Extrapolation

Extrapolation Method

The process in which you estimate the value of given data beyond its range is called an extrapolation method. In other words, the extrapolation method means the process that is used to estimate a value if the current situation continues for a longer period. Extrapolation Method is a vital component in Mathematics. It also has its branches in Statistics, Sociology, Psychology, and other fields of study too. In this section, we shall discuss what extrapolation means, the definition, extrapolation formula, and have a better understanding of the extrapolation method with the help of a few example problems. In addition to that, you should also know about interpolation. This is the process of estimating the value of the given data. 

What is Extrapolation?

Extrapolation Method is a process in which you estimate value by understanding the known factors beyond a particular area. The other definition would be the data values are assumed to be points like a1, a2,…………an. It exists as statistical data and when this data is tested periodically,  it can give you the necessary information or the next data point. An easy example of extrapolation that you can observe in your everyday life can be driving a car or riding a bike. You usually tend to look further at your sight to understand the condition of the road better.

Extrapolation in Statistics

Here you try to understand the future or you predict the future with the available data or information. With the historical data present, you can estimate the rate at which some incident might increase or some time it might decrease. For example, the rate at which the population of the world is increasing or it can be determining the weather conditions. 

Extrapolation Method

Extrapolation method is of three types - linear, conic, and polynomial extrapolation. Given below is a brief description of these methods:

  1. Linear Extrapolation

When you want to predict the value that is not too far away from the existing data, linear extrapolation will help you for any linear function. When you have been given a graph, you use this method to draw a tangent line at the last point and extend this line beyond its limits. 

  1. Conic Extrapolation

This type of extrapolation helps you create a conic section with the last five endpoints of the data. When you have a para or a hyperbola, the conic section’s curve is relative to the x-axis and doesn’t curve, but in case of an ellipse or a circle, it curves on itself. 

  1. Polynomial Extrapolation

You can create a polynomial curve using all the data points given to you. This method is applied using Newton’s System of Finite Series or Lagrange Interpolation. With the associated points, you can find the required data. 

Extrapolation Formula

In a linear graph, let’s assume two endpoints (a1, b1 ) and (a2, b2 ). Here you’ll have to find the value of the point “a” that has to extrapolate. Therefore, the extrapolation formula goes by:

b(a) = b1 + ((a – a1)/(b – b1))(b2 – b1)

Extrapolate Graph

We know that The process in which you estimate the value of given data beyond its range is called an extrapolation method. See the example below to understand the extrapolate graph. Here the unknown values are  a1, a2,  a3 and you need to find  a4. Now finding the a4 is the called extrapolation point. 

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Solved Examples

Example 1: The two points on a straight line are (2, 6) and (4, 10). Find the value of b when a = 6 using linear extrapolation. 

Solution:

Given, 

a1 = 2

b1 = 6

a2 = 4

b2= 10

a = 6

b(a) = b1 + ((a – a1)/(a2 – a1))(b2 – b1)

Substituting the values: 

b(6) = 6 + ((6-2)/(4-2)) (10-6)

b(6) = 6 + (4/2) (4)

b(6) = 6 + (2) (4)

b(6) = 6 + 8

 b(6) = 14


Example 2: The two points on a straight line are (4,8 ) and (10, 6). Find the value of b when a = 8 using linear extrapolation. 

Solution:

Given, 

a1 = 4

b1 = 2

a2 = 10

b2= 6

a = 8

b(a) = b1 + ((a – a1)/(a2 – a1))(b2 – b1)

Substituting the values: 

b(8) = 8 + ((8-4)/(10-4)) (6-2)

b(8) = 6 + (4/6) (4)

b(8) = 6 + (0.667) (4)

b(8) = 6 + 2.667

b(8) = 8.667


Example 3: The two points on a straight line are (6, 3) and (8, 9). Find the value of b when a = 12 using linear extrapolation. 

Solution:

Given, 

a1 = 6

b1 = 3

a2 = 8

b2= 9

a = 8

b(a) = b1 + ((a – a1)/(a2 – a1))(b2 – b1)

Substituting the values: 

b(12) = 3 + ((12-6)/(8-6)) (9-3)

b(12) = 3 + (6/2) (6)

b(12) = 3 + (3) (6)

b(12) = 3 + 18

b(12) = 21

FAQ (Frequently Asked Questions)

1. What is Extrapolation?

Extrapolation Method is a procedure wherein you estimate an incentive by understanding the known factors beyond a specific region. It exists as statistical data and when this data is tried occasionally, it can give you the vital data or the future data point or it can be used to predict the future point. 

2. What is Extrapolation Formula?

In a linear graph, let’s assume two endpoints (a1, b1 ) and (a2, b2 ). Here you’ll have to find the value of the point “a” that has to extrapolate. Therefore, the extrapolation formula goes by:

b(a) = b1 + ((a – a1)/(b – b1))(b2 – b1)

3. How Can You Implement Extrapolation in Statistics?

In Statistics, you try to understand the future or you predict the future with the available data or information. With historic data available, you can understand the rate at which some occasions can change the future. For example, the rate at which the population of the world is increasing or it can be determining the weather conditions.