Linear Interpolation

Estimating a Value Between Two Known Points

Linear interpolation estimates an unknown value between two known points, assuming a straight-line relationship between them. Enter two known (x, y) points and the x value you want an estimate for, and this calculator finds the corresponding y.

The Formula

$$\vC{y} = y_1 + (x - x_1) \cdot \frac{y_2 - y_1}{x_2 - x_1}$$

where $(x_1, y_1)$ and $(x_2, y_2)$ are your two known points and $x$ is the value you want an estimate for.

Worked Example

A car is at 50 miles at 2 hours, and 140 miles at 5 hours. Where was it at 3.5 hours?

1. $y = 50 + (3.5 - 2) \cdot \frac{140 - 50}{5 - 2} = 50 + 1.5 \cdot 30 = \vC{95 \text{ miles}}$

Key Factors to Consider

• Interpolating between two points is generally more trustworthy than extrapolating beyond them. A target x value that falls between your two known points stays within the range the relationship was actually observed, while a target outside that range assumes the same straight-line trend continues unchanged — an assumption that gets riskier the further outside the known range you go.
• This method is exact for a truly linear relationship, and only an approximation otherwise. A physical process with constant speed, a fixed rate, or any genuinely straight-line relationship gives an exact answer through this formula — but most real-world processes (like compound growth or acceleration) actually curve, so the further apart your two known points are, the more a straight-line estimate can diverge from reality partway between them.
• Reference tables (like lookup tables in engineering, steam tables, or actuarial tables) commonly rely on linear interpolation between listed values. Rather than publishing every possible input value, a table lists a manageable set of known points and expects the reader to interpolate between them — this is one of the most common practical uses of this exact technique.
• Adding a third known point can reveal whether linearity is actually a reasonable assumption. If a third point you know falls close to the line drawn between the other two, linear interpolation is probably reliable in that range — if it falls far off that line, the underlying relationship likely curves and a straight-line estimate should be treated with more caution.

Common Mistakes

• Extrapolating far beyond the known range and treating the result with the same confidence as interpolation. A target value outside your two known points assumes the trend keeps going unchanged — a much riskier assumption than estimating a value that actually falls between them.
• Assuming a straight-line relationship without checking whether the real process is actually linear. Processes like compound growth or acceleration curve rather than follow a straight line, so a linear estimate can be noticeably wrong, especially far from the known points.
• Mixing up which known point is which. Swapping x1/y1 with x2/y2 in the formula produces a different (and wrong) slope — double-check that each known point’s x and y values are entered together correctly.
• Using two known points that are very far apart when the underlying relationship isn’t linear. The wider the gap between your two points, the more room there is for a curve to diverge from the straight line connecting them — closer, more relevant known points generally produce a more trustworthy estimate.

Useful to Know

• Have several data points instead of just two, and want the single best-fitting line through all of them? The Least Squares Regression Calculator calculator handles that broader case.
• Need the slope between two points rather than an estimated value at a third? The Distance, Midpoint & Slope Calculator calculator covers that directly.