 60 cartons of 8. It did not say which size was in which carton, because the buyer's order specified the split as a quantity against a seat height and never asked for the split to be carried onto the carton marks. So all 60 cartons printed the same thing: model code, quantity, carton number, order number. Not one carton mark said which size was inside it.

And the substitute was invisible at every level a check normally reaches. A Size 4 chair measured against a Size 4 tolerance passes, is red, and is labelled red. It is not a defective chair. It is a good chair in a room it does not belong to.

The sample that was drawn from one pallet

The buyer's pre-shipment inspection was a Level 2 check, 1.5 to 2 man-days, sampling under ISO 2859-1 general inspection level II. With a lot of 480 the code letter is H, the sample is 50 units, and at AQL 2.5 a lot is accepted with 3 or fewer defects and rejected at 4.

The inspector opened 7 cartons and measured seat height on 50 chairs. Seven cartons hold 56 chairs, so the 50 units came out of the first carton group the workshop had staged for loading. Every one of the 50 measured 380 mm. Every one fell inside the EN 1729 tolerance for Size 4, which is plus or minus 10 mm, so 370 to 390. The report recorded zero defects.

The chairs in those 7 cartons were all Size 4 because they were all off the same pallet, and the pallet was the Size 4 batch, because Size 4 legs were in stock and Size 5 legs were not.

Here is the number that should have stopped the report. If the 50 chairs had been drawn at random from the whole lot, and 200 of the 480 really were Size 5, the chance of measuring none of them is about one in four trillion. It is not a remote risk. It is arithmetically impossible. The sample found none because the sample was never in a position to find any.

The sample was a fair and competent sample of the pallet in front of it. It was not a sample of the lot, and a report that records it as one is not describing what it measured.

A wrong proportion produces zero defects

There are 2 separate things wrong with that report, and the second one is the part worth taking away.

The first is selection, and it is the one anybody would notice. But underneath it sits a structural point that applies to every mixed-size order a buyer will ever write. AQL counts units that do not conform to a specification. It answers a question of the form: how many of these chairs are bad? The Size 4 chairs in this consignment were not bad. Each one conformed to the Size 4 specification on every dimension the order named, and a sample of 50 confirmed it. A wrong proportion of 2 perfectly good chairs produces a defect count of zero, and no acceptance number in ISO 2859-1 will ever turn zero into a failure.

So the lot could not have been rejected under the sampling plan the inspector was working to, whatever had been found. The lot passes. What the lot actually was, is a different question, and the acceptance number in the plan belongs to the first question.

The second thing wrong is precision, and it is subtler. The inspector's data was sound: zero of 50 measured chairs were Size 5. Treated as 50 independent observations, that supports a one-sided 95 per cent upper bound of 5.8 per cent on the Size 5 share of the lot. A report written to that standard could have said, truthfully, that no more than about 6 per cent of the consignment was Size 5. The truth was 42 per cent. The statement was understated by a factor of 7.

And the inspector had no way of knowing it was overstated, because the 50 chairs did not come from 50 independent places. Every chair in a carton is the same size, because a carton is filled from one batch. The 7 cartons carried 56 chairs and all of them were one height, so the real number of independent observations was 7, not 50. For a cluster sample of that shape the design effect is one plus (m minus one) times the intra-cluster correlation, and with the correlation at its maximum of one and m at 7.14, the design effect is 7.14. The effective sample size was 7. Apply the same one-sided bound at an effective n of 7 and the upper limit comes out at 34.8 per cent, still below the truth of 42 per cent. Correcting the variance does not repair the selection. Once the sample comes from one pallet, no amount of statistics recovers the lot.

50 is a decent number for a per-unit characteristic. For a characteristic that lives in the relationship between units, it is 50 of 7.

A carton is the population, not a sample of it

The obvious question is: how many cartons should the inspector open? It is the right question and the arithmetic gives an answer nobody likes.

Take 12 cartons at random from the 60 and count how many are Size 5. On a correct consignment of 30 Size 5 cartons, rejecting the lot when fewer than 4 of the 12 are Size 5 falsely rejects a good lot about 5 times in 100. On a consignment like this one, with 25 Size 5 cartons, the same rule catches the problem about 1 time in 6.

20 cartons does not fix it. The false rejection rate falls to almost nothing, and so does the detection rate, because the error is small: 5 cartons out of 60, or 8.3 percentage points of the consignment. The sampling error on a share that small is nearly 28 points at 12 cartons, and still 12.5 points even if you read every single carton. The signal is smaller than the noise for almost the whole way.

To reach 80 per cent detection with a 1 per cent false rejection rate, you need to open 53 of the 60 cartons.

Which is the tell. 53 out of 60 is not a sample, it is a census with a bad name. The reason is not that inspectors are slow. It is that the characteristic being verified is a property of the population and not of a unit, and populations do not get sampled cheaply.

The correct instrument is a census of the carton marks. 60 cartons, 60 labels, a minute each, zero sampling error, and the split becomes a fact rather than an estimate. The number it produces, 25 out of 60, does not need a confidence interval, because it is not an estimate.

And that census costs 1 man-day at $199, which is 1.1 per cent of the order value. Compare it with what the same consignment cost when nobody counted.

Two classrooms with nothing in them

The school found out at the point of installation, 19 days after the balance of $12,768 had been released on the strength of a report that recorded zero defects.

The caretaker team unloaded 60 cartons and began to sort. There were red chairs and green chairs in the consignment and no carton mark to say which was which, so every carton was opened and counted by eye. 280 Size 4 chairs came out. The Size 5 count was 200.

12 secondary classrooms at 20 chairs each is 240. 200 left 2 classrooms with nothing in them, and the 40 Size 4 chairs that had arrived in their place could not go in. A Size 4 chair at a Size 5 desk leaves 330 mm between seat and table, where the EN 1729 pairing calls for 280 mm. That is 50 mm more clearance than an 11 to 14 year old is meant to have, and for that age group the popliteal height, the floor-to-under-knee measurement, runs 405 to 435 mm. On a 380 mm seat the feet do not reach the floor: the gap between the seat and the underside of the knee is 25 to 55 mm. The chair that came instead is a chair for a smaller child, in a room for larger ones.

The school hired 40 chairs locally for the 3 weeks before replacements arrived, and bought the replacements from the factory at $38 each and air freighted them. $1,520 for the chairs and the freight, plus $880 for the hire. $3,600 against an order value of $18,240.

19.7 per cent of the order. And the balance had already gone.

Where the report stopped

A verification report is worth what its limits are worth, so it is worth being plain about what this one established and what it did not.

Recorded: model, carton count of 60, quantity of 480 against the order, seat height measured on 50 sampled units against the per-unit tolerance, and zero defects found at AQL 2.5, with the code letter, the sample size and the accept and reject numbers stated.

Not recorded, and not establishable from the work done: the size split. Only 7 of 60 cartons were opened, and no carton mark carried a size, so the proportion of Size 4 to Size 5 in the consignment was not a fact the report could state.

Also not established: why. Whether the shortfall came from a leg-set shortage, a scheduling error or a change of intent is a question for the factory's production and purchasing records, not for an inspector at the end of the loading dock. The report says so rather than guessing.

Also outside what any inspection could say: whether the chair is right for the child. EN 1729-1 sets the dimensions and the size marks, and EN 1729-2 is a product safety and durability standard requiring accredited test reports on the exact model. An order verification statement is not a product test report, has no legal standing, and cannot be used for tender compliance. Matching a size mark to a pupil population is the buyer's and the school's work. The inspection can establish what was in the container.

One limit is worth holding onto. The report recorded 50 units checked, and the reader hears 50 chairs verified. 7 of the 56 chairs in those 7 cartons came from the same pallet as each other, and the report did not say so. A report that records the sampling frame as carefully as it records the numbers is worth more than one that records only the numbers.

Six lines to put in the next purchase order

Nothing in this case needed a laboratory. It needed 6 lines in a purchase order, and every one of them is a line a buyer can write.

A school furniture inspection that treats the split as a sampled characteristic passes this consignment every time, and no extra sampling on the chairs themselves changes that.

• Carry the split onto the carton mark. Each carton shall state the seat height in millimetres and the EN 1729 size mark colour, and the packing list shall state the carton number range for each size. Model code and quantity alone do not identify a classroom.
• No mixed cartons. A carton shall contain 1 size only, and the size on the carton mark shall be the size on every chair inside it.
• No substitution across sizes. Where a size cannot be produced in the ordered quantity, the shortfall shall not be made up from another size. The buyer shall be notified, the line re-quoted, and the balance on that line withheld until it is settled.
• Verify the split as a census, not a sample. The size split shall be established by reading every carton mark on the consignment. AQL applies to the appearance and dimension characteristics of the chairs and does not apply to the split.
• Check the size mark on the chair at every unit. A colour code is a pass or fail label applied in-house, not a tolerance, and a wrong colour on a correct chair is a labelling defect no dimensional check will ever find. 1 seat height measurement identifies the size with certainty here, because the Size 4 band of 370 to 390 and the Size 5 band of 420 to 440 are separated by 30 mm.
• Release the balance against the packing list, not against the goods. The buyer's own check of the factory's carton-mark template and packing list happens before loading and costs nothing but 3 minutes. On this order it would have surfaced the blank size field before a single chair was packed.

What the check would have cost against what the mix-up cost

The inspection that was booked cost 1.5 to 2 man-days. It produced a report with zero defects in it and a consignment that was 19.7 per cent of the order value wrong.

The check that would have caught it is smaller than the one that was booked. Reading 60 carton marks and opening only the ones whose mark is missing or unreadable is 1 man-day at $199, 1.1 per cent of the order value. A census carries no sampling error, so there is nothing to interpret afterwards and nothing to argue about.

The ratio is 18 to 1, and the ratio is not the interesting part. The interesting part is the third line of the table that is not money: a set of carton marks carrying the size would have been printed by the factory at no cost to anybody, would have been checked against the packing list in 3 minutes before loading, and would have made the consignment self-identifying for the caretaker team as well. The expensive check was never going to be the thing that saved this order. The paperwork was.

Four questions to ask before the next delivery

Every school order carrying more than 1 chair size has this failure available to it, and it is not an unusual way to fail. It fails quietly, it fails inside the tolerance, and it gets caught by a person counting chairs in a corridor rather than by an instrument.

• Does my purchase order state the size split per chair, or per carton? A split written only as quantities against a seat height cannot be checked at the container door, because nothing at the container door carries the height.
• If 1 size cannot be made in time, is it permitted to make up the number from another size, and who signs that off? That answer belongs in the purchase order, before the legs run short, not in an email afterwards.
• Does my report verify the split, or does it only report that the sample passed? And does it record the sampling frame, so a reader can see whether 50 units came from 50 places or from 7?
• Who counts the cartons at my end, and what does the carton mark have to tell them? If the answer is that the caretaker team opens every carton and sorts by eye, then the carton mark is missing a field that costs nothing to print.