- Category: Mathematics

In Statistics, **Extrapolation** is a process of estimating the value beyond the distinct range of the given variable. The meaning of this term is the process of estimating something if the present situation continues. It is an important concept not only in Mathematics but also in other disciplines like Psychology, Sociology, Statistics etc with some categorical data.

**Extrapolation** is defined as an estimation of a value based on extending the known series or factors beyond the area that is certainly known. In other words, extrapolation is a method in which the data values are considered as points such as x_{1}, x_{2}, ….., x_{n}. It commonly exists in statistical data very often, if that data is sampled periodically and it approximates the next data point. One such example is when you are driving, you usually **extrapolate** about the road conditions beyond your sight.

Extrapolation is a statistical method beamed at understanding the unknown data from the known data. It tries to predict future data based on historical data. For example, estimating the size of a population a few years in the future based on the current population size and its rate of growth.

Extrapolation is classified into three types, namely

- Linear extrapolation
- Conic extrapolation
- Polynomial Extrapolation

Let us discuss the three types of extrapolation in detail.

For any linear function, linear extrapolation provides a good result when the point to be predicted is not too far from the given data. It is usually done by drawing the tangent line at the endpoint of the given graph and that will be extended beyond the limit.

A polynomial curve can be created with the help of entire known data or near the endpoints. This method is typically done by means of Lagrange interpolation or Newton’s system of finite series that provides the data. The final polynomial is used to extrapolate the data using the associated endpoints.

A conic section can be created with the help of five points nearer to the end of the given data. The conic section will curve back on itself if it is a circle or ellipse. But for parabola or hyperbola, the curve will not back on itself because it is relative to X-axis.

Let us consider the two endpoints in a linear graph (x_{1}, y_{1}) and (x_{2}, y_{2}) where the value of the point “x” is to be extrapolated, then the extrapolation formula is given as

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

As we know, extrapolation is a process of predicting the data point about the outside of a curve when a few points are given. In the example given below, the known data are x_{1}, x_{2}, x_{3}. Finding the point x_{4} is known as extrapolation point.

**Question: The two given points that lie on the straight line is (1, 5) and (4, 10). Determine the value of y at x = 5 on the straight line using a linear extrapolation method.**

**Solution: **Given: x_{1} = 1 ,y_{1} = 5

and x_{2} = 4, y_{2} = 10

The linear extrapolation formula is given as;

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

Substitute the known values in the given formula,

y(5) = 5 +((5-1)/(4-1))(10-5)

y(5) = 5 + (4/3) (5)

y(5) = 5 + 6.65

y(5)= 11.65

Therefore y(5) = 11.65