COMMON MISTEAKS MISTAKES IN USING STATISTICS: Spotting and Avoiding Them

# What is a Random Sample?

## Sources of Confusion

Many people misunderstand what  the phrase "random sample" means.  There are two reasons why this is the case.

First, the word "random" in the phrase "random sample" does  not have its ordinary, everyday meaning -- that is, does not refer to the first definition you would find in a dictionary. Here are the first definitions from a couple of dictionaries:

"Having no specific pattern or objective; haphazard" (The American Heritage Dictionary, Second College Edition, Houghton Mifflin, 1985)

"Proceeding, made, or occurring without definite aim, reason, or pattern (Dictionary.Com, http://dictionary.reference.com/browse/random, accessed 11/19/09)

One common mistake that arises from applying this ordinary, everyday meaning of "random" to the phrase "random sample"1 is concluding that a sample is not random because it has a pattern. In fact, a random sample (using the technical meaning of the phrase) might have a pattern (or it might not). In fact, there is no way we can tell from looking at the sample whether or not it qualifies as a random sample. 2

This brings up the second reason why the phrase is often misunderstood: The adjective "random" refers to the method by which the sample is chosen. Thus the phrase "random sample" really means "randomly chosen sample." 3

The next sections elaborate on what "randomly chosen" means.

## Why Is Random Sampling Important?

Notes
1. Adding to the confusion, the dictionary definition may apply in the phrases "random process" and "random variable"; see Note 1 in Example 6 on the page Random Variables and Probability Distributions for an example.

2. Many "tests of randomness" for strings of numbers have been developed. However, none of them is perfect; they can just help screen out some strings that are not random. Often a "suite" of such tests is used to check "random number generators". See Turiel, Thomas (2007), Quantum Random Bit Generators, The American Statistician, v. 61 no. 3, pp. 255 - 259, for further discussion and references.

3. W. Edwards Deming, in his obituary of Walter Shewhart,  reflects this idea, saying, "There was to Shewhart no such thing as a random sample. There was and is, however, such a thing as a sample selected by a random operation." (Quoted from p. 20 of  an excerpt of the obituary that appeared in Amstat News, September, 2009. The original article appeared in The American Statistician, v. 21, pp. 39 - 40.)  Shewhart defined a random sample as "A sample drawn under conditions such that the Law of Large Numbers applies" (p. 256) in his article Random Sampling, The American Mathematical Monthly, Vol. 38, No. 5, May, 1931, pp. 245 - 270.

Updated August 28, 2012