Interpolate and Extrapolate With Fitted Curves

Interpolation is estimating y at an x that sits between your known data points. You fit a function (linear segment, polynomial, spline) through the data and evaluate it at the missing x. As long as the underlying relationship behaves smoothly across that gap and the surrounding points are reliable, the estimate inherits roughly the precision of those neighbors.

Extrapolation is not interpolation, and treating the two as if they were is the standard mistake. Going past the boundary of observed x's is a fundamentally different problem: you're assuming the model that fit your known range continues to hold outside it, with no data to check the assumption. Polynomials in particular blow up outside their fit range. A cubic that bends gently through five points can shoot to ±10⁴ a few units past the last observation (Runge's phenomenon, since you'll see it cited that way). Linear extrapolation is more stable than polynomial but still rests on the linearity assumption holding off-domain. The page flags every query as either interpolation or extrapolation based on x's position relative to min(x) and max(x), and surfaces residuals so you can sanity-check the fit before trusting any prediction inside the range, let alone outside.

Enter (x, y) pairs representing known measurements. Each x-value must be distinct—two points at the same x with different y values creates ambiguity. If you have repeated measurements, average them first or pick the most reliable reading.

Specify the query x where you want an estimated y. The tool determines automatically whether this x falls within your data range (interpolation) or outside it (extrapolation). The distinction matters: interpolation leverages surrounding data, while extrapolation projects a trend into unknown territory.

Data quality drives result quality. Outliers, transcription errors, or measurement noise propagate through the fitted curve. Inspect your points visually before trusting any estimate. A single misplaced point can tilt a polynomial dramatically.

Linear interpolation connects adjacent points with straight segments. For a query between x₁ and x₂, it weights the y-values by distance: y ≈ y₁ + (y₂ - y₁) × (x - x₁) / (x₂ - x₁). Simple, fast, and robust when data changes smoothly. It won't capture curvature but avoids wild swings.

Polynomial interpolation fits a single curve through all points. A degree-n polynomial can pass exactly through n + 1 points. Quadratic (degree 2) handles one bend, cubic (degree 3) handles two. Higher degrees risk oscillation between points (Runge's phenomenon), producing unrealistic wiggles.

Pick linear when data trends are roughly straight or when you have few points. Pick polynomial when you see clear curvature and have enough points to justify the degree. A degree-4 polynomial through five points will pass exactly through each, but that doesn't guarantee accuracy—it may overfit noise.

Extrapolation asks the model to venture past observed boundaries. Linear extrapolation assumes the slope continues unchanged—often reasonable for short steps but unreliable over long distances. Curves bend; systems saturate; relationships shift.

Polynomial extrapolation is far worse. Outside the data window, polynomials can shoot toward infinity or dive negative depending on leading-term signs. A cubic that fits five points nicely may predict absurd values one unit beyond the last observation.

Treat any extrapolated value as tentative guidance, not a reliable forecast. Flag it in reports. If critical decisions hinge on that estimate, collect additional data closer to the region of interest rather than stretching a model beyond its foundation.

Residuals are observed y minus fitted y at each data point. Small, randomly scattered residuals suggest the model captures the relationship adequately. Patterns—a systematic curve, increasing spread—indicate the model misses structure.

For polynomial interpolation passing exactly through all points, residuals at those points are zero by construction. That doesn't prove the curve is "right"—it just fits the data exactly. The curve's behavior between points may still oscillate or stray from the true underlying function.

Inspect residuals especially when using least-squares regression curves (not exact-fit polynomials). A parabolic residual pattern means a linear fit missed curvature; a funnel shape signals heteroscedasticity. Residual analysis guides model refinement.

The chart overlays your data points with the fitted curve and marks the query location. Seeing where your estimate sits relative to the data helps you judge plausibility. An estimate that falls amid clustered points feels more credible than one perched on a curve bending away into empty space.

Zoom out to view curve behavior beyond the data window. Polynomial curves often behave wildly outside their domain—visualizing this reinforces caution about extrapolation. A curve shooting upward at the edge warns you not to trust values projected further out.

Visualization also reveals outliers. A point far from the curve might be a measurement error or a genuine anomaly worth investigating. Either way, its influence on the fit deserves attention.