# Introduction to Interpolation

The time series data which is recorded after a regular or irregular interval of time consists of values of a phenomenon at a certain point in time, or when the values of some hypothetical function corresponding to a few values of the independent variable are not sufficient to provide information regarding the values of the same phenomenon in between the specified intervals. Examples include determining the production of wheat in different years, the export of cotton goods during the last ten years, yearly enrollment in a university, finding the values of logarithms to a specified base corresponding to natural numbers, the demand of a product at different levels of prices for a period of time within the specified time not mentioned in the series, the values of $\log X$, the demand corresponding to the values of $X$ or $P$ from within the specified values. The process and technique of estimating such values is known as interpolation.

This unit is restricted to cases where these related variables follow an approximately linear trend, so the formulae are developed on the assumption of linearity. A number of different formulae are available for interpolation, mostly by Newton. Here we shall look at only a few of them which satisfy the needs of BBA, Economics and MBA students. The application of these formulae shall be explained with the help of a few examples. The derivation of these formulae is beyond the scope of this tutorial.

In addition to linearity the data should satisfy two more assumptions for the better application of these formulae:

• No sudden jumps are present in the data from one time period to another; this implies that the data are in the form of continuous or smooth curves.
• The data have a uniform rate of change; this assumption is equivalent to the assumption of linearity.