Introduction

Section 2.1 Introduction

When computing means, medians, variances, etc. in the previous chapter, you took given data and create measures that in some sense describe the data using a single value. These single values can be called "descriptive statistics" or perhaps "point estimates" that help understand the properties of the original data set. In this chapter, you will instead take a data set and create a mathematical model that can be used to predict or infer properties of the underlying problem. Statistical procedures such as in this chapter that are used to predict are often lumped into the world of "inferential statistics".

So, given a set of data points \((x_0,y_0), (x_1,y_1), ... (x_n,y_n)\text{,}\) it is often desirable to have a nice continuous formula \(y = f(x)\) that expresses the general nature of those data points. Such a formula "interpolates" the data points if

\begin{equation*} y_k = f(x_k), \end{equation*}

that is the formula gives a graph that exactly passes through each of the given data points.

On the other hand, sometimes the data points are known to be only approximate or the complexity of the formula needed to interpolate all of the data points exactly is too large. In this case, the formula may only be required to return values that are relatively close to the data points. Such a formula is said to "approximate" and gives

\begin{equation*} y_k \approx f(x_k). \end{equation*}

Let's consider ways to create useful models that approximate the data points.

From basic algebra, if you are given two distinct points then there is one line which passes exactly through (i.e. interpolates) both. There are many ways to create this linear model but for points \((x_0,y_0), (x_1,y_1)\text{,}\)

\begin{equation*} y = \frac{y_1 - y_0}{x_1 - x_0}(x - x_0) + y_0 \end{equation*}

is the linear function which passes through both points if the x-values are distinct. If the x's are equal then

\begin{equation*} x = x_0 \end{equation*}

is linear and interpolates both data points. However, once you collect three or more points it is likely that there is no line which exactly "interpolates" all of the points. If we desire a linear model then we must settle for a model that approximates. In this chapter, you will investigate how to create polynomial functions which in some manner approximate a collection of data point in some "best" manner.