**Interpolation**

Interpolation is a helpful statistical and mathematical tool that we use to estimates the values between two points. Also, in this topic, you will learn about the definition, example, and formula of interpolation.

**Definition of Interpolation**

It refers to the process of finding a value between two points on a curve or line. Furthermore, for remembering its meaning we should think of it as two words first ‘inter’ which means ‘enter’ and ‘polation’ which means to look ‘inside’.

So, in this way interpolation means to look inside the data we originally have. Moreover, this tool is not only useful in statistics but is also useful in science, business or at any time there is a need to predict the values that fall within the two existing data points.

**Example of Interpolation**

Interpolation can be easily understood with the help of an example. Also, an example will show the concept of it more clearly. Suppose a gardener planted a chilly plant and he measured its height and kept track of its growth every other day.

Moreover, the gardener is a curious person and he would like to estimate how tall his plant will be on the fourth day. Besides, the gardener’s observation table looks like this:

Day |
Height (in mm) |

1 |
0 |

3 |
4 |

5 |
8 |

7 |
12 |

9 |
16 |

11 |
20 |

Most noteworthy, if you see that chart closely then you will find that it is not very difficult to figure out the height of the plant on the fourth day. Moreover, the plant height on the fourth day will probably be 6 mm. Furthermore, it happens because a disciplined chilly plant grows in a linear pattern.

Also, there is a linear relationship between the numbers of the days and the height of the plant. Besides, linear pattern means it creates a straight line and we could even estimate it by plotting the data on the graph.

But, what if the plant does not grow with a linear pattern? What if the plant growth was in the shape of a curve?

Moreover, to solve this problem the interpolation formula is made, which could come quite handy.

**Interpolation Formula**

Its formula looks like this:

y – y1 = \(\frac{y2 – y1}{x2 – x1}\) (x – x1)

Now let’s see how we can use this formula. Also, there are two set points where we can find the estimated value that are:

(x1, y1) and (x2, y2)

Let’s go back to the chilly plant example the first set of value for day fifth are (3, 4) and the second set of value for day seventh are (5, 8). Moreover, the value of x is 6 since we want to find the height ‘y’ on the fourth day. Now put these values in the formula and guess the height of the plant on the fourth day.

y – y1 = \(\frac{y2 – y1}{x2 – x1}\) (x – x1)

Next, y – 4 = \(\frac{8 – 4}{5 – 3}\) (x – 3)

y – 4 = \(\frac{4}{2}\) (x – 3)

Next, y – 4 = 2 (x – 3)

y = 2 (x – 3) + 4

Then, y = 2 (4 – 3) + 4

y = 2 (1) + 4

Therefore, y = 6

**Interpolation Methods**

We can construct other forms of the interpolation by picking up a different class of interpolants. For example, rational interpolation is the rational functions using the Padé approximant, Trigonometric polynomials that use the Fourier series trigonometric polynomials. Besides, another possibility is to use wavelets.

**Solved Question for You**

**Question. **How many types of interpolation methods are there?

- 4
- 6
- 10

**Answer.** The correct answer is option B. Besides, the names of these six types of interpolation method are Linear, cubic spline, nearest neighbour, Biharmoic (v4), thin-plate spline, and shape-preserving.

In problem two, the sum of standard deviations should be 4.4 not 3.16.