# A Guide to Interpolation vs. Extrapolation (Plus Examples)

By Indeed Editorial Team

Published 19 May 2022

The Indeed Editorial Team comprises a diverse and talented team of writers, researchers and subject matter experts equipped with Indeed's data and insights to deliver useful tips to help guide your career journey.

Interpolation and extrapolation are techniques you can use to estimate values from existing data. You can apply the two methods to analyse trends or understand patterns in data. Understanding the differences between interpolation and extrapolation can help you decide when each strategy is suitable for a particular application. In this article, we list terms that relate to interpolation and extrapolation, define the two techniques and provide examples of their application.

## What terms relate to interpolation vs. extrapolation?

Data analysts may use the following terms when talking about interpolation vs. extrapolation:

• Hypothetical data: data points based on estimations instead of actual observations

• Data prediction: statistical techniques to analyse historical facts and estimate future events

• Curve: a path the equation generates when you give it different input values

• Polynomial function: a mathematical expression that contains at least one term that has a power, such as x^2 + y

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## What is interpolation?

Interpolation is a statistical method that estimates an unknown point in a dataset. It calculates the values within a specific range of data points. You can use a function to predict the value of a variable in the data. For example:

The expression y = 2x + 10 produces a line showing different values of y for different values of x. If x ranges from 0 to 10 and you estimate a value within this range, you are interpolating. For instance, you can estimate the value of y when x=6 using the expression y = 2(6) + 10.

## What is extrapolation?

Extrapolation is a method that estimates unknown values outside a dataset's range. You can apply an extrapolating function to calculate a weight outside the boundaries of existing data. For example:

You can use the function y = 2x + 5 to extrapolate different values of y when x is outside the range of the present data. Extrapolation can occur if the data ranges from 0 to 10, while you estimate the value of y at x = 50. Since 50 is outside the current range, the function interpolates the value of y.

## Interpolation vs. extrapolation

Interpolation makes an estimate within the range of the known data, while extrapolation estimates outside the span of the data. Statistical experts may prefer interpolation because it can give a more accurate assessment of an unknown value than extrapolation. This is because extrapolation assumes that the trend in the data continues for data points you collect in the future. For example, if you collect data about cars and observe white cars, extrapolation may assume all cars you see in the future are white. Here are examples to demonstrate how interpolation and extrapolation differ:

### Example of interpolation

A group of scientists collects information about the population of dogs to study different dog breeds in a specific location. They record the number of dogs in the area between January and August. If the data for May is missing, the scientists can predict the dog population for that month using the current data. This approach interpolates data points within the range of the existing data points.

### Example of extrapolation

A school plans to estimate the number of students it may admit in the next three years. The administrators collect data about the number of students they've accepted since opening the school. Using this information, the school can estimate the number of new students it's likely to admit in three years. Since the prediction is for a time outside the school's existing data range, it uses the extrapolation method.

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## Types of interpolation

Some methods of interpolating data include:

### Linear interpolation

Linear interpolation is a method of constructing new data points within the range of a discrete set of known points. Statisticians can use it to approximate a function value between observations. This type of interpolation assumes that the function is approximately linear between the present data instances. You can trace a line between two known points to estimate a missing data point. Finding where this line intersects with the function shows the value of the expression at the desired point.

### Polynomial interpolation

Polynomial interpolation is a method of estimating values between known data points by creating a polynomial equation. A polynomial is an equation with a degree over one. You can define this equation by finding the line of best fit for the data. This process creates a new polynomial with an output that is a weighted sum of the current data. The function calculates the value estimates using present data point values as variables in the equation.

### Spline interpolation

Spline and polynomial interpolation are non-linear interpolation techniques. Since polynomial methods can yield inaccurate results for high-degree polynomials, the spline approach is an alternative to polynomial interpolation. Data analysts may prefer low-degree polynomials, as they combine many polynomials to model the existing data points.

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### Multivariate interpolation

Multivariate interpolation is a method that allows data experts to interpolate multiple variables simultaneously. This feature contrasts with univariate interpolation techniques, such as linear interpolation, which only allow for the interpolation of one variable at a time. Analysts can use multivariate interpolation when they wish to interpolate multiple variables that relate to each other. For example, in meteorological data, you can interpolate various aspects of the weather, such as temperature, air pressure, humidity and wind speed, at the same time.

## Types of extrapolation

Some extrapolation techniques include:

### Linear extrapolation

Linear extrapolation estimates a value based on extending a line from a known set of weights. You can use this technique when there is a linear relationship between data variables. You can select a point and extend the graph line to estimate a value. The point where the line intersects the graph is the desired value.

### Polynomial extrapolation

Polynomial extrapolation allows for the estimation of non-linear values outside the known value range. It's based on a set of known data and relies on a polynomial to estimate the unknown values. You can make predictions by inserting preferred values in the unknown variables of the polynomial. The estimate's accuracy depends on how well the polynomial equation fits the data.

### Conical extrapolation

Conical extrapolation is a technique for predicting the value of a function outside the range of present points using a conic function. The method assumes that the function is extendable beyond the known scope smoothly and continuously. You can perform conical extrapolation by placing a cone over the known data points and predicting the expression by extending the cone outwards.

### Fast Fourier transform extrapolation

Fourier extrapolation converts the data from the time domain into the frequency domain. It does this by taking the Fourier transform of the data. You can extrapolate the resulting data by adding or subtracting harmonics. The last step involves converting the data back into the time domain.

## Real-world applications of interpolation and extrapolation

Here are examples of interpolation and extrapolation applications:

### Communication

For a device to transmit a signal over a long distance, it may first convert the signal to a digital format. For example, when you make a call, your phone converts your voice to a digital signal. As digital data is typically discrete in ones and zeros, the device samples the data at regular intervals. Transmission cables send the sampled signal to the receiving device.

The recipient converts the signal back into analogue format. In a phone call, the receiver converts the digital data into a voice signal. The converter applies interpolation to translate the zeros and ones into the original voice signal. When devices transmit audio signals, noise may alter the information. You can use extrapolation to predict the original data. This prediction can provide clear audio during the reconstruction of the original audio by removing noise.

### Forecasting

You can use extrapolation to forecast future trends. These estimations rely on historical data about the tendency you wish to predict. Some examples of forecasting using extrapolation include:

• forecasting the global population by extrapolating population data from the last decade

• estimating how much a disease may spread by using data from a previous outbreak

• predicting business profits and growth by analysing past strategies

• estimating stock market prices by extrapolating historical stock data

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### Aerospace engineering

When sending a rocket to space, engineers can use a known mathematical formula to represent each of its parameters—for example, speed, acceleration or temperature. They may record the readings from these functions at regular intervals. If an unexpected event occurs, scientists can use the existing data to interpolate values that might be missing. For example, scientists may lose the signal to the rocket for a few minutes. They can interpolate the readings to estimate the rocket's speed or acceleration when the system lost the call.

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