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Johnson’s S_{B} distribution is a system of curves for bounded data. The “S_{B}” 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

Where:

*ln*= the natural logarithm,*Z*~*N*(0, 1) (i.e., a standard normal distribution with mean 0 and standard deviation 1).

This leads to the equation for Johnson’s S_{B} distribution:

or

Olsson [2] explained the fitting of Johnson’s S_{B} system and *S _{U}* 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 S_{B} 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.

## References

[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 *S _{B}* and

*S*Systems of Curves using the Method of Maximum Likelihood, Journal of Quality Technology, 11, 211-217.

_{U}[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 *S _{B}* distribution. USDA.

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**Stephanie Glen**. "Johnson’s SB Distribution" From

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