Transformation between log-normal distributions

If the number distribution is log-normal, the mass distribution is also lognormal with the same geometric standard deviation. Using the same treatment as was used to derive equation (2.77) gives, for a mass analysis  

Since the relations between the number average sizes and the number geometric mean are known [equations (2.77) to (2.80)], these can now be expressed as relationships between number average sizes and the weight geometric mean to give a similar set of equations. 

Other average sizes can be derived from the above using a similar procedure to that used to derive equations (2.84) and (2.85) to give  

Similarly, for a surface distribution, the equivalent equation to equation (2.80) is  

Substituting this relationship into equations (2.86) to (2.89) yields the equivalent relationships relating surface average sizes with the surface geometric mean diameter. 

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