Linear Interpolation Calculator
Estimate a y-value between two known coordinate pairs, see exactly where the target lies along the interval, and reverse the same line to estimate x from a target y. Extrapolation is labeled separately so it cannot quietly masquerade as interpolation.
Define the bracket
Known point A
Known point B
Bracket result
y = 32.0000 + [(18.0000 − 10.0000) / (25.0000 − 10.0000)] × (77.0000 − 32.0000) = 56.0000
Both x and y increase from A to B; the line is monotone increasing.
What linear interpolation assumes
Linear interpolation estimates a value between two known points by treating the segment joining them as straight. If the target x lies 40% of the way from x₁ to x₂, the estimated y lies 40% of the way from y₁ to y₂. This proportional-distance idea is the heart of the method.
The assumption is local, not universal. It says the relationship is adequately represented by a line across this bracket. It does not claim that the underlying process is linear everywhere, that the two endpoints are free of measurement uncertainty, or that the calculated value is an observed fact. Short, well-chosen brackets are generally more defensible than long intervals across a strongly curved process.
t = (x − x₁)/(x₂ − x₁)
y = y₁ + t(y₂ − y₁)
Equivalent line form: y = mx + b, where m = (y₂ − y₁)/(x₂ − x₁).
Worked example from two temperature readings
Suppose a response is 32 units at x = 10 and 77 units at x = 25. To estimate the response at x = 18, first calculate the horizontal interval: 25 − 10 = 15. The target has traveled 18 − 10 = 8 units from the first endpoint, so t = 8/15 ≈ 0.5333. It is 53.33% of the way across the bracket.
The vertical change is 77 − 32 = 45. Taking 53.33% of 45 gives 24. Adding that change to y₁ gives 32 + 24 = 56. The segment’s slope is 45/15 = 3 response units per x unit, and its intercept is 2, so the equivalent line equation is y = 3x + 2. Substituting x = 18 confirms y = 56.
The reverse target demonstrates the inverse operation. Given y = 56 on this nonhorizontal line, the vertical fraction is (56 − 32)/(77 − 32) = 24/45, again 0.5333. Applying that fraction to x gives 10 + 0.5333 × 15 = 18. Forward and reverse answers agree because both use the same two-point segment.
How to enter endpoints safely
- Use two observations on the same scales. Both x-values need one unit and both y-values another consistent unit. Convert before calculating if one point uses Fahrenheit and another Celsius, or one uses minutes and another hours.
- Use distinct x-values. If x₁ equals x₂, the denominator is zero and the segment is vertical. A single-valued y(x) interpolation is not defined.
- Choose a target x. A target between the endpoints is interpolation. A target outside them is extrapolation, which the result labels or blocks according to the selected policy.
- Optionally enter a reverse target y. Reverse interpolation requires y₁ and y₂ to differ. A horizontal segment cannot identify one unique x from y.
- Retain source precision. Avoid rounding endpoints before the calculation. Round the final result according to the measurement resolution and purpose.
Endpoint order does not matter. Swapping A and B changes the signs of both numerator and denominator in t, leaving the interpolated value unchanged. The fraction is then measured from the newly labeled A, so the track percentage becomes 1 − the former percentage.
Interpolation, extrapolation, and regression are different
Interpolation
Estimates within the observed x bracket. The target is surrounded by the two supplied points, limiting how far the straight-segment assumption is extended.
Extrapolation
Extends the same slope beyond an endpoint. Error can grow quickly when the actual relationship bends, saturates, changes regime, or has physical bounds.
Regression
Fits a relationship to several noisy points and generally does not pass through every observation. It estimates a trend rather than merely joining two anchors.
For two exact bracketing values in a lookup table, linear interpolation is simpler and more transparent. For a smooth curve represented by many points, splines or a domain-specific equation may better preserve curvature.
Fraction t and signed changes
The interpolation fraction t is dimensionless. When x is between x₁ and x₂, t lies from zero through one. At x₁, t = 0 and y returns y₁. At x₂, t = 1 and y returns y₂. A negative t means the target lies beyond A in the opposite direction from B; a value above one means it lies beyond B.
Signed differences make decreasing relationships work without a special formula. If y₂ is lower than y₁, Δy is negative, so the interpolated change is negative. If the x endpoints are entered in descending order, Δx is negative too; the ratio still places the target correctly. The visual track normalizes the location between the smaller and larger x for readability, while the algebra retains the entered A-to-B direction.
Units follow the slope. If y is dollars and x is hours, slope is dollars per hour. The fraction has no unit, and the interpolated y remains dollars. Writing units beside the inputs and final estimate is a quick dimensional-analysis check.
Monotonicity and physical boundaries
A straight segment between two endpoints is monotone: it only increases, only decreases, or stays flat. An interpolated y should therefore lie between y₁ and y₂ whenever target x lies between x₁ and x₂. If a manual result falls outside that vertical range, inspect the signs, parentheses, and endpoint order. This range test is one of the fastest ways to catch an arithmetic error.
Domain limits can be stricter than the mathematical line. Percentages may need to stay from 0% through 100%, concentrations cannot be negative, and a tank level cannot exceed physical capacity. Interpolation between valid endpoints automatically stays between their y-values, but extrapolation can violate those bounds. Clamp a result only when the governing method explicitly requires it; silently replacing a calculated value hides that the linear model has left its meaningful range.
Rounded or censored endpoints create another boundary issue. A published value of “less than 5” is not the exact number 5, and a value rounded to the nearest ten represents an interval. Treating such labels as exact anchors can give a neat estimate with unjustified precision. Preserve qualifiers from the source table in the final report.
Calibration and lookup-table use
Interpolation often appears in calibration tables, engineering charts, time series, tax or rate schedules, and experimental measurements. A bracket should use the nearest trustworthy point below and above the target when the relationship is locally smooth. Skipping closer points makes a longer chord that may conceal curvature.
Instrument calibration adds uncertainty. Endpoint measurements can have errors, and the assumption of straight behavior between them contributes model error. This calculator reports the deterministic line through the entered values; it does not propagate endpoint uncertainty or supply a confidence interval. A formal calibration analysis may require multiple reference standards, repeat measurements, residual diagnostics, and traceability records.
Some schedules are step functions rather than continuous lines. Shipping zones, tax brackets, eligibility cutoffs, and tiered prices may require selecting a band, not interpolating between published thresholds. Read the governing instructions before assuming that values between table entries should be prorated.
Range warning: A mathematically valid extrapolation is not automatically a credible forecast. Check physical limits and use a model supported beyond the bracket.
Manual checks and edge cases
Verify that the result returns each endpoint exactly: target x = x₁ must produce y₁, and target x = x₂ must produce y₂. For a midpoint x = (x₁ + x₂)/2, the result should be the midpoint (y₁ + y₂)/2. Forward and reverse calculations should reconstruct the same target when the segment is not horizontal.
If both y-values are equal, every x on the segment has the same y. Forward interpolation still works, but reverse interpolation is not unique, so the calculator reports it as undefined. If x-values are nearly equal, the slope can be extremely large and sensitive to input rounding. A precise-looking answer may then be numerically fragile.
Never interpolate dates or categories by treating labels as arbitrary numbers unless the numeric spacing is meaningful. Dates can be converted to elapsed time, but weekdays, product grades, and nominal category codes do not automatically form a continuous scale.
Frequently asked questions
Can the endpoints be entered in reverse order?
Yes. The formula uses signed differences, and swapping endpoints leaves the estimated point unchanged.
What if target x equals an endpoint?
The result is exactly that endpoint’s y-value: t is zero at A and one at B.
Why is reverse interpolation undefined for a horizontal line?
When y₁ equals y₂, the same y corresponds to every x on the segment, so there is no unique inverse x.
Does interpolation create a confidence interval?
No. This deterministic two-point calculation does not model measurement noise or uncertainty in the endpoints.
Is a value outside the bracket wrong?
The arithmetic is extrapolation rather than interpolation. It may be useful, but it relies on a stronger continuation assumption.
Should I use a spline instead?
A spline can represent curvature across several smooth points. Use it when the data and domain support that complexity; two-point linear interpolation remains appropriate for a single local bracket.
When the estimate should come from a function’s derivative rather than two surrounding data points, switch to the linear approximation calculator.
References
NIST Dataplot — linear interpolation reference
NIST Digital Library of Mathematical Functions — interpolation