In the previous chapter we began to develop ways of inferential thinking. In particular, we learned how to use data to decide between two hypotheses about the world. But often we just want to know how big something is.
For example, in an earlier chapter we investigated how many warplanes the enemy might have. In an election year, we might want to know what percent of voters favor a particular candidate. To assess the current economy, we might be interested in the median annual income of households in the United States.
In this chapter, we will develop a way to estimate an unknown parameter. Remember that a parameter is a numerical value associated with a population.
To figure out the value of a parameter, we need data. If we have the relevant data for the entire population, we can simply calculate the parameter.
But if the population is very large – for example, if it consists of all the households in the United States – then it might be too expensive and time-consuming to gather data from the entire population. In such situations, data scientists rely on sampling at random from the population.
This leads to a question of inference: How to make justifiable conclusions about the unknown parameter, based on the data in the random sample? We will answer this question by using inferential thinking.
A statistic based on a random sample can be a reasonable estimate of an unknown parameter in the population. For example, you might want to use the median annual income of sampled households as an estimate of the median annual income of all households in the U.S.
But the value of any statistic depends on the sample, and the sample is based on random draws. So every time data scientists come up with an estimate based on a random sample, they are faced with a question:
“How different could this estimate have been, if the sample had come out differently?”
In this chapter you will learn one way of answering this question. The answer will give you the tools to estimate a numerical parameter and quantify the amount of error in your estimate.
We will start with a preliminary about percentiles. The most famous percentile is the median, often used in summaries of income data. Other percentiles will be important in the method of estimation that we are about to develop. So we will start by defining percentiles carefully.