Linear interpolation is a way to estimate a value that falls between two known points on a line. The formula takes two known data points and finds the missing value that sits proportionally between them. You can write it as: y = y1 + (x – x1) × (y2 – y1) / (x2 – x1). This works when you know two coordinates and want to find a third one that lies on the straight line connecting them.
How To Interpolate Between Two Values Formula Examples?
The linear interpolation formula is straightforward once you break it down. You have two known points: (x1, y1) and (x2, y2). You want to find the y-value that corresponds to some x that sits between x1 and x2.
The formula is:
y = y1 + (x – x1) × (y2 – y1) / (x2 – x1)
Here is what each part means:
The fraction (x – x1) ÷ (x2 – x1) gives you the proportion of the distance you have traveled from point 1 toward point 2. Multiply that proportion by the total change in y, and add it to y1. That gives you your estimated y-value.
Let us walk through a concrete example. Suppose you know that at 2:00 PM the temperature was 68°F and at 6:00 PM it was 80°F. You want to estimate the temperature at 4:00 PM.
Your points are (2, 68) and (6, 80). Your target x is 4.
Plug the numbers into the formula:
The estimated temperature at 4:00 PM is 74°F. Notice that 4:00 PM is exactly halfway between 2:00 and 6:00, and 74 is exactly halfway between 68 and 80. That is linear interpolation working as intended.
When Should You Use Linear Interpolation?
Linear interpolation is appropriate when the relationship between your two variables is approximately straight. This happens often in tables, charts, and data sets where values change at a fairly steady rate.
Common real-world uses include:
For example, a car speedometer calibration table might list speed at 1,000 RPM intervals. If your engine runs at 2,500 RPM, you can interpolate between the 2,000 and 3,000 RPM rows to estimate the speed. As long as the engine speed and vehicle speed follow a roughly linear relationship in that range, the estimate will be close.
Linear interpolation assumes a straight line between your two points. It does not account for curves, acceleration, or sudden changes. If your data curves sharply between the points you know, the interpolated value will be less accurate.
The Difference Between Interpolation and Extrapolation
Interpolation estimates a value between two known data points. Extrapolation estimates a value outside the range of your known data. This distinction matters because the two methods carry very different levels of reliability.
When you interpolate, you are working within the range of data you have observed. The estimate is grounded in actual measurements on both sides of your target value. Even if the relationship is not perfectly linear, interpolation between two nearby points is usually a reasonable approximation.
When you extrapolate, you extend the line beyond your known data. You are assuming the trend continues in the same direction at the same rate. That assumption can fail badly. A growth trend that looks straight over a short window may flatten out or accelerate beyond the data you have seen. Extrapolation is inherently riskier than interpolation.
Use the same formula for both, but treat the results differently. Interpolated values are estimates you can generally trust if your data is sound. Extrapolated values are guesses that should be labeled as such.
How To Interpolate When the Points Are Not Equally Spaced
The linear interpolation formula does not require your known points to be evenly spaced. It works for any two points, regardless of the distance between them. The proportion calculation handles uneven spacing automatically.
Consider this example. A plant grows to 10 cm on day 3 and 22 cm on day 10. You want to estimate its height on day 7.
Your points are (3, 10) and (10, 22). Your target x is 7.
The days are not equally spaced from the endpoints. Day 7 is 4 days after day 3 and 3 days before day 10. The formula correctly weights the estimate closer to day 10 because day 7 is nearer to it. The result, 16.86 cm, reflects that weighting.
This is the key advantage of the formula. You do not need to eyeball a graph or guess a proportion. The math does it for you.
Using Linear Interpolation in Spreadsheets
If you use Excel, Google Sheets, or another spreadsheet program, you do not need to write the formula by hand every time. You can build it once and reuse it.
In Excel, you can use the FORECAST function for linear interpolation. The syntax is:
=FORECAST(x, known_y’s, known_x’s)
For the temperature example, if your known y-values are in cells B1 and B2 and your known x-values are in cells A1 and A2, the formula would be:
=FORECAST(4, B1:B2, A1:A2)
This returns 74, the same result you got by hand.
In Google Sheets, the same function works. You can also use TREND or LINEST for more complex linear regression, but FORECAST is the simplest option for basic interpolation between two points.
If you prefer to see the math in your spreadsheet, you can type the full formula directly:
=B1 + (4 – A1) × (B2 – B1) / (A2 – A1)
Both approaches give the same answer. The choice is about whether you want to see the calculation or hide it behind a function.
Limitations of Linear Interpolation
Linear interpolation is a simple tool, and it has simple limits. The biggest limitation is the straight-line assumption. Real-world data often curves. When it does, linear interpolation between distant points produces errors.
The error grows as the distance between your known points grows. If you are interpolating between two points that are very close together, the straight-line approximation is usually excellent. If the points are far apart and the data curves significantly between them, the estimate can be off.
Temperature over a day is a good example. Temperature follows a curve — it rises in the morning, peaks in the afternoon, and falls at night. If you interpolate between 8 AM and 8 PM, your midday estimate will be wrong because the actual temperature curve bulges above the straight line. Interpolating between two points an hour apart gives a much better estimate.
Linear interpolation also assumes your data has no noise or measurement error. If either of your known points is inaccurate, the interpolation inherits that inaccuracy. The formula cannot correct bad data. It only estimates what the line between those two points would predict.
For data that curves, you have alternatives. Polynomial interpolation fits a curve through multiple points. Spline interpolation uses piecewise polynomials for smoother results. These methods are more accurate for curved data, but they are also more complex. For most everyday estimation tasks, linear interpolation is sufficient.
Frequently Asked Questions
What is the linear interpolation formula?
The formula is y = y1 + (x – x1) × (y2 – y1) / (x2 – x1). It estimates a y-value that falls proportionally between two known points on a straight line.
Can I interpolate between more than two points?
Linear interpolation only uses two points at a time. For more than two points, you would use a method like polynomial interpolation or spline interpolation, which can fit curves through multiple data points.
Does linear interpolation work for negative numbers?
Yes. The formula handles negative values the same way it handles positive ones. The arithmetic works regardless of the signs of your numbers.
Is interpolation the same as extrapolation?
No. Interpolation estimates a value between known data points. Extrapolation estimates a value outside the range of known data. Extrapolation is less reliable because it assumes a trend continues beyond what you have observed.
