The rule
y = y₁ + (x − x₁) × (y₂ − y₁) ÷ (x₂ − x₁)
Why it works
Linear interpolation draws a straight line between two known points and reads off any value in between. It is the simplest possible assumption about what happens between measurements, and often a perfectly good one.
How to do it by hand
- Work out the slope between the two known points
- Measure how far along the range your x sits
- Multiply that distance by the slope
- Add it to the first point's y value
What is worth knowing
The distinction between interpolating and extrapolating matters far more than the arithmetic. Interpolating between two measurements is usually safe, because you have evidence on both sides. Extrapolating past the last one assumes the pattern continues, and that assumption has no evidence behind it — which is why the calculator flags it. Linear interpolation is what every lookup table with intermediate values does, from tax brackets to steam tables, and it is the crudest member of a family that goes up through polynomial and spline interpolation when a straight line will not do.