Linear Interpolation Calculator

Estimate an unknown value between two known data points on a straight line.

Provide the coordinates of two points and a third point's x-coordinate to find its corresponding y-coordinate through linear interpolation.

Examples

Click on any example to load its data into the calculator.

Temperature Estimation

standard

Estimate the temperature at an altitude of 1500m.

P₁: (1000, 15)

P₂: (2000, 10)

Find for x: 1500

Stock Price Analysis

standard

Estimate a stock's price at 10:30 AM.

P₁: (10, 250)

P₂: (11, 255)

Find for x: 10.5

Sales Growth Projection

standard

Project sales for the 5th month of the year.

P₁: (3, 12000)

P₂: (6, 18000)

Find for x: 5

Scientific Data Interpolation

standard

Find the pressure at a depth of 25 meters.

P₁: (10, 2.0)

P₂: (50, 6.0)

Find for x: 25

Other Titles
Understanding Linear Interpolation: A Comprehensive Guide
Learn the principles of linear interpolation, its mathematical formula, and its wide-ranging applications in various fields.

What is Linear Interpolation?

  • A method for estimating new data points.
  • Assumes a linear relationship between known points.
  • Fundamental for data analysis and function approximation.
Linear interpolation is a mathematical technique used to find a new data point within the range of a discrete set of known data points. It operates on the assumption that the value being estimated follows a straight-line progression between two known points. This method is one of the simplest forms of interpolation and is widely used due to its ease of implementation and calculation.
The Core Concept
Imagine you have two points on a graph, (x₁, y₁) and (x₂, y₂). Linear interpolation allows you to pick any x-value between x₁ and x₂ and find the corresponding y-value that lies on the straight line connecting the two original points. It's like 'reading between the lines' of your data.

Conceptual Examples

  • If temperature is 10°C at 1000m and 5°C at 2000m, you can estimate it's 7.5°C at 1500m.
  • If a plant is 2cm tall on Day 5 and 4cm on Day 9, you can interpolate its height on Day 7 was 3cm.

Step-by-Step Guide to Using the Linear Interpolation Calculator

  • Entering the coordinates of your known data points.
  • Specifying the point you wish to find.
  • Interpreting the calculated result and its meaning.
Our calculator simplifies the process, but understanding the steps is key to using it effectively.
Input Fields
1. Point 1 (x₁, y₁): Enter the coordinates of your first known data point. This is your starting reference.
2. Point 2 (x₂, y₂): Enter the coordinates of your second known data point. This is your ending reference.
3. Value to Find (x): Enter the x-coordinate of the point for which you want to find the y-value. This value should ideally be between x₁ and x₂.
Calculation and Output
After clicking 'Calculate', the tool will instantly provide the 'Interpolated Value (y)'. This is the estimated y-value that corresponds to your input x-value, lying on the straight line defined by your two points.

Practical Usage

  • Given P1=(2, 4) and P2=(6, 8), finding y for x=4 results in y=6.
  • Given P1=(0, 0) and P2=(10, 20), finding y for x=5 results in y=10.

Real-World Applications of Linear Interpolation

  • Finance: Estimating interest rates or stock prices.
  • Computer Graphics: Resizing images and creating smooth transitions.
  • Engineering & Science: Analyzing data from experiments.
Linear interpolation is not just a theoretical concept; it has numerous practical uses.
In Finance
Analysts use it to estimate values from yield curves or to determine stock prices at times between discrete ticks.
In Computer Graphics
When you resize an image, linear interpolation (or a more complex variant like bilinear interpolation) is used to calculate the color values of the new pixels.
In Science and Engineering
When experimental data is collected at intervals (e.g., temperature readings every hour), interpolation can be used to estimate the values at any point in time between readings.

Industry Use Cases

  • Estimating a country's population in 2015 based on census data from 2010 and 2020.
  • Calculating the fuel level in a tank at a specific time based on two separate readings.

Common Misconceptions and Correct Methods

  • Interpolation vs. Extrapolation.
  • The assumption of linearity.
  • When not to use linear interpolation.
Interpolation vs. Extrapolation
A common mistake is to confuse interpolation with extrapolation. Interpolation is estimating a value within the range of your known data (between x₁ and x₂). Extrapolation is estimating a value outside that range, which is significantly less reliable as it assumes the linear trend continues indefinitely.
The Linearity Assumption
Linear interpolation is only accurate if the underlying relationship is truly linear. If the data follows a curve (e.g., exponential growth), linear interpolation will produce an error. For curved data, polynomial or spline interpolation methods are more appropriate.

Points of Caution

  • Using data from 9am and 10am to estimate a value at 11am is extrapolation, not interpolation.
  • Applying linear interpolation to predict a rocket's trajectory (a curve) will lead to inaccurate results.

The Mathematical Derivation and Formula

  • Understanding the slope-intercept form.
  • Deriving the interpolation formula step-by-step.
  • A worked example with detailed calculation.
The formula for linear interpolation is derived from the equation of a straight line.
The Formula
The classic formula is: y = y₁ + (x - x₁) * (y₂ - y₁) / (x₂ - x₁). This equation essentially finds the fraction of the horizontal distance (x - x₁) relative to the total horizontal distance (x₂ - x₁) and applies that same fraction to the total vertical distance (y₂ - y₁), adding it to the starting vertical position (y₁).
Worked Example
Let's find y for x=3, given the points (2, 5) and (4, 9).
x₁=2, y₁=5
x₂=4, y₂=9
x=3

y = 5 + (3 - 2) (9 - 5) / (4 - 2)
y = 5 + (1) (4) / (2)
y = 5 + 4 / 2
y = 5 + 2
y = 7

Behind the Math

  • The term (y₂ - y₁) / (x₂ - x₁) is the slope of the line.
  • The term (x - x₁) is the horizontal distance from the first point to the point of interest.