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# Extrapolation|Definition & Meaning

## Definition

**Extrapolation** means calculating a number **outside** the given range of data of some kind. It involves guessing the value by figuring out and using the relationship between the variables over the given set of data. It is a crucial idea in mathematics and other academic fields, such as psychology, sociology, and statistics.

The term “extrapolation” refers to an assessment of a price based on expanding the existing series or components outside the region that is unquestionably known. In other words, extrapolation is a technique where the data values are treated as points, such as x1, x2, …, xn.

If statistical data is **periodically** sampled and resembles the next data point, it frequently appears in that data. One such instance is how you often extrapolate the state of the roads while driving.

Figure 1 above shows the extrapolation graph of x = 2y.

**Extrapolation Statistics**

A statistical extrapolation technique aims to interpret the unseen data from the existing data. It uses previous data to attempt to anticipate future data. For instance, it uses the current population and growth rate to project the sample size in a few years.

**Extrapolation Formula**

Consider the two ends of a linear graph, (x1, y1) as well as (x2, y2), from which the number of the point “x” is to be extrapolated. The extrapolating formula is then given as

\[ \mathsf{y(x)=y_{1}+\frac{x-x_{1}}{x_{2}-x_{1}}(y_{2}-y_{1})} \]

**Extrapolation Methods**

Three categories of extrapolation exist, including

- Linear extrapolation
- Extrapolation Using Polynomials
- Conic extrapolation

Discussing each one of them in detail.

**Linear Extrapolation**

The process of **linear** **extrapolation** is drawing a **tangent line** style=”font-weight: 400;”> from the known data’s end and extending it past that point. When used to extend the graphs of a roughly linear combination or not beyond the existing data, linear extrapolation would only produce satisfactory results.

**Extrapolation Using Polynomials**

**Conic Extrapolation**

**French Curve Extrapolation**

**Extrapolation Graph**

Figure 2 below shows a graphical representation of extrapolation.

**Geometric Extrapolation With Error Prediction**

Figure 3 below shows the geometrical explanation of extrapolation.

**Extrapolation Arguments**

**Examples of Extrapolation**

We will look at some solved examples of **extrapolation** to understand the concept further.

**Example 1**

**Solution**

x1 = 1 , y1 = 2 , x2 = 3 , y2 = 4

We know that the linear extrapolation formula is:

\[\mathsf{y(x)=y_{1}+\frac{x-x_{1}}{x_{2}-x_{1}}(y_{2}-y_{1})} \]

Replace the specified values with the known values:

y(5) = 5 + [(5 – 1) / (3 – 1)](4 – 2)

**Example 2**

**Solution**

x1 = 1 , y1 = 7 , x2 = 2 , y2 = 8

We know that the linear extrapolation formula is;

\[\mathsf{y(x)=y_{1}+\frac{x-x_{1}}{x_{2}-x_{1}}(y_{2}-y_{1})} \]

Replace the specified values with the known values:

y(9) = 9 + [(9 – 1) / (2 – 1)](8 – 7)

Therefore y(9) is equal to 17.

*All images are made using GeoGebra.*