3.2: Parameter and Statistic

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In inferential statistics, we will be trying to learn more about the population from a sample, in other words, we will be using the numerical summaries obtained from a sample to estimate the corresponding numerical summary of the population.

The following terminology will help us to distinguish between a sample and the population. A parameter is a numerical summary of the population of size $N$ which can only be obtained by a census, and it remains constant although most of the time unknown. A statistic is a numerical summary of a sample of size $n$ which can only be obtained by sampling, and it varies from sample to sample.

Parameter	Statistic
Population Size $N$ Census Constant Unknown without the census	Sample Size $n$ Sampling Varies Can be found by obtaining a sample

The most common use of a statistic is to estimate the value of the corresponding parameter that is usually unknown. The number that is used to estimate the parameter is called an estimator or an estimate and an attempt to estimate the parameter is called an estimating. Some estimates are better than others. While a random guess may satisfy the definition of an estimate, we have an intuition that using a statistic is better than just guessing. An estimator is called an unbiased estimator of the parameter if the mean of its values, in the long run, equals the parameter; otherwise, it is said to be a biased estimator. In other words, an unbiased estimator yields, on average, the correct value of the parameter, whereas a biased estimator does not. For example, the sample mean is an unbiased estimator of a population mean while a random guess is a biased estimate of the population mean.

This diagram compares the estimating process with a target shooting.

$Ch 3-2-1a.png$

$Ch 3-2-1b.png$

$Ch 3-2-1c.png$

$Ch 3-2-1d.png$

Can you guess which ones are biased and which ones are not? Clearly, any unbiased estimator is better than any biased estimator, however, some biased estimators are better than others! For example, which of the two biased shooting patterns on the right you would want your hunting partner to have? A shooting instructor can easily fix the first shooting pattern by telling the shooter to aim lower and to the right, but there is no easy fix for the other one! Please understand the difference between being correct and being correct on average none of the unbiased shooting patterns hits the bull’s eye all the time but on average it appears that they do!

For popular numerical summaries, the distinguishing between being the parameter or statistic is so important that we introduce a different notation to denote the same measure, even when we use the same formula. For example, we label the population mean with a Greek letter $\mu$, and the sample mean with $\bar{x}$ although the formulas are the same. For example, the average age of all US presidents at inauguration is $\mu$ and is equal to 55.02 and the average age of the last 5 US presidents at inauguration is $\bar{x}$ and is equal to 67.2.

Note that the same data set can be treated as the population or a sample depending on the context and depending on the context, we will use either $\mu$ or $\bar{x}$to label the average of the data set. For example, we will treat the 5 starting players as the population when we are interested to learn the average height of the five starting players. We will treat the same group of five as a sample when we are interested to learn the average height of the team but only observed the heights of the starting five players.

In this section we started to develop the vocabulary and notation to distinguish between the parameters and statistics.

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