Chained Measurements Versus One Datum

Why this matters

Lay out twelve marks by measuring each one from the mark before it, and every small error rides along into every mark that follows. Twelve marks each up to a thirty-second of an inch off can put the last one 3/8 in from where it belongs, on a wall where a sheet of drywall needed it within a fraction of that. Lay out the same twelve marks all from one end, and each mark is only as wrong as its own reading. Nothing about the tape or the person changed. Only the reference did. This card is about that choice: a datum (one fixed reference every measurement starts from) versus a chain (each measurement starting from the last), why the chain accumulates error, and how to work from a datum on jobs where the tape will not reach.

What actually accumulates

Every mark has a small error: where the tape's hook sat, how the eye read the graduation, which side of the line the pencil went. Call it up to a thirty-second of an inch per mark on a careful job.

In a chain, each mark inherits all the error before it. Mark 2 is measured from mark 1, so mark 1's error is already built into mark 2's starting point, and mark 2 then adds its own. By mark 12, the position carries twelve readings' worth of error.

From a datum, each mark carries only its own. Mark 12 is read straight off the tape at 192 in from the same zero as mark 1. Whatever went wrong at mark 5 stays at mark 5.

16 IN CENTERS, TWELVE SPACES, 192 IN

CHAINED: each space measured from the last mark
         each step lands 1/32 in long, same way

intended   0    16     32         176       192
           |----|------|-- ... ---|---------|
actual     0  16 1/32 32 1/16  176 11/32  192 3/8
error         1/32   1/16       11/32      3/8

DATUM: every mark read from the same zero

intended   0    16     32         176       192
           |----|------|-- ... ---|---------|
actual     0  within 1/32 of its own number, each
error      never more than 1/32 at any mark

The 3/8 in at the end is 12 x 1/32 in = 12/32 = 3/8 in. That is the worst case: every mark wrong by the full amount in the same direction.

Systematic error versus random error

Whether a chain really reaches that worst case depends on what kind of error each step carries, and the two kinds add differently.

• Systematic error leans the same way every time. A pencil line drawn on the far side of the graduation every time, a tape held with the same slight sag, a hook with a bent tip. Each step adds the full amount in the same direction, and the chain reaches the full 3/8 in by mark 12.
• Random error leans either way. Reading the graduation a hair long on one mark and a hair short on the next. Some steps cancel each other, so the accumulated error grows more slowly, roughly with the square root of the number of steps. Twelve random errors of up to 1/32 in typically leave the last mark somewhere around the square root of 12 times 1/32, which is 3.46 / 32 = about 0.11 in, a bit under 1/8 in. Typically, not at most: 3/8 in is still possible.

Real layout has both, and the systematic part dominates, because a person marking twelve times in a row does it the same way twelve times. So plan for the linear sum, not the hopeful square root. A datum makes the question irrelevant, because there is nothing to sum.

Worked example: a stud wall and a sheet edge

A 16 ft wall is framed at 16 in on center: twelve spaces of 16 in, 192 in total, so a 4 ft sheet of drywall or sheathing breaks on a stud every three spaces, at 48, 96, 144 and 192 in. The sheet edge needs to land on the center of a 1 1/2 in wide stud, leaving 3/4 in of nailing for each of the two sheets that meet there.

Lay the studs out by chaining, each step landing 1/32 in long:

• Mark 3, at 48 in: 3 x 1/32 = 3/32 in long.
• Mark 6, at 96 in: 6 x 1/32 = 6/32 = 3/16 in long.
• Mark 9, at 144 in: 9 x 1/32 = 9/32 in long.
• Mark 12, at 192 in: 12 x 1/32 = 3/8 in long.

The sheets do not drift; they come off a factory edge at exactly 48 in. So at the last break the stud center is 3/8 in past the sheet edge. One sheet lands on 3/4 - 3/8 = 3/8 in of stud, and the other on 3/4 + 3/8 = 1 1/8 in. A 3/8 in edge leaves almost no wood for a fastener set at the edge distance the sheet's maker calls for; the crew either adds a stud or cuts the sheet. Neither is a disaster. Both are time that the layout method spent before anyone picked up a hammer.

Lay the same wall out from one datum, reading 16, 32, 48 and on up to 192 off a single tape hooked on the end of the plate, and every mark is within 1/32 in of its number. At 192 in the stud center is within 1/32 in of the sheet edge, and both sheets get close to their 3/4 in. Most tapes print the 16 in centers in a contrasting color for exactly this reason: they are meant to be read from the end, not stepped.

The slips that turn a small error into a large one

Burning an inch. When the hook is awkward, a common practice is to hold the 1 in mark on the starting point and subtract one from every reading. In a datum layout, forgetting the subtraction once puts one mark an inch off. In a chain, the same slip moves that mark and every mark after it, because each later step starts from the wrong place. The habit that prevents it is to burn from a round number like 10 in, so a forgotten subtraction produces a mark that is obviously wrong, and to write "burned 10" at the top of the layout.

Re-hooking on a mark. A tape cannot hook a pencil line. Holding the end of the tape "about on" the last mark adds the full width of the line plus the hold, every step, in the same direction. It is the commonest source of the systematic error above.

Mixing references mid-layout. Running the first half from the left end and the second half from the right end puts two datums on one line, and the error between them lands in the middle as an odd-sized space. That can be a deliberate choice, with the odd space put where it is hidden, but it should be a choice, not a discovery.

When the tape will not reach

Long layouts outrun a tape: a run of fence posts, pipe hangers down a corridor, a ceiling grid, a row of recessed lights. You still want one datum, and the way to keep it is to transfer it deliberately, once, and check it, rather than chaining.

• Set a secondary datum. Measure from the primary datum to a clearly marked secondary point, say at 50 ft, and label it with its distance. Every mark beyond it is measured from that secondary point, not from the mark before.
• Check the secondary datum two ways. Measure it again from the far end, or confirm it against a total you know. A secondary datum that is wrong moves everything after it, so it gets the one careful check the chain never gives any of its marks.
• Use a line or a long tape for the run. A string line or a long tape laid out once and read at each station gives one reference for the whole run.
• Worked: a fence line longer than the tape. Posts go at 8 ft centers along a 160 ft line, twenty spaces, with a 100 ft tape. Hook the tape on the corner post (the datum) and mark posts at 8, 16, 24 and on to 96 ft, all read off that one tape. Post 12, at 96 ft, becomes the secondary datum: drive a stake beside it and write "96 ft from corner" on it. Before using it, measure back from the far corner post, which should read 160 - 96 = 64 ft. If it reads 64 ft, mark posts 13 to 20 at 8, 16 and on up to 64 ft from the stake. If it reads 63 ft 10 in, the 2 in is found now, at one stake, rather than spread across eight post holes. Chaining the same line post to post would have meant twenty separate starts.
• Center outward when symmetry matters. Tile, a ceiling grid and a row of fixtures are commonly laid out from a center line, so that the cut pieces at both ends match. The center line is the datum.

What a datum also buys: a check at the end

A datum layout has a built-in check that a chain does not. Measure the last mark from the datum, compare it with the planned figure, and the difference is the actual error at the far end. In a chain, the last mark has nothing to compare against except the previous mark, which carries the same accumulated error; the chain is consistent with itself and wrong as a whole, and nothing on the wall says so.

On a long chained layout already done, run a tape from the datum to the last few marks before anything is fixed to them. If the last mark is out by more than the job can tolerate, re-mark from the datum. That is cheaper before the studs, posts or hangers are set than after.

Where the same idea appears with other names

Machinists call it baseline dimensioning versus chain dimensioning, and drawings are dimensioned from a datum edge for the same reason. A plan that gives running dimensions all from one wall is a plan built for datum layout; a plan that gives only the space between each pair of items is inviting the chain, and the person laying it out can convert it, adding the spaces into running totals from one end, before marking anything. Write those running totals down on the plan first; adding them up on the fly at the wall is a chain in arithmetic form.

The chain problem is about positions taken one from another. When physical parts stack (a set of washers, shims, a run of fittings each with its own length tolerance), the parts themselves accumulate, and choosing a datum for the tape does not help. That is a different problem, covered in The Tolerance Stack-Up That Causes the Fault.

References

• Fastener edge-distance requirements from the sheet product's installation instructions
• See related: The Tolerance Stack-Up That Causes the Fault; Tape Measure Reading + Fractions Reference; Measurement Resolution and Honest Rounding; How to Square a Layout with Diagonals and a Triple