Standard Deviation and Variance

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Definition:                           

Equation

Parameters

Diagram

 

Calculate Standard Deviation and Variance:              top

Parameter

Value

 

Inputs:

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

Outputs:

N (count)

 
∑xi (sum)

 
  min (minimum value)  
  max (minimum value)  
  R (range=max-min)  
x (mean)

 
s2N (variance)

 
  sN (standard deviation)  
  s2N-1 (unbiased variance)  
  sN-1 (unbiased std. dev.)  
       

Output Format:                                                        top

Select Format: Scientific Engineering Fixed

Notes:                                                                       top

bulletThe 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".
bulletIn 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".
bulletIn Excel, the STDEVA function gives sN-1 while STDEVP gives sN.  (The "P" at the end of STDEVP is for "population" and highlights some of the terminology problems).
bulletYou can also use the range, R, of a sample to estimate the standard deviation using order statistics.

 

Copyright 2004-2005  Raymond J. Allen