To interpolate, use y = y₁ + (x − x₁) × (y₂ − y₁) ÷ (x₂ − x₁): find how far x sits between x₁ and x₂ as a fraction, then move the same fraction of the way from y₁ to y₂. Between (10, 50) and (20, 80), x = 14 is 40% of the way across, so y = 50 + 0.4 × 30 = 62. This linear interpolation calculator does it for two points, for a whole pasted table, and in 2D (bilinear), and it flags when you are extrapolating outside your data.
Linear interpolation is how you read between the lines of a printed table — steam tables, tax tables, statistical tables, calibration charts — or fill a gap in a series of measurements. It assumes the values change at a constant rate between the two known points.
How to use the calculator
- Two points: enter (x₁, y₁), (x₂, y₂) and the x you want. You get y, the slope, the fraction t and the equation of the line, plus a chart. Switch “Solve for” to x to find the x that gives a known y.
- Table: paste x, y pairs, one per line (commas, spaces or tabs — copy two columns from a spreadsheet). Enter x; the calculator finds the bracketing rows and interpolates between them. Outside the table it extends the end segment and labels the answer as extrapolated.
- Bilinear: enter the four corners of a grid cell and a point (x, y) inside it.
- Copy the link to share or save the exact inputs.
The linear interpolation formula
For two known points (x₁, y₁) and (x₂, y₂):
y = y₁ + (x − x₁) × (y₂ − y₁) ÷ (x₂ − x₁)
An equivalent, often cleaner, form uses the fraction t = (x − x₁) ÷ (x₂ − x₁):
y = (1 − t) × y₁ + t × y₂
This is equation 3.3.14 in the NIST Digital Library of Mathematical Functions, §3.3(ii). The weights (1 − t) and t add to 1, so the answer is a weighted average of y₁ and y₂; the nearer point gets more weight. When 0 ≤ t ≤ 1 you are interpolating; t below 0 or above 1 means extrapolation.
How to interpolate, step by step
- Pick the two known points on either side of your x.
- Subtract: Δx = x₂ − x₁ and Δy = y₂ − y₁.
- Find the fraction t = (x − x₁) ÷ Δx.
- Multiply: t × Δy.
- Add to the starting value: y = y₁ + t × Δy.
A quick check: the answer must sit between y₁ and y₂ whenever x sits between x₁ and x₂. The table shows how t maps to y for the points (0, 20) and (10, 60); at t = 0.5 the result is simply the average of the two y values.
| Position between the points | t | y for (0, 20) → (10, 60) |
|---|---|---|
| 10% of the way | 0.1 | 24 |
| 25% of the way | 0.25 | 30 |
| 50% of the way | 0.5 | 40 |
| 75% of the way | 0.75 | 50 |
| 90% of the way | 0.9 | 56 |
Worked examples
1. Between two points
(10, 50) and (20, 80), find y at x = 14. Slope = 30 ÷ 10 = 3. y = 50 + (14 − 10) × 3 = 62.
2. Interpolating from a table
Table (x, y): (0, 1.2), (5, 3.4), (10, 4.1), (20, 6), (40, 7.9). For x = 12 the bracketing rows are (10, 4.1) and (20, 6): y = 4.1 + (12 − 10) × (6 − 4.1) ÷ (20 − 10) = 4.48. Paste the same rows into Table mode to check.
3. Extrapolation
A value was 150 in 2020 and 170 in 2024. Extending the line to 2030 gives 150 + 10 × 5 = 200. For the same table as example 2, x = 50 extrapolates from the last segment to 8.85. Treat both as trend guesses, not measurements.
4. How big is the error on a curve?
Interpolating y = x² between (2, 4) and (3, 9) at x = 2.5 gives 6.5; the true value is 6.25, so the error is 0.25. The NIST DLMF bound for linear interpolation is h² ÷ 8 × max|f″| = 1 ÷ 8 × 2 = 0.25, and the midpoint of a parabola hits it exactly. Tighter spacing in the table shrinks the error with the square of the gap.
5. Bilinear interpolation on a grid
A 0–10 × 0–10 cell has corners f(0,0) = 20, f(10,0) = 30, f(0,10) = 40, f(10,10) = 60. At (4, 7): along x at y = 0, 20 + 0.4 × 10 = 24; along x at y = 10, 40 + 0.4 × 20 = 48; then along y, 24 + 0.7 × (48 − 24) = 40.8. Doing y first gives the same answer.
Solving for x: inverse interpolation
Sometimes you know the output and need the input — the temperature that gives a certain reading, or the dose that gives a target response. The NIST DLMF (§3.3(v)) describes inverse interpolation as interchanging the roles of the points and the function values. For a straight line between two points that simply rearranges the formula:
x = x₁ + (y − y₁) × (x₂ − x₁) ÷ (y₂ − y₁)
With (10, 50) and (20, 80), a known y of 62 is 12 ÷ 30 = 40% of the way from 50 to 80, so x = 10 + 0.4 × 10 = 14. Set “Solve for” to x in Two points mode to do this directly. Two cautions: the two y values must differ, and if the data go up and then down between your points, more than one x can give the same y — pick an interval where the data change in one direction only.
Worked example: reading between rows of a statistics table
Printed tables skip rows, and interpolation fills the gaps. The NIST t-distribution table lists the 0.975 critical value (used for a two-sided 95% interval) as 2.042 at 30 degrees of freedom, 2.021 at 40 and 2.000 at 60. For 35 df: t = (35 − 30) ÷ (40 − 30) = 0.5, so the estimate is 2.042 + 0.5 × (2.021 − 2.042) = 2.0315. NIST's own row for 35 df is 2.030, so the error is only about 0.0015. For 50 df, interpolating between the 40 and 60 rows gives 2.0105, against the table's 2.009. The error grows with the gap between rows, exactly as the h² term in the error bound predicts. Need the df first? Use the degrees of freedom calculator.
Interpolation vs extrapolation vs curve fitting
| Method | Where x is | Passes through the data? | Use it for |
|---|---|---|---|
| Linear interpolation | Between known points | Yes | Reading tables, filling gaps |
| Linear extrapolation | Beyond the data | Yes | Short-range trend estimates only |
| Regression (line of best fit) | Anywhere | Not necessarily | Noisy data, prediction with uncertainty |
NIST’s Dataplot manual draws the same line: an interpolated function goes through every original point, while a fitted function may not. For smooth curved data, the same manual calls cubic spline interpolation “normally the preferred type”; straight segments are simpler and never overshoot between points.
Common mistakes
- Mixing up which y goes with which x. Keep each pair together; swapping only y₁ and y₂ flips the slope.
- Using rows that don’t bracket x. Pick the rows immediately above and below. Using distant rows adds error on curved data.
- Interpolating on a log scale linearly. Values that grow exponentially (pressure vs temperature, compound growth) often interpolate better on log(y).
- Trusting a long extrapolation. The further past the data, the less the straight-line assumption holds.
To summarise a whole data set with the box plot maker and five-number summary calculator, find the right degrees of freedom for a t, chi-square or F table, or turn scores into percentages with the grade calculator. Preparing for the exam? Try the AP Statistics score calculator.
Frequently asked questions
What is the linear interpolation formula?
y = y₁ + (x − x₁) × (y₂ − y₁) ÷ (x₂ − x₁). It is the equation of the straight line through (x₁, y₁) and (x₂, y₂), evaluated at x. NIST’s Digital Library of Mathematical Functions writes the same thing as f = (1 − t)·f₀ + t·f₁, where t = (x − x₁) ÷ (x₂ − x₁).
How do you interpolate between two numbers?
Work out how far x is between x₁ and x₂ as a fraction t, then move the same fraction from y₁ to y₂. For (10, 50) and (20, 80) at x = 14: t = 4 ÷ 10 = 0.4, and y = 50 + 0.4 × 30 = 62.
How do you interpolate a value from a table?
Find the two rows whose x values bracket your x, then apply the two-point formula to just those rows. Paste the whole table into Table mode and the calculator picks the right pair — for x = 12 in the example below it uses the rows at 10 and 20 and returns 4.48.
What is the difference between interpolation and extrapolation?
Interpolation estimates a value inside the range of your known x values; extrapolation extends the line beyond them. Both use the same formula, but extrapolation assumes the trend keeps going, which is much riskier. NIST’s Dataplot LINEAR INTERPOLATION command refuses points outside the data range for this reason; this calculator allows it and labels the result as extrapolated.
How accurate is linear interpolation?
It is exact when the data really lie on a straight line and approximate otherwise. For a smooth function, NIST DLMF bounds the error by h² ÷ 8 × the largest |f″| on the interval, where h is the gap between the two x values. Halving the gap cuts the worst-case error to a quarter. Example: interpolating x² between 2 and 3 gives 6.5 at 2.5 instead of 6.25, an error of 0.25 — exactly the bound 1² ÷ 8 × 2 = 0.25.
What is bilinear interpolation?
Linear interpolation in two directions on a rectangular grid: first along x on the two grid rows, then along y between those two results. NIST describes it as a weighted average of the 4 surrounding grid points. With corners 20, 30, 40, 60 on a 0–10 grid, the value at (4, 7) is 40.8.
Does the order of the two points matter?
No. Swapping (x₁, y₁) and (x₂, y₂) gives the same line and the same answer. In Table mode the rows can be in any order; the calculator sorts them by x.
Can I interpolate x from a known y?
Yes. In Two points mode set “Solve for” to x and enter the y you know; the calculator uses x = x₁ + (y − y₁) × (x₂ − x₁) ÷ (y₂ − y₁). For (10, 50) and (20, 80), y = 62 gives x = 14. NIST DLMF calls this inverse interpolation: the roles of the points and the function values are swapped. It needs y₁ ≠ y₂, and it only has one answer if the data rise or fall steadily across the interval.
How do you interpolate a value in a t-table?
Find the rows either side of your degrees of freedom and interpolate between their critical values. In NIST’s t-table the 0.975 column reads 2.042 at 30 df and 2.021 at 40 df, so 35 df gives 2.0315. The printed value for 35 df is 2.030, so the straight line is off by about 0.0015 because the table values curve.
What is the interpolation formula in words?
Start at the lower known y, then add the same fraction of the y gap as x covers of the x gap. That fraction is (x − x₁) ÷ (x₂ − x₁); multiply it by (y₂ − y₁) and add y₁.
What if the two x values are the same?
The formula divides by x₂ − x₁, so it is undefined when they are equal (a vertical line). Use two points with different x values.
Sources & method
- NIST Digital Library of Mathematical Functions §3.3(ii) — Linear interpolation, eq. 3.3.14 and error constant 3.3.15
- NIST Dataplot Reference Manual — LINEAR INTERPOLATION (points connected with straight lines)
- NIST Digital Library of Mathematical Functions §3.3(v) — Inverse interpolation (interchange the roles of the points and the function values)
- NIST/SEMATECH e-Handbook §1.3.6.7.2 — Critical values of the Student’s t distribution (table used in the worked example)
- NIST Dataplot Reference Manual — BILINEAR INTERPOLATION (weighted average of the 4 surrounding grid points)
Results are estimates for general information. Found an error? It helps everyone — see our methodology.