Interpolation Calculator

What Is Interpolation?

Interpolation estimates unknown values between known data points by assuming a smooth relationship between them. Linear interpolation — the simplest form — draws a straight line between adjacent known points and reads off the intermediate value. It is widely used in engineering lookup tables, calibration curves, and data analysis.

When the target x lies between two known points (x₁, y₁) and (x₂, y₂), linear interpolation computes y by proportional weighting: y = y₁ + (x − x₁)×(y₂ − y₁)/(x₂ − x₁). This assumes a linear relationship within each segment.

Extrapolation extends the estimate beyond the data range. While the same formula can be applied, accuracy degrades rapidly because the true relationship outside the observed range is unknown. Results should always be flagged and treated with caution.

Formula: y = y₁ + (x − x₁) × (y₂ − y₁) / (x₂ − x₁) where (x₁, y₁) and (x₂, y₂) are the bracketing data points and x₁ ≤ x ≤ x₂ for interpolation

Example Calculation

Known points: (10, 50) and (20, 100). Find y at x = 15. y = 50 + (15 − 10) × (100 − 50)/(20 − 10) = 50 + 5 × 5 = 75. The result is exact because the relationship is linear. For x = 25 (extrapolation): y = 50 + 15 × 5 = 125.

When to Use This Calculator

• An engineer looking up a value between entries in a calibration table or material property chart
• A scientist estimating a measurement at a point between observed data samples without fitting a full regression model
• A technician reading between lines of a pressure-temperature chart or sensor calibration curve
• A data analyst filling in missing values in a time series where surrounding data points are known and reliable

Common Mistakes to Avoid

• Assuming linear interpolation is accurate for non-linear relationships — if the underlying function curves significantly between data points, linear interpolation introduces error; use more data points or higher-order interpolation
• Treating extrapolated values with the same confidence as interpolated values — extrapolation accuracy degrades rapidly outside the data range; always flag extrapolated results and validate against domain knowledge
• Using interpolation on noisy data — interpolation passes through every data point, including noise; if data contains measurement error, regression (which smooths through noise) is more appropriate
• Not sorting data by x-values before interpolating — the algorithm requires ordered data to find the correct bracketing interval; unsorted data produces incorrect results

How to Interpret Results

• If the extrapolation warning appears, treat the result as a rough estimate only — the further the target x is from the data range, the less reliable the result
• For well-spaced data with a smooth underlying relationship, linear interpolation is typically accurate to within a few percent of the true value
• If you need higher accuracy between widely spaced points, consider adding more data points in the region of interest rather than relying on interpolation

Related Standards & References

• Burden & Faires, Numerical Analysis (Cengage) — standard textbook treatment of polynomial and piecewise-linear interpolation with error bounds
• Press, Teukolsky, Vetterling & Flannery, Numerical Recipes (Cambridge University Press) — practical interpolation and extrapolation algorithms and their caveats

Frequently Asked Questions

When should I use interpolation vs. regression?

Use interpolation when your data points are exact (e.g., from a calibration table or known physical law) and you need values between them. Use regression when your data contains noise or measurement error and you want to find the best-fit trend. Interpolation passes through every data point; regression may not.

How reliable is extrapolation?

Extrapolation is inherently risky because you are assuming the observed trend continues beyond the data range. For short extrapolations (within ~10% of the range), linear extension may be reasonable. For larger extrapolations, the error grows rapidly and the result should be validated against domain knowledge or additional data.