Interpolation Vs. Extrapolation: What’s The Difference?

Statistics tend to take on a mystical quality for most people, meaning that the concepts seem beyond reach. For most of it, that isn’t true – the basic concepts and ideas behind statistics aren’t complex. And the more you know about them, the harder it is for someone to use them to deceive or mislead you.

Both interpolation and extrapolation have definitions outside of mathematics, but the way that they’re directly compared to each other and confused with one another is their statistical definition. The way to tell them apart is to focus on the prefixes.

Interpolation is a way to predict values that fall within the data set, hence the prefix inter-. Extrapolation, on the other hand, is a way to predict values that fall outside the data set, hence the extra- prefix.

Key Takeaways:

What Is Interpolation?

Interpolation is a way to find a value that exists inside a data set. This value is hypothetical, as it hasn’t been recorded – it’s going to be one of the values between the other data points that you’ve recorded. Which would mean that it would be one of the values that sit in the gap between two points on a graph.

There are several different types of interpolation.

• Piecewise constant interpolation. Also known as nearest neighbor interpolation or proximal interpolation, this form of interpolation is most often used in higher-dimensional multivariate interpolation. It’s extremely simple; the idea is to assign the same value as a known data point to the next nearest value.

• Linear interpolation. This is a type of interpolation that everyone will be familiar with. The notion behind this is to take two data points and draw a line connecting them. The points you pass through to make the line are your interpolations.

• Polynomial interpolation. As opposed to just drawing a line, this method uses a polynomial of a higher degree. The different calculation leads to a curved line rather than a straight one.

• Spline interpolation. This is similar to a polynomial interpolation, save that it uses a lower-degree polynomial. The result of the calculator is called a spline, hence the name.

• Mimetic interpolation. This type of interpolation is most often used with vectors. It’s much more abstract than the other types, relying more on projection.

While interpolation is most often used in its mathematical definition, it is also used to mean to insert something. However, It is much more specific than insert and has a much more negative connotation.

It’s most often used in reference to, say, historical documents that have been altered. If information or words are added to a book, then they’ve been interpolated. However, it even goes so far as to say “to alter or corrupt” by inserting new or foreign matter.”

The word can be used simply to mean that you added words or text or to “insert between other things or parts,” however, due to the nature of the first definition, using interpolate in that way has an implication of underhandedness.

It comes directly from the Latin interpolare, which means to alter, refurbish, or interpolate. The prefix inter- is from Latin – even being used in the Latin word itself – which means to in, amongst, or between. It first came into English usage in 1612.

What Is Extrapolation?

Extrapolation is a mathematical form of estimation. However, unlike interpretation, extrapolation is estimating outside the data points that you have. So instead of filling in between two points you have, you take the data that you have and keep going with it.

There are a few different ways to extrapolate.

• Linear extrapolation. This is extremely similar to linear interpolation, even using the same equation. The difference is that it extends past the data points you have.

• Polynomial extrapolation. As with interpolation, a polynomial curve can be created to fit the data. It becomes extrapolation when you extend it beyond the values that you have. There are generally several options in this case, and you have to be careful you don’t end up with unusable values.

• Conic extrapolation. If you take five points near the end of the data to work with, you can create a conic section. Depending on the shape of it, it may or may not loop back to rejoin itself. An ellipse or circle, obviously, will, while a parabola or hyperbola won’t.

• French curve extrapolation. This method is most effective if your distribution tends towards exponential. It’s not exactly an exponential curve, as it works best if the data has accelerating or decelerating factors.

• Geometric extrapolation with error prediction. This is most effective when you have a large number of known values. You need at least three points of a sequence and the index.

Extrapolate is more often used in the sense of prediction than it is in its mathematical term. Though, of course, its mathematical meaning also involves prediction.

In both its math sense and its conversational sense, it involves taking known factors or data and using that to predict what will happen. And as is often the case with prediction, it’s less accurate the farther you get from the known quantities.

Extrapolate is a versatile verb; it can be used as both a transitive verb and an intransitive verb. It’s also possible to to extrapolate from existing data, or extrapolating data, or extrapolating new data.

This means that you don’t need to worry about using the word incorrectly. As long as you know the basic meaning of the word, how you put it in a sentence in relation to the noun isn’t specific.

Extrapolate is also from the Latin word polare, which means to polish. That’s likely where the idea of interpolare got the meaning of “refurbish” from.

However, instead of the prefix inter-, it got the prefix extra-. Extra is a Latin adverb and preposition that means outside, except, or beyond. It comes from exter or exterus, which means being on the outside or foreign, which itself came from ex. Extrapolate first came into use in 1874.

Interpolation vs Extrapolation FAQ

1. What is the root word of interpolation?

   The root word for interpolation comes from the Latin word interpolate, which means to refurbish or alter. Interpolire came from polire, which means “to polish.” Interpolate had the prefix inter- added to it, with the meaning of “between, among, [or] in the midst.”

2. What’s a real-world example of extrapolation?

   A real-world example of extrapolation is tracking the expected curve of the spread of disease. Almost all of the predictions involving the spread of covid were a type of extrapolation. It’s also been used with other pandemics, such as HIV and various flu.

3. Can extrapolation be used to predict the value of a point?

   Extrapolation can certainly be used to predict the value of a point. However, the more specific you get with your extrapolation, the less likely it is to be accurate. This is even more true if the point you’re trying to predict is a long way outside of the known data.

   None of the extrapolation methods listed above specifically generate points; however, a point can certainly be selected from the resulting line.

4. What’s the difference between linear interpolation and linear extrapolation?

   The difference between linear interpolation and linear extrapolation is that linear interpolation predicts values between the data points you have, while linear extrapolation predictions them outside the data you have.

   The two types of prediction use the same equation and the same concept in order to make their predictions. However, due to its nature of it, interpolation is typically much more accurate than extrapolation.

Author

Di Doherty

Di has been a writer for more than half her life. Most of her writing so far has been fiction, and she’s gotten short stories published in online magazines Kzine and Silver Blade, as well as a flash fiction piece in the Bookends review. Di graduated from Mary Baldwin College (now University) with a degree in Psychology and Sociology.