The attached .pdf file "Notes on bias correction of sample variance.pdf" is a demonstration of the theory that explains why we divide by "n-1" instead of "n" when calculating sample variance. The ratio "n/(n-1)" is a correction for bias when calculating the point estimate of the population variance using sample data. See "Wackerly, Mendenhall, Scheaffer" Sixth Edition page 372.
The attached R script enables a demonstration of the presence of bias and bias correction.
Note that a different bias correction is needed for calculating sample standard deviation. (need to develop further)