Notes

A sample is a subset of observations obtained from a larger set, termed a population.

A random sample of a specified size is one for which every possible sample of that

size has the same chance of being selected from the population.

There are four underlying assumptions:

If the randomness assumption does not hold, the:

http://itl.nist.gov/div898/handbook/eda/section2/eda251.htm

For many experiments and observations concerning natural phenomena—such as measuring the speed of light—one finds that performing the procedure twice under (what seem) identical conditions results in two different outcomes. Uncontrollable factors cause “random” variation. In practice one tries to overcome  his as follows: the experiment is repeated a number of times and the results are averaged in some way. In this chapter we will see why this works so well, using a model for repeated measurements. We view them as a sequence of independent random variables, each with the same unknown distribution.

It is a probabilistic fact that from such a sequence—in principle—any feature of the distribution can be recovered. This is a consequence of the law of large numbers.

Averages vary less in a sample:

Scientists and engineers involved in experimental work have known for centuries that more accurate answers are obtained when measurements or experiments are repeated a number of times and one averages the individual outcomes.1 For example, if you read a description of A.A. Michelson’s work done in 1879 to determine the speed of light, you would find that for each value he collected, repeated measurements at several levels were performed. In an article in Statistical Science describing his work ([18]), R.J. MacKay and R.W. Oldford state: “It is clear that Michelson appreciated the power of averaging to reduce variability in measurement.”

Variance in a sample of size 1: (I recently gave a talk at Stanford about turning points in my work, "An Academic and Otherwise Life, an n =1", and included the remark about the infinite sample variance for n = 1 in my introduction to indicate that there were no sound inferences to be made from my personal narrative. At the end, I went ahead and made an inference, which was that I thought it was a good idea to have turning points throughout one's work--and thus one's life--and not just up to age 30.)