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# Interpolation|Definition & Meaning

**Definition**

**Interpolation** is the process of **creating** a **new** value based on the values already present in a **set** of data. One further way to **explain** it is to think of it as the act of putting in or **interjecting** a value that is **somewhere** in the **middle** of two other values.

The **process** of **fitting** the data points to the value of a function is known as interpolation.Â

It **can** be utilized in a wide range of **technical** and scientific **applications,** such as the **construction** of new **data points** within the bounds of a discrete data set of **existing** data **points** or the calculation of the formula for the **function** that will flow from the given set of points **(x,y)**.

We will go into great detail about **interpolation’s definition, formulas,** and **applications** in statistics in this post.

Figure 1 – Interpolation of two points

## Explanation of Interpolation With an Example

**Interpolation** is the **process** of **deriving** a new value from the **current** values in a data **collection.** The act of injecting or **interjecting** some **intermediate** value **among** two other values is **another** way to describe it.

**Interpolation** in data **science** and **mathematics involves computing** the value of a **function** depending on the values of many other data points in a particular sequence. This **function** can be expressed as f(x), as well as the **range** of **known** x values is xo to xn.

**Consider,** for instance, a **regression** line with the equation **y = 3x + 4**. To construct this **“best-fit”** line, we **know** that the **value** of x **must** be between 0 and 10 **Suppose** that we select x = 6 We **may estimate** the **value based** on the best-fit line and **equation.**

The **following** figure **represents** the **interpolation** of **sin** wave between two points **A and B**.

Figure 2 – Interpolation of a sine wave

The **illustration** below **represents** the cos wave with **interpolation** between two **points on** cos wave.

Figure 3 – Interpolation on a cosine wave

## What Is the Difference Between Extrapolation and Interpolation?

**Extrapolation** is the **estimation** of an **unknown** value by extending a series of known values or facts. To draw a **conclusion** that is not directly supported by the **available evidence** is to engage in the **process** of extrapolation. Interpolation is the **estimation** of a value that exists between two **known** values in a set of values.

## Standard Interpolation Techniques

The following are the three most **prevalent interpolation** methods:

**Interpolation linearization****Polynomial interpolation****Spline interpolation**

### Interpolation Linearization

**Interpolation** using **linear** relationships is one of the simplest **methods** available. In this part of the process, two **points** on such a graph are used to calculate the other values by drawing a straight line between them. **Estimates** are typically flawed when a straightforward **approach** is **utilized.**

### Polynomial Interpolation

**Polynomial interpolation** is a method for **estimating** the values of **missing** data points in a data **collection** by **plotting polynomial** functions on a graph. It is a **method** that is both more **precise** and **accurate.** The graph of the **polynomial** completes the **curve** by filling in the **gaps between** known **points** so that data can be found between those locations.

There are many different **approaches** to polynomial interpolation, including the following:

**Polynomial interpolation** is a method for **estimating** the values of missing data **points** in a data collection by **plotting polynomial functions** on a graph. It is a method that is both more **precise** and **accurate.** The graph of the polynomial completes the curve by **filling** in the **gaps** between **known points** so that data can be found **between** those **locations.**

There are many different **approaches** to **polynomial interpolation,** including the following:

**Lagrange interpolation****Interpolation using Newton’s**method,**polynomial interpolation,**and**spline interpolation**

**Newton’s** split differences **interpolation polynomial** is another name for the procedure that is **commonly** known as the **Newton method.**

The **Lagrange** and **Newton interpolation** methods both provide the same result, which is a **polynomial** function that has the **lowest feasible degree** and is able to pass through all of the data points in the set. This is known as the **smallest polynomial** function.

However, the **computations** that are used to arrive at the identical **conclusion** by the two **methodologies** are rather **different.**

### Spline Interpolation

**Piecewise** functions are put to use in the **process** of **spline** interpolation in order to make estimates for values that are absent from a data set and to complete it.

The **Lagrange** & **Newton** methods **estimate** a single polynomial that applies to the full data set. Spline interpolation, on the other hand, **defines** numerous simpler **polynomials** that apply to **different** subsets of the data. Because of this, it is often thought to offer more **detailed findings** and is regarded as a procedure that is more reliable.

## Methods of Extrapolation That Are Commonly Used

The **following** are **examples** of three of the most prevalent types of extrapolation methods:

**extrapolation along a linear path****polynomial extrapolation****conic extrapolation**

A **technique** that is very similar to linear **interpolation, linear extrapolation** includes **drawing** a straight line and making use of a **linear function** to make **predictions** about values that are not **contained** within a data set.

**Polynomial extrapolation** is a method that **determines** the values on a graph by using **polynomial** forms and functions.

The **process** of **determining** unknown values by employing conic sections that already have data for them is called **conic extrapolation.**

## Interpolation Example in Real Life

**Interpolation** is useful if items need to be **scaled** up or down. Perhaps you are familiar with the catering costs for events with 10 guests, **50** **guests,** and 100 guests, but you require an accurate estimate for events with **25 or 75 guests**. This is a useful fact to have while throwing a party.

Or **maybe** you’re an **entrepreneur** whose tiny business is suddenly expanding. It is professional to anticipate what you will require as you expand your **organization.**

How **many** new employees have been hired? **How** large is the room? Who is **responsible** for cleaning this area? How long will it take to clean the space? How **much** longer would it take to **complete** tasks with more employees? What will you be able to **accomplish** in a day? These **questions** might all be solved by interpolation.

**Perhaps** you work as a server in a **restaurant.** You need not only a good **concept** of how long it takes the kitchen to **prepare food** for your clients, regardless of their number.

**Additionally,** you should have a solid concept as to how it takes the group to **consume** meals so that you may check on them during the **proper moment.** This is an example of **interpolation.**

Acquiring an **automobile.** How much **money** will you invest in a vehicle with a given amount of miles? Will it be **beneficial?** Could you afford it? This is an example of interpolation.

## A Numerical Example of Linear Interpolation

### Example

**Determine** the value of the **unknown variable** by **computing** it using the **interpolation** equation **found** in the data set. **Consider** the **following** scenario: the **value** of X is **60.** What is the y **value** in this **case?**

### Solution

**With** the use of **interpolation,** we can get the value of Y **when** X is 60 **using** the **following formula:** â€“

**Y = Y1 + [ (Y2 – Y1) / (X2 – X1) ] * (X – X1)**

X **has** been set at 60, and it is now **necessary** to find Y.

**Y = Y1 + [ (Y2 – Y1) / (X2 – X1) ] * (X – X1)**

**= 80 + [ (120 – 80) / (70 – 50) ] * (60 – 50)**

**= 80 + (40/20) * 10**

**= 80 + 2*10**

**= 80 + 20**

**= 100**

*All mathematical drawings and images were created withÂ GeoGebra.*