Lecture 8: Sampling distributions

September 29, 2022 (4th week of classes)
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Estimating under uncertainty with certainty (i.e., with some confidence)

Statistics is the science of assisting in decision making with incomplete knowledge because the values of statistics of interest (mean, variance, median, etc) different among samples and from the intended statistical population. This concept in critical to statistics and is known as sampling variation.

"While nothing is more uncertain than a single life,
nothing is more certain than the average duration of 
a thousand lives." Elizur Wright

Critical understanding:

[1] We usually work with one single sample, and therefore only one mean value \(\bar{Y}\).

But, understanding how sampling distributions are built are necessary to understand the process of estimating uncertainty (i.e., sample mean values vary from one sample to another) to determine confidence on inferences.

[2] The sampling distribution makes it obvious that although the population mean \(\mu\) is assumed as a “constant”, its estimate \(\bar{Y}\) is a variable.

[3] The mean of all sample estimates of the mean equals the population mean.

[4] The mean of all sample estimates of the mean equals the population mean and is centered exactly on the true (population) mean \(\mu\)!

This means that the sample statistic mean \(\bar{Y}\)̅ is an unbiased estimate of \(\mu\).

[5] Sample values for the standard deviation (and any other statistic) also vary among samples (this will be discussed in our next lecture). Standard deviations of samples are key to estimate uncertainty of a sample mean.

What is a Sampling Distribution? By Mike Marin ()

Sampling from a normally distributed population. This is a great interactive tutorial that demonstrates the properties of samples, sampling distributions and estimators. Project leader: Mike Whitlock; programmers: Boris Dalstein, Mike Whitlock & Zahraa Almasslawi.