Distributions of Data

Part 1: Introduction

The world of public discourse is awash in data and conclusions drawn from data. For example, what can we reasonably conclude from intelligence tests? Are IQ tests accurate predictors of scholastic achievement or job performance? Are these tests biased?

The discussions of these issues are filled with terms such as "standard deviation" and "bell-shaped curve." What exactly is a bell-shaped curve, and why does it have anything to do with these issues? The tools used in these debates come from the disciplines of probability and statistics. The development and understanding of many of these tools rely heavily on calculus. In this module, we will examine some of these connections.

We begin with a discussion of the reliability of electrical/electronic components. How can one predict the probability of the failure of a computer on the space shuttle or the average lifetime of the light bulb in your attic? We will end with a description bell-shaped curves and discussion their role in the modeling of data. In between, we will discuss other models for the distribution of data, as well as, ways of describing and displaying the data itself.

When you pick up a package of light bulbs in the store, you might notice a statement about the average life of a typical bulb -- say, 1000 hours. What does this mean? Suppose you took a sample of 100 of these bulbs, burned them continuously until they failed, and recorded the life of each bulb. How could you use this data to check the claim?

Suppose you measured the heights of all the women students in your school. What fraction of these women would you expect to have heights between 5'2" and 5'7"?

These are questions about the modeling of the distribution of data. In this module we will use calculus to obtain models for a number of different types of distributions of data -- including models for the types of data given above.