Why is the degrees of freedom?

Question

What is the degrees of freedom and why in the denominator of the sample variance formula we use the factor of n-1 instead of n?

In: Mathematics

Answer 1

“Degrees of freedom” is basically “number of different inputs”. It’s a little more complex than that that, but that’s fine for ELI5.

We use n-1 instead of n for sample variance because we already “used up” one calculation to calculate the sample mean, which is an input to the sample variance calculation. We don’t really have as many independent inputs as we think we do. If we knew the true population mean, or our sample was the full population, we’d divide by n instead of n-1. Switching to n-1 accounts for the fact that we’re using the sample mean, not the population mean, and the sample mean itself is an estimated calculated from the same data.