Editing Multimodal distribution (section)

===Graphical methods===

In the study of sediments, particle size is frequently bimodal. Empirically, it has been found useful to plot the frequency against the log(&nbsp;size&nbsp;) of the particles.<ref name=Folk1957>{{cite journal | last1 = Folk | first1 = RL | last2 = Ward | first2 = WC | year = 1957 | title = Brazos River bar: a study in the significance of grain size parameters | url = https://doi.pangaea.de/10.1594/PANGAEA.896129| journal = Journal of Sedimentary Research | volume = 27 | issue = 1| pages = 3–26 | doi=10.1306/74d70646-2b21-11d7-8648000102c1865d|bibcode = 1957JSedR..27....3F }}</ref><ref name=Dyer1970>{{cite journal | last1 = Dyer | first1 = KR | year = 1970 | title = Grain-size parameters for sandy gravels | journal = Journal of Sedimentary Research | volume = 40 | issue = 2| pages = 616–620 |doi=10.1306/74D71FE6-2B21-11D7-8648000102C1865D}}</ref> This usually gives a clear separation of the particles into a bimodal distribution. In geological applications the [[logarithm]] is normally taken to the base 2. The log transformed values are referred to as phi (Φ) units. This system is known as the [[Grain size|Krumbein]] (or phi) scale.

An alternative method is to plot the log of the particle size against the cumulative frequency. This graph will usually consist two reasonably straight lines with a connecting line corresponding to the antimode.

;Statistics

Approximate values for several statistics can be derived from the graphic plots.<ref name=Folk1957/>

<math display="block">\begin{align}
\text{mean} &= \frac{ \phi_{16} + \phi_{50} + \phi_{84} }{ 3 } \\[1ex]
\text{std. dev.} &= \frac{ \phi_{84} - \phi_{16} }{ 4 } + \frac{ \phi_{95} - \phi_5 }{ 6.6 } \\[1ex]
\text{skewness} &= \frac{ \phi_{84} +  \phi_{16} - 2  \phi_{50} }{ 2 ( \phi_{84} -  \phi_{16} ) } + \frac{ \phi_{95} +  \phi_{ 5 } -  2 \phi_{50} }{ 2( \phi_{95} - \phi_5 ) } \\[1ex]
\text{kurtosis} &= \frac{ \phi_{95} - \phi_5 }{ 2.44 ( \phi_{75} - \phi_{25} ) }
\end{align}</math>

where ''φ''<sub>x</sub> is the value of the variate ''φ'' at the ''x''<sup>th</sup> percentage of the distribution.