Interpolation and Data Estimation

Linear interpolation assumes that the change between two known data points is approximately straight when viewed locally. It is a way of estimating within the range of measured values rather than beyond them.

That distinction matters because interpolation is usually more defensible than extrapolation. Inside the known interval you are filling a gap. Outside it, you are betting that the pattern continues.

A linear estimate is reasonable when the interval is small and the underlying relationship is fairly smooth across that span. It becomes weaker when the data are curved, threshold-based, oscillatory, or sparse.

This is why plotting or at least mentally sketching the data helps. If the trend clearly bends, a linear estimate should be framed as a rough local convenience rather than a faithful model of the process.

Interpolation takes a known starting value and adds a fraction of the difference between the two surrounding values. The fraction comes from how far the target x-value sits between the two known x-values.

That means the formula is really a weighted average of the two y-values, with weights determined by position in the interval.

Suppose a value is 12 at x = 4 and 20 at x = 8. At x = 5, the target sits one quarter of the way across the interval, so add one quarter of the y-change of 8. The interpolated value is therefore 14.

This example shows why interpolation is intuitive: it is just proportional movement between two neighbouring points.

Interpolation sits naturally beside descriptive statistics and probability because it is part of responsible data reasoning: summarise what you know, estimate cautiously where justified, and describe the uncertainty honestly. Use it when you need a missing in-range estimate, not as a substitute for collecting better data.