Math & Statistics

Linear Interpolation Calculator

Calculate the straight-line y value at a target x from two entered coordinate endpoints. Review x and y spans, fractional position, slope, target offset, comparison gap, and whether the target lies inside or outside the segment.

Starting xStarting yEnding xEnding yTarget xEntered comparison y

Linearly interpolated y-

Entered x interval-

Entered y interval-

Target position within x interval-

Segment slope-

Target x minus start-

Interpolated y minus comparison-

Absolute target distance from segment midpoint-

Decision view

Coordinate interpolation with projection guides

Coordinate interpolation with projection guidesThe two entered endpoints define a line segment, and the target x projects to the interpolated y and comparison reference.

Exact scenario comparisonTarget x changes while all other entered assumptions remain constant.

Target x	Linearly interpolated y	Entered x interval	Entered y interval	Target position within x interval	Segment slope	Target x minus start	Interpolated y minus comparison	Absolute target distance from segment midpoint

How to use Linear Interpolation Calculator

Enter two distinct x endpoints with their corresponding y values.
Enter the target x in the same coordinate system.
Check whether the target fraction lies between zero and one.
Compare the calculated y with the entered reference only after confirming a straight-line assumption.

Calculator guide

Understanding Linear Interpolation Calculator

Linear interpolation locates a target x between two known endpoints and carries the same fraction across the y span. This page exposes the coordinate fraction, slope, offset, interpolated point, comparison gap, and extrapolation warning distance.

One shared fraction Horizontal position determines vertical position.

Slope is preserved The target lies on the endpoint line.

Inside is interpolation Fractions from zero through one remain on the segment.

Outside is extrapolation The arithmetic continues but the assumption is stronger.

Calculation method

How the calculation works

Locate the target x as a fraction of the entered endpoint interval and apply the same fraction to the y interval, preserving slope and offsets. In the Linear Interpolation Calculator, the live scenario varies target x and tracks linearly interpolated y while the remaining results preserve the reconciliation path. Subtract the starting x from the target x and divide by the full x span. Multiply that fraction by the y span and add the starting y.

Detailed calculation process

Map one fractional position across both coordinate spans

The default segment runs from (10, 24) to (18, 52), with target x = 14.5.

General formula: t = (x - x0)/(x1 - x0); y = y0 + t(y1 - y0); m = (y1-y0)/(x1-x0) The target's horizontal fraction t is reused vertically. This preserves the endpoint slope and places the interpolated point on the connecting line.

What each symbol means

x0, y0 Starting endpoint coordinates in x and y units.

x1, y1 Ending endpoint coordinates in x and y units.

x Target x coordinate in the same unit as x0 and x1.

t Unitless fractional position across the x interval.

y Interpolated target value in y units.

m Line slope in y units per x unit.

Worked substitution with the default inputs

1. Measure the endpoint spans: Delta x = 18 - 10 = 8; Delta y = 52 - 24 = 28 The x endpoints must differ so the fraction and slope are defined.

2. Locate the target horizontally: x - x0 = 14.5 - 10 = 4.5 The target is 4.5 x units from the start.

3. Convert the offset to a fraction: t = 4.5 / 8 = 0.5625 The target lies 56.25% of the way from the start to the end.

4. Carry the fraction into y: y = 24 + 0.5625 x 28 = 24 + 15.75 = 39.75 The same fractional progress is applied to the y span.

5. Reconcile slope and comparison: m = 28/8 = 3.5; 24 + 3.5x4.5 = 39.75; 39.75 - 40 = -0.25 The target is 0.5 x units from the segment midpoint and remains inside the endpoints.

The default target fraction is 0.5625, the segment slope is 3.5 y-units per x-unit, and the interpolated point is (14.5, 39.75), which is 0.25 below the comparison.

Geometry check

Read the target directly on the coordinate segment

Projection guides make the substitution visible instead of hiding it in one result.

Start point Defines fraction zero.

End point Defines fraction one.

Target projection Connects x target to interpolated y.

Comparison point Shows the entered reference at the target x.

Worked situations

Practical examples

A target fraction of 0.5 would land at the coordinate midpoint.
The default target fraction 0.5625 produces y = 39.75.
Targets outside the x endpoints use the same formula but become extrapolation.

Better inputs

Useful tips

Sort endpoints for easier interpretation, though the algebra works in either direction.
Plot the target and endpoints to inspect extrapolation.
Use a nonlinear model when the underlying relationship is curved.

Before relying on the result

Limitations and common mistakes

Interpolation assumes a straight relationship between the two endpoints.
Equal x values make the slope and fraction undefined.
Sparse or noisy endpoints, discontinuities, and targets outside the segment can make the result unreliable.

Reference

Key terms

Interpolation fraction: Target position as a share of the endpoint x span.
Slope: Change in y per unit change in x.
Extrapolation: Applying the line beyond the entered endpoint interval.

Important note

Calculated directly from the entered values using the displayed formula and rounding settings.

Frequently asked questions

Must the target be between the endpoints?

For interpolation, yes. The formula also calculates outside values, but that is extrapolation.

Can the line slope downward?

Yes. A negative y span produces a negative slope and a descending segment.

Why can x endpoints not be equal?

The formula divides by their difference, which would be zero.

Does linear interpolation fit all data?

No. It is appropriate only when a straight change between the selected endpoints is defensible.