In sample variance we divide by n-1 and not n. I know a couple of arguments for this - one is that this is sort of a normalization to ensure that the expected value of sample variance is equal to population variance (I calculated this mathematically and it is sort of a good argument) and the other argument is an intuitive one where we argue that the outliers in a population i.e. "Samples" with extreme values actually have a very low probability of getting selected in our Sample and hence our sample is more closer to mean than it should be scaled up. But a few days back our stats teacher gave a new argument related to degrees of freedom. Just to be sure I checked and got this definition for degrees of freedom: the number of variables we can vary without changing the constraints of the system. But my question is in Sample Variance we actually have n+1 variables - n independent random variables and one mean of these. And therefore I believe that I can vary all these n random variables and get the mean accordingly or vary n-1 random variable AND the mean and get the nth random variable - either way i will get n degrees of freedom. What is the flaw in this argument because my teacher argued that there would be n-1 independent variables and n-1 degrees of freedom?
2026-03-30 00:18:27.1774829907
Sample variance: degree of freedom argument
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Estimates of statistical parameters can be based upon different amounts of information or data. The number of independent pieces of information that go into the estimate of a parameter are called the degrees of freedom. In general, the degrees of freedom of an estimate of a parameter are equal to the number of independent scores that go into the estimate minus the number of parameters used as intermediate steps in the estimation of the parameter itself (i.e. the sample variance has N-1 degrees of freedom, since it is computed from N random scores minus the only 1 parameter estimated as intermediate step, which is the sample mean).
-Wikipedia
The sample mean is an estimated parameter, not a random variable. Constructing the mean as a linear combination of your existing variables does not add to the dimensionality of your system (I'm not sure if you have any prerequisite linear algebra).
I hope that makes sense