Deriving the two to one, rather than believing it
The string is a straight line from the nut slot to the top of the saddle. Call the distance from the nut to the saddle the scale length S, and let the string sit at some height above the fret plane at each end: a small height at the nut, a larger one at the saddle. Anywhere between the two, the height above the fret plane is whatever it is at the nut plus the rest of the climb in proportion to how far along you are.
Now lower the saddle by an amount, without touching the nut. The nut end has not moved, so the whole line pivots about it. At a point that is a fraction u of the way from nut to saddle, the line drops by that fraction of the saddle movement. So the action at that point changes by u times the saddle change, and the saddle change is the action change divided by u.
The twelfth fret sits at exactly half the scale length. That is not an approximation and it is not scale-dependent: fret positions are the scale length minus the scale length over two to the power n over twelve, and at n equal to twelve that is the scale length minus half of it. So u is exactly 0.5 and the divisor is exactly 2. Twenty thousandths off the action at the twelfth fret takes forty off the saddle, on any instrument with any scale length.
Go to another fret and the number changes. The seventeenth fret on a 25.5 inch scale sits 15.948 inches from the nut, which is 0.625 of the scale, so the ratio there is 1.60. The fifth fret sits at 0.251 of the scale and the ratio is 3.99. Measure at the first fret and it is 17.8, which is a fair warning that the first fret is not the saddle department.
The nut height does not change the saddle figure
This one surprises people who expect everything to interact with everything. Raise the nut and the whole string line lifts at the low frets, the action at the twelfth goes up a little, and it looks as though the saddle arithmetic must change. It does not. The nut end is the pivot, and lowering the saddle rotates the line about it. How high the pivot sits does not affect how much the line drops at the halfway point when you move the far end.
What the nut height does change is the action at every fret, which is why the page takes it as an optional input for the table. A guitar with a 20 thou clearance at the first fret and one with a 6 thou clearance can have identical action at the twelfth and feel completely different in the first position. That difference belongs to the nut and no amount of saddle work will address it.
Reading the table honestly
The table draws a straight line from nut to saddle, which is what the string is when the frets lie in a plane. Real necks are not planes. They carry relief, which is a bow that lowers the frets in the middle of the board relative to a straight line, and every thou of relief adds a thou of clearance over the frets near the middle of the bow while adding nothing at either end.
So the middle rows of the table read lower than a measurement will. That is the model being honest about what it left out, not a bug: the relief contribution is a separate quantity, measured separately, and adding a guessed value would make the table look more precise and be less true. Measure relief on its own page and add it to the middle rows if you want the full picture.
The other thing the table cannot see is fret levelling. One fret standing a few thou high is invisible to every calculation on this site and audible immediately as a buzz, and it is the reason a setup starts with the frets rather than the saddle.
Thousandths, millimetres and sixty-fourths
Setup work runs in three units that all describe the same distances, and mixing them causes real mistakes. A thousandth of an inch is 0.0254 mm. A sixty-fourth of an inch is 15.6 thou or 0.397 mm. Common action figures land where all three are awkward: 6/64 is 93.75 thou is 2.38 mm.
This page takes thousandths because that is the unit a feeler gauge set is marked in and because it divides sensibly. It shows the same numbers in millimetres and sixty-fourths so you can carry them into whatever the rest of your notes use. The one habit worth keeping is writing the unit down every time, because a note that says the action is 6 means nothing a month later.
Where this connects
The other two things holding the string off the frets are relief, on the neck relief calculator, and the nut, on the nut slot depth calculator. When the saddle runs out of height, the question moves back to the neck angle and what the fret plane projects to at the bridge, which is the neck angle and saddle height calculator. Action height and string gauge together decide how the strings feel, and the tension half of that is the string tension calculator. Fret positions, which every distance here is measured to, come from the fret position calculator.
Questions people ask
Why does the saddle move twice as far as the action?
Because the twelfth fret is exactly halfway along the string and the nut end does not move. Lowering the saddle rotates the string line about the nut, so a point halfway along drops half as much as the end does. To take twenty thousandths off the action at the twelfth fret you have to take forty off the saddle. This is the single most common setup mistake, because sanding the amount you wanted to drop the action feels like the obvious move and produces half the result. The ratio is exact and does not depend on the scale length.
Does the ratio change if I measure action at a different fret?
Yes, and substantially. The ratio is the scale length divided by the distance from the nut to the fret you measured at. At the twelfth it is exactly 2. At the seventeenth on a 25.5 inch scale it is about 1.6, so a twenty thou change there takes 32 off the saddle rather than 40. At the fifth it is about 4. Measure at the first fret and the ratio is nearly 18, which is a signal rather than a number to use: that end of the neck is the nut department.
Does the nut height change how much I take off the saddle?
No, and this catches people out. The nut is the pivot the string line turns about when you move the saddle, so how high the pivot sits does not affect how far the halfway point drops for a given movement at the far end. What the nut height does change is the action everywhere, especially in the first few frets, which is why the page takes it as an input for the table. If the first position feels stiff, that is the nut, and taking material off the saddle will not fix it.
Why is my measured action higher than the table says at the middle frets?
Almost certainly relief. The table draws the string as a straight line from nut to saddle over a flat fret plane, and a real neck is bowed. That bow lowers the frets in the middle of the board relative to the straight line, which adds clearance at exactly the frets in the middle of the table and none at either end. The model leaves relief out on purpose, because guessing it would make the numbers look more precise without making them more true. Measure it separately and add it to the middle rows.
What if the saddle runs out before I reach the action I want?
Then the saddle is no longer the thing that decides it. A saddle sitting flush with the bridge has no more to give, and the remaining variables are behind it: the angle the neck is set at, the height the fret plane projects to at the bridge, and on some instruments the bridge height itself. Those are structural changes rather than setup adjustments and they belong to whoever does that kind of work. The page flags the case rather than suggesting how to get around it, and there is a separate page for the geometry involved.