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I received a question today via email about what statistics you can use when your sample equals the entire population. This issue also comes up when people ask about the use of the finite population correction factor. Both topics are controversial, and there is no clear consensus on what to do. I touched on this issue briefly in my discussion about population sizes (why does sample of 100 people from a population of 10 thousand have roughly the same precision as a sample of the same size from a population of 10 million people?).
The issue boils down to the following question: are you interested in this population without any desire to extrapolate to different populations. In particular, are you uninterested in how your results might apply to people in a different location? Are you interested only in the current process being studied and uninterested in future output from this process?
If there is no interest in different locations or in the future values, then you have an enumerative study. With an enumerative study, a sample that includes the entire population has no sampling error. Tests of hypothesis and confidence intervals are meaningless in this situation.
If there is some effort to extrapolate, however, then you have an analytic study. With an analytic study, the sample could be considered to be a smaller part of conceptual population. This population might be all data at your particular location and at locations that are similar to yours. Or it might be all data past, present, and future from a process. Here you would want to use confidence intervals and hypothesis tests, but you have to provide some assurance that there is similarity to other locations or stability in the process across time.
Think about what actions you will take on the basis of your study.
- If your actions will only be on the population of interest, then you have an enumerative study.
- If your actions will extend beyond the population you are studying, then you have an analytic study.
Consider two conceptual samples taken of the entire populations at Children's Mercy Hospital. The first sample is all of the billing records for 2001 through 2003. We have some evidence of overbilling, and we want to estimate the overbilling in order to repay the insurance companies. We don't care about bills prior to 2001 (statute of limitations) or beyond 2003 (because we have corrected the billing process to prevent further overbilling). We are also not interested in overbilling at similar children's hospitals (let them pay their own bills). This is an enumerative study.
The second sample is an observation of the hand washing practices of all physicians at the hospital during the month of December 2003. Here we do care about future handwashing of these physicians and our findings might apply to other children's hospitals as well. This is an analytic study.
Walter Shewhart and W. Edwards Deming developed the concept of Enumerative and Analytic studies and they argue that statistical process control tools are the best way to handle analytic studies. Some resources are May 2001 presentation by Lloyd Provost, Use of experimental Design in Analytic Studies, and a September 1997 presentation by Anthony Cutler, Deming's Vision Applied to Probabilistic Risk Analysis and a web page by Jonathan Siegel on the Shewhart-Deming Critique of Classical Statistics.