22.1: Importantly Confusing Points
- Page ID
- 57822
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)So, here is something that is extremely confusing because we use the same word to mean/imply different things. This section can be summarized as "Make sure you know what the terms mean in the different contexts." However, since I like to write, here is a whole section of the book trying to make this idea manifest. While I focus on the variance, the same issue arises with things like "mean" and "skew."
Note that in statistics (in all disciplines), few things are as fundamental as terminology. We use the same word to describe the spread of a fixed set of numbers, the variability within a specific data sample, and a fixed property of a probability distribution. All three of these are termed the "variance." However, while each is a valid measure of dispersion, they represent fundamentally different ideas: a calculated value, a random statistic, and a deterministic parameter.
This section unpacks this critical terminology.
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A set of numbers has a variance. A sample from a random variable has a variance. A distribution has a variance. All three measures are termed "variance;" however, they mean different things.
One can also say that a "random variable has a variance." While a random variable is not a distribution (it follows one, however), we will often conflate the random variable and its distribution.
Here is a set of numbers I specify (i.e., they are not randomly generated): \(\{1, 2, 3, 4, 5, 6, 7\}\). They have a mean and a variance. The mean is \(4\). The variance is \(4.667\). Those measures only tell me about the numbers I specified. Since they are not realizations of a random variable, it does not make sense to use those values for anything other than mathematical calculations on those particular values.Here is a set of numbers generated from a Poisson distribution (see the Poisson Section) with mean \(\lambda = 1\): \(\{1, 2, 1, 2, 0\}\). These values (this sample) also have a mean (\(\mu = 1.20\)) and a variance (\(\sigma^2 = 0.70\)). Since these numbers are a realization of a random variable, these two values can be called "statistics." They are functions of observed random variables. Because they are statistics, the mean should be close to the expected value of the random variable, and the variance should be close to the variance of the random variable.
Since these statistics are functions of a random sample, those statistics are also random variables. As such, they have distributions. In fact, understanding and knowing the distribution of sample statistics is one of the goals of statistics. This study led you to confidence intervals and test statistics back in your previous statistics course.
Here is a Chi-square distribution with parameter "4 degrees of freedom": \(\chi^2_{\nu=4}\). It has an expected value (\(\mu=4\)) and a variance (\(\sigma^2=8\)). These are parameters and are not a function of a sample (hence they are represented by lowercase Greek letters). These are genuine numbers that have no distribution.
They have no distribution because they do not vary. Our understanding of them may vary, but the value does not.
Population parameters are also numbers that we do not know when we are applied statisticians. These are, however, numbers that we want to know — or at least obtain a good estimate of. To get those estimates, we obtain a sample from the distribution and measure the corresponding sample statistics. While the sample statistics will not (usually) exactly equal the population parameters, they will tend to be close. How close depends on the sample size and the statistic measured.
This "issue" with the variance is not unique to the variance.
The lesson to take away from this is: You need to be aware of the symbols and what they mean. This is especially true when it comes to values, random variables, and distributions.