# Johnson’s SB Distribution

Probability Distribution: List of Statistical Distributions >

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

Johnson  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).
Z has a probability density function (PDF) of This leads to the equation for Johnson’s SB distribution: or Olsson  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  introduced the distribution to the forestry literature. Several authors have expanded on the distribution in this field, including Parresol , who demonstrated recovery of parameters for the 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 .

## References

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

 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.

 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.

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

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Stephanie Glen. "Johnson’s SB Distribution" From StatisticsHowTo.com: Elementary Statistics for the rest of us! https://www.statisticshowto.com/johnsons-sb-distribution/
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