What are Interpolation And Extrapolation?
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, ……, xn. 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.
FAQs on Extrapolation: Types And Methods
1. What is the fundamental concept of extrapolation in mathematics?
In mathematics, extrapolation is a statistical technique used to estimate or predict the value of a variable beyond its original, observed range. It involves assuming that an established trend in the data will continue to apply for values that have not been measured. For example, if you know a city's population for the last 10 years, you could extrapolate that data to predict its population in 5 years from now.
2. What are the two primary types of extrapolation methods?
The two main methods of extrapolation are based on the type of trend observed in the data:
- Linear Extrapolation: This method is used when the data points appear to follow a straight-line trend. It extends this line to estimate values outside the known data range. It is simpler but assumes a constant rate of change.
- Polynomial Extrapolation: This method is used when the trend is curved. It fits a polynomial curve to the existing data points and extends this curve to make predictions. This can account for non-constant rates of change but can become highly inaccurate if extended too far.
3. How does extrapolation differ from interpolation?
The key difference lies in where the estimation is made. Interpolation is the process of estimating a value that falls within the range of known data points. In contrast, extrapolation is the process of estimating a value that lies outside the range of known data points. For instance, if we have temperature data for every hour from 9 AM to 5 PM, estimating the temperature at 10:30 AM is interpolation, while predicting the temperature at 7 PM is extrapolation.
4. What are some important real-world applications of extrapolation?
Extrapolation is a critical tool for forecasting and planning in various fields. Key applications include:
- Economics and Finance: To predict future stock prices, GDP growth, or inflation rates based on historical trends.
- Science and Engineering: To estimate how a material might behave under extreme temperatures or pressures not yet tested in a lab.
- Demography: To project future population growth or resource demand for urban planning.
- Climate Science: To forecast the long-term effects of rising CO2 levels on global temperatures based on current and past data.
5. Why is extrapolation often considered less accurate or reliable than interpolation?
Extrapolation is generally less reliable because it makes a significant assumption: that the trend observed within the data set will continue indefinitely outside of it. Real-world systems are often complex and can change unexpectedly due to new factors. Interpolation is safer because you are estimating within a range where the trend is already known and bounded by data points, leaving less room for drastic, unforeseen changes.
6. What underlying assumptions must be true for an extrapolation to be valid?
For an extrapolation to be considered valid, two critical assumptions must hold true. Firstly, it assumes that the underlying relationship or pattern in the data is correctly identified (e.g., it is truly linear or follows a specific curve). Secondly, and more importantly, it assumes that this relationship will remain constant beyond the observed range. Any change in the system, such as new market competition in business or a change in physical laws under extreme conditions, can invalidate the extrapolation.
7. How is the extrapolation method used in statistical research and forecasting?
In statistical research, the extrapolation method is a form of predictive modelling. Researchers first build a model (like a regression line) based on an existing dataset. They then use this model's equation to forecast future outcomes or estimate data for unobserved scenarios. For example, a market researcher might use sales data from the first three quarters of a year to extrapolate the total sales for the entire year, which helps in inventory and budget planning.
