Research Design and Statistical Consulting
George M. Diekhoff, Ph.D.

# Un-Besseling the Standard Deviation

I recently worked on a project that requires measuring several aspects of diversity in corporate boards of directors. The measure of diversity I chose to use with both dichotomously-scored (i.e., 0/1) nominal scale variables and continuous variables was the standard deviation. I aggregated the data from the board members associated with each firm and calculated the standard deviations for each diversity variable. For instance, the standard deviation of the ages of the board members served as a measure of age diversity and the standard deviation of the gender variable (0=female, 1=male) served as a measure of gender diversity, and so on. The problem I found is that SPSS calculates Bessel’s corrected standard deviation (with N-1 in the denominator), not the actual standard deviation (with N in the denominator). Consequently, the Bessel-corrected standard deviations reflected not only diversity, but also sample size. Take two companies, A and B. Company A has 1 male and 1 female. Company B has 2 males and 2 females. Both companies show equal gender diversity, but Bessel’s corrected standard deviation as computed by SPSS is 0.71 for Company A and 0.58 for company B, and both values are higher than the theoretical upper limit for the standard deviation for binary data which is .50! What’s a boy to do?! Un-Bessel the standard deviations (called “SD” below) as follows:

Transform > Compute Variable
Target Variable: New_Std_Dev
Numeric Expression: ((SD**2)/(N/(N-1)))**.5

In words: (1) Square the Bessel-corrected standard deviation (SD). (2) Divide #1 by N/(N-1). (3) Find the square root of #2.Having un-Besseled in this way, both Company A and Company B now show a standard deviation of .50.

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