Course Outline

segmentGetting Started (Don't Skip This Part)

segmentStatistics and Data Science: A Modeling Approach

segmentPART I: EXPLORING VARIATION

segmentChapter 1  Welcome to Statistics: A Modeling Approach

segmentChapter 2  Understanding Data

segmentChapter 3  Examining Distributions

segmentChapter 4  Explaining Variation

segmentPART II: MODELING VARIATION

segmentChapter 5  A Simple Model

segmentChapter 6  Quantifying Error

segmentChapter 7  Adding an Explanatory Variable to the Model

segmentChapter 8  Models with a Quantitative Explanatory Variable

segmentPART III: EVALUATING MODELS

segmentChapter 9  Distributions of Estimates

9.5 Notation and Terminology

segmentChapter 10  Confidence Intervals and Their Uses

segmentChapter 11  Model Comparison with the F Ratio

segmentChapter 12  What You Have Learned

segmentFinishing Up (Don't Skip This Part!)

segmentResources
list Introduction to Statistics: A Modeling Approach
9.5 Notation and Terminology
We should pause for a second and introduce some new terminology and notation that applies especially to sampling distributions. Recall that earlier in the course we discussed the importance of notation for keeping straight the distinction between the distribution of data (that is, a sample) and the distribution of the population (that is, the longrun result of a DGP). So, for example, \(\bar{Y}\) represents the mean of a sample of data, whereas \(\mu\) represents the mean of the population.
Sampling distributions are a third kind of distribution—completing our Distribution Triad—and so we need notation to specifically indicate when we are talking about a sampling distribution. Let’s fill out our notational toolbox, therefore, to include sampling distributions.
Sample / Data  Population / DGP  Sampling Distribution of Means  

Mean  \(\bar{Y}\)  \(\mu\) (mu)  \(\mu_\bar{Y}\) (mu sub ybar) 
Standard Deviation  \(s\)  \(\sigma\) (sigma)  \(\sigma_\bar{Y}\) (sigma sub ybar) 
Model statement 
Estimated from sample \(Y_i=b_0+e_i\) 
Parameters being estimated \(Y_i=\beta_0+\epsilon_i\) 
Distribution of estimates A lot of \(b_0\)s 
Also note: we use a special word to refer to the standard deviation of the sampling distribution of the mean: the standard error, or standard error of the mean.
We can also have sampling distributions of other estimates besides the mean. For example, we could have a sampling distribution of standard deviations or sampling distributions of SS, PRE, F or any other statistic.
In general, Greek letters (e.g., \(\mu\) or \(\sigma\)) are used to describe parameters that are unknown and estimated. The population mean (\(\mu\)) is unknown, for example, and so represented with a Greek letter. Things we calculate based on samples are generally represented with Roman letters. So, \(\bar{Y}\) is the mean of a sample of data.
Sampling distributions are unknown (imaginary, in fact) and so the mean and standard deviation of a sampling distribution are represented with Greek letters. But the subscript (for example, \({\sigma_{\bar{Y}}}\)) represents the statistic that the sampling distribution is made out of. Because it’s a statistic, it is represented with a Roman letter. For example, if we analyze the notation for standard error (\({\sigma_{\bar{Y}}}\)), the \(\sigma\) represents the standard deviation of this distribution and the tiny \(\bar{Y}\) represents that the distribution is made up of sample means.