Extrapolation in Mathematics and Statistics

Interpolation, in statistics, is defined as an estimation between the given data or observations. Whereas, extrapolation, in statistics, is a process of estimating the value beyond the separate range of the given variable on the basis of its relationship with another variable. It is a crucial concept not only in Mathematics but also in other subjects like Sociology, Psychology, Statistics, etc., with some categorical data.

What is Extrapolation?

Extrapolation is known as an estimation of a value based on extending the known factors or series beyond the area that is known. Or in other words, extrapolation is a technique in which the data values are considered as points such as x₁, x₂, ……, x_n. It mostly exists in statistical data very often, if that data is sampled periodically and it is near the next data point. One example of such is when you are driving, you typically extrapolate about road conditions outside your sight.

Interpolation vs. Extrapolation

We could utilize our function to predict the value of the dependent variable for an independent variable that is in the midst of our data. For this case, we should perform the interpolation.

Assume that data with x between 0 and 10 is used to produce a regression line y = 2x + 5. We can utilize this line of best fit to estimate the y value corresponding to x = 7. Simply put this estimate into our equation and we see that y= 2(7) + 5 =19. Because our x value is amongst the range of values used to create the line of best fit, this is an example of interpolation.

Whereas we could use our function to predict the value of the dependent variable for an independent variable that is beyond the range of our data and in this case we are doing extrapolation. We have included an example of extrapolation below.

Extrapolation Method

Extrapolation is mainly categorized into three types:

• Linear Extrapolation

• Conic Extrapolation

• Polynomial Extrapolation

We have explained all the categories below.

A.   Linear Extrapolation:

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

B.Conic Extrapolation:

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

C.Polynomial Extrapolation:

A polynomial curve can be created with the assistance of full given (or known) data or near the endpoints. This technique is usually done using Lagrange interpolation or Newton’s system of finite series that provides the data. The final polynomial is utilized to extrapolate the data using the associated endpoints.

What is Extrapolation Statistics?

Extrapolation is a statistical technique beamed at understanding the unknown data from the known data. It attempts to predict future data based on historical data. Such as, estimating the size of a population after a few years based on the current population size and its rate of growth.

How To Extrapolate Numbers?

To successfully extrapolate data, you must have correct model information and if possible, utilize the data to find a best-fitting curve of the appropriate form (for example, linear, exponential) and calculate the best fitting curve on that point.

Extrapolation Formula

Let us assume that the two endpoints in a linear graph (x1, y1) and (x2, y2) where the value of the point “x” is to be extrapolated and then the extrapolation formula is given as: $y(x) = {y_1} + {\frac{x - {x_1}}{{x_2} - {x_1}}} \times ({y_2} - {y_1})$

Graph of Extrapolation

From the below diagram x₁, x₂, and x₃ are known data, whereas x₄ is finding the extrapolation point.

Solved Examples

1. The two given points lie on the straight line (2, 6) and (5, 11). Find out the value of y at x = 5 on the straight line with the help of a linear extrapolation method.

Solution:

Given: x₁ = 2, y₁ = 6

And x₂ = 5, y₂ = 11

The linear extrapolation formula is given by:

$y(x) = {y_1} + {\frac{x - {x_1}}{{x_2} - {x_1}}} \times ({y_2} - {y_1})$

Replace the given values in the formula,

$y(5) = {6} + {\frac{5 - 2}{5 - 2}} \times (11 - 6)$

$y(5) = {6} + {\frac{3}{3}} \times (5)$

y(5) = 6 + 5

y(5) = 11

Therefore, y(5) = 11

2. On a straight line two points are given are (4, 8) and (10, 6). Determine the value of ‘b’ where the value of a = 8 using linear extrapolation.

Solution:

Given a₁ = 4, b₁ = 2, a₂ = 10, b₂ = 6, and a = 8.

Substituting the values in the equation 

$b(a) = {b_1} + {\frac{a - {a_1}}{{a_2} - {a_1}}} \times ({b_2} - {b_1})$

we get,

$b(8) = {2} + {\frac{8 - 4}{10 - 4}} \times (6 - 2)$

$b(8) = {6} + {\frac{4}{6}} \times (4)$

b (8) = 6 + 2.667.

b (8) = 8.667.

Conclusion

The estimation of the data set value is termed extrapolation. The future data is predicted based on historical data. One of the relatable examples will be the estimation of the population over the next 10 years based on the current population.