Method
Between two marks: how to interpolate honestly
How to estimate a reading between millimetre lines, and where the estimate stops being real information.
Guides on Method: Adding up the errors, Repeatability and agreement, start to finish, Bone-pressed length, start to finish, Stretched flaccid length, and why the clinics take it, Girth, done the way the studies do it, The measurement glossary
You can honestly estimate a reading to about half a division on a millimetre ruler, so a mark-and-a-half reads as .5 mm. Finer than that, your eye is inventing digits. Record the nearest mark and any half-division separately, and keep the rule for converting them the same every time.
What a ruler gives you
A ruler marked every millimetre has a resolution of 1 mm. That is the spacing between the lines, and it is the smallest difference the instrument displays without your help.
The Joint Committee for Guides in Metrology's Guide to the expression of uncertainty in measurement (JCGM, 2008) treats the finite resolution of an indicating device as an uncertainty source in its own right. An instrument cannot be more precise than the smallest step it shows, unless the reader contributes something. The reader contributes by interpolating, and the question is how much of that is information.
The half-division
When a mark falls close to a line, you read the line. When it falls clearly between two lines, you can usually tell whether it is nearer the lower, nearer the upper, or about in the middle. That is three states, which is why the half-division is the natural unit of interpolation: line, middle, next line.
On a millimetre ruler that gives a practical reading unit of 0.5 mm. Some people can split to a quarter division with a good light, a magnifier and a fine, thin line, and that is a claim you should test on yourself rather than assume. A repeat-reading test will tell you, because an interpolated digit that is real shows up as repeatable and one that is not shows up as scatter.
When interpolation stops being real
A simple test: take several readings of the same mark, repositioning each time, and write each to the finest division you think you can resolve. If the last digit is scattered at random while the rest holds still, that digit carries nothing. Drop it.
Three things make the last digit untrustworthy.
- Line thickness. A printed line can be a significant fraction of a millimetre wide, so where inside the line you read is arbitrary. Marking thickness and resolution covers this.
- Parallax. A ruler read at an angle shifts the apparent position, and a thick or transparent ruler does it more, as transparent rulers and parallax describes.
- The object. The thing you are reading against is soft and not perfectly still, so the position you are interpolating to may move by more than a tenth of a division between looks.
Where any of these is larger than the interpolation step, the step is not real.
What to write down
Write the raw observation, not the converted number.
| What you saw | What to record |
|---|---|
| Mark sits on the 13.2 line | 13.2 |
| Mark sits roughly midway between 13.2 and 13.3 | 13.2, half |
| Mark is just past the 13.2 line | 13.2, slight |
Convert later, under a rule you set in advance. For instance, "half" becomes +0.05 cm and "slight" becomes 0, or you round half-divisions to the nearer even millimetre. The rule matters less than applying it every time and writing it into the protocol, which is how rounding to the nearest millimetre approaches the same decision.
Deferring the conversion keeps your eye and your rounding rule separate. If you round as you read, you cannot tell afterwards whether a number moved because the object did or because your rule did.
How much to keep in the final figure
Once the readings are in, the final figure should not claim more resolution than the readings support. Averaging three half-division readings can legitimately give a number between the marks, but it will not support a hundredth of a millimetre. Quoting 13.033 from a ruler marked every millimetre is false precision in decimals, and the usual rule is to report to roughly the size of your interpolation step and no finer.
An easy form of that rule: report to the nearest 0.05 cm if you honestly read half-millimetres, to the nearest 0.1 cm if you read to the mark.
A note on digital instruments
A digital calliper or tape displays digits automatically, and the temptation is to treat every displayed digit as real. The same discipline applies, since the displayed resolution is a floor on the error, not a statement of accuracy. For that case, digital calliper resolution is the relevant page.
Where other sites pick up
The estimate between two marks is one reader's judgement, and judgement varies between people, a subject the judges hub takes up for ratings. A photo-based tool states its limits at its accuracy hub, and first steps for rating are covered elsewhere. None of that is interpolation, and the number a service like Rate Cock gives has no ruler marks behind it at all.