“Why is the denominator in the sample mean n, but the denominator for the sample variance is n−1?” a reader asked me. My answer needs to be comprehensible to his grand-daughter, who we can safely say is not doing an advanced degree in statistics at an institution of higher learning. All of us have had to answer this question at some time in our careers, either for our students or for ourselves. How do you answer it, and how helpful is your answer? Do you feel obliged to introduce distinctions such as populations vs samples, description vs inference, parameters vs statistics, Greek vs Roman letters? Or more advanced concepts, such as degrees of freedom, dimensions of subspaces, unbiasedness or maximum likelihood? Or do you think we should just use n as the divisor in the sample variance and move on, perhaps with a footnote stating that half the world uses n, and the other half uses n−1, while a couple of people with PhDs in statistics from Berkeley use n+1?
In the old days, when we wanted a variety of approaches to answering a question like this, we’d leaf through a selection of introductory texts, and fix on the answer we like best. These days we may not need to leave our desk to carry out this task. We can search the web, we can often LOOK INSIDE texts, and find the answer we like, at any desired level. Or can we? I must confess that I have never found an answer I liked to the “n vs n−1” distinction, not a simple, intuitive, but correct explanation, that makes sense to students at all levels. There are some good tries out there, but none that I find entirely satisfactory. I encourage you to look.
Following my introduction to statistics over fifty years ago, I noticed that from time to time, my teachers seem to lose it, and us, and “go off with the fairies”. Those who insist on clarifying the distinction of my title hit this very early on. They want to introduce the familiar $s^2$, and they want to do it right. If the price to pay for this is that we must leave the world of rational thought, so be it, they reason. In her lovely 1940 paper on degrees of freedom (d.f.) cited in the excellent Wikipedia article on the same topic, Helen M Walker (1891–1983) wrote, “this concept often seems almost mystical, with no practical meaning.” Sadly familiar to so many of us.
Can we look to history for insight on this matter? Readers of Walker’s historical review of d.f. will find little help for their pedagogical task. Gauss clearly understood the notion, but then we probably had to wait until “Student” (1908) and of course R.A. Fisher for further clarification, while Karl Pearson was famously not so clear on the concept. This is not stuff for intro courses. What we can learn from history is that people have been arguing about ways of presenting the n vs n−1 distinction for many decades now. On this point, I’d be happy to offer a small cash prize for the earliest reference in the statistics literature to my title. (Exactly how I will decide who wins, so that I can award the prize, I leave for another time.) Certainly the education and psychology literature has several excellent contributions to our topic, as they should, for they have been inflicting our subject on their students for nearly a century now. There was a valuable burst of activity in the American Educational Research Journal forty years ago, and doubtless there have been many similar exchanges at other times and in other places. Do you think a clear winner has emerged? I don’t.
Can we look to statistical theory to help in our explanation of the use of n−1? If we want to achieve unbiasedness—of our estimate of $σ^2$ but not of our estimate of $σ$ — then we can justify the n−1. That’s not too hard to explain, but is it worth the effort? If we are willing to introduce maximum likelihood estimation (under normality), we can justify the n, but that’s even more effort, and, I think, beyond my reader’s grand-daughter. We can even justify n+1 if we seek a minimum mean square error estimate of $σ^2$ (within a certain class). My conclusion is that at best, invoking theory leads to a draw between n and n−1. You pays yer money, and you takes yer choice.
I can’t see any real problem with introductory courses using the divisor of n for the sample variance. My reader wrote, “…the use of n instead of n−1 would make one of my grandchildren happy.” Me too!