Standard Deviation and Variance

Definition:

Equation

Parameters

Diagram

	The terms "sample" and "population" are inconsistent across sources so be careful! This is because it is not always clear if your "sample" represents the entire "population" of random values, or is truly just a "sample" of a larger "population".
	In most cases, one wishes to estimate the properties of a large population based on a subset of samples. Usually in this case, the actual mean of the population, μ, is not known and must be estimated from the sample as the sample mean, x. This may seem like a small point, but causes problems when trying to estimate the variance of the population given a small sample. It turns out from k-statistics that the best estimate for the population variance is N/(N-1) times the sample variance. So, you would want to use the "N-1" values listed above. If, on the other hand, your list includes the entire population, then you would probably want to use the "N" values. If you have a large number of samples then this difference is trivial. Note again that several sources invert the meanings of "population variance" and "sample variance".
	In Excel, the STDEVA function gives s_N-1 while STDEVP gives s_N. (The "P" at the end of STDEVP is for "population" and highlights some of the terminology problems).
	You can also use the range, R, of a sample to estimate the standard deviation using order statistics.

	Parameter	Value
Inputs:	Enter list of samples (x_i) below (one per line):
			If your numbers are all on one line, you can:




Outputs:	N (count)
Outputs:	∑x_i (sum)
	min (minimum value)
	max (minimum value)
	R (range=max-min)
	x (mean)
	s²_N (variance)
	s_N (standard deviation)
	s²_N-1 (unbiased variance)
	s_N-1 (unbiased std. dev.)