Johnson’s SB Distribution

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Johnson’s SB distribution is a system of curves for bounded data. The “SB” is short for “system bounded.”

Johnson [1] showed that a bounded random variable X (i.e., one with upper and lower limits) could be transformed to an approximately normal distribution with the transformation Johnson’s SB Distribution
  • ln = the natural logarithm,
  • Z ~ N(0, 1) (i.e., a standard normal distribution with mean 0 and standard deviation 1).
Z has a probability density function (PDF) of Johnson’s SB Distribution

This leads to the equation for Johnson’s SB distribution:

Johnson’s SB Distribution
or Johnson’s SB Distribution

Olsson [2] explained the fitting of Johnson’s SB system and SU Systems of curves to grouped data using the method of maximum likelihood. This computer program Olsson developed uses the Nelder-Mead simplex subroutine and gives ML estimates for the four parameters required in the fitting procedure.

The SB distributions have the occasional practical application. For example, Hafley and Schreuder [3] introduced the distribution to the forestry literature. Several authors have expanded on the distribution in this field, including Parresol [4], who demonstrated recovery of parameters for the distribution.

Johnson’s SB Distribution
Plot of skewness squared-kurtosis values for the 527 pine plot observations. The SB distribution covers the region between the lognormal line and the impossible region [4].


[1] Johnson, N.L. 1949. Systems of frequency curves generated by methods of translation. Biometrika. 36: 149-176.

[2] Olsson, D. (1979). Fitting Johnson’s SB and SU Systems of Curves using the Method of Maximum Likelihood, Journal of Quality Technology, 11, 211-217.

[3] Hafley, W.L.; Schreuder, H.T. 1977. Statistical distributions for fitting diameter and height data in even-aged stands. Canadian Journal of Forest Research. 7: 481-487.

[4] Parresol, B. (2003). Recovering Parameters of Johnson’s SB distribution. USDA.

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