What is log normal distribution

Log normal distribution, LNVT distribution

The logarithmic normal distribution (LNVT for short) can be derived from the GaussianError function (linear normal distribution) can be derived by substitution, because the logarithm of the particle size (ln x) is mostly normally distributed.

Normal distribution - distribution density

First we consider the distribution density of the normal distribution. This is formally described with

Click here to expand $ q_r (x) = \ frac {1} {s_x \ cdot \ sqrt {2 \ pi}} \ cdot e ^ {- \ frac {1} {2} (\ frac {x - \ overline { x}} {6 x}) ^ 2} $

Characteristic values: $ s_x $ = standard deviation of size x, $ \ overline {x} = x_ {50, r} $ = median of the r-distribution.

By taking the logarithm and integrating, the Distribution sum function the LN distribution:

Click here to expand $ q_r (t) = \ frac {1} {\ sqrt {2 \ pi}} \ cdot exp (- \ frac {t ^ 2} {2}) $ with

$ t = \ frac {x - \ overline {x}} {s_x} $

The Average $ \ overline {x} _r $ determines the position of the distribution and the standard deviation $ s_x $ determines the width of the distribution.

Log normal distribution - distribution density

The Distribution density functionthe logarithmic normal distribution results from

$ q_r (x) = \ frac {1} {\ sigma_x \ sqrt {2 \ pi}} exp [- \ frac {1} {2} (\ frac {x - x_ {50, r}} {\ sigma_x} ) $ by the following substitution:

$ t = \ frac {1} {s_x} ln \ frac {x} {x_ {50, r}} $.

The Distribution density function is then:

Click here to expand $ q_r (ln x) = \ frac {1} {s_x \ sqrt {2 \ pi}} \ cdot exp [- \ frac {1} {2} \ cdot (\ frac {ln (\ frac { x} {x_ {50, r}})} {s_x}) ^ 2] $

The logarithmic normal distribution network is divided in such a way that value pairs of the distribution sum $ Q (x) $ belonging to the above equation result in a straight line for all types of quantities $ r $.

Logarithmic normal distribution network

 

A logarithmic normal distribution therefore exists when measured value pairs $ (x, Q) $ of a cumulative distribution form a straight line after being entered in the logarithmic probability network. In contrast to the linear normal distribution network, the logarithmic probability network has a division of the x-axis.

Click here to expandParticle collectives with logarithmic normal distribution always occur when there are larger fines in the collective and the proportion is rather small.

Median values

The given median value $ X_ {50, k} $ and the standard deviation $ s_k $ allow arbitrary median values ​​to be calculated. A determination is made as follows:

Click here to expand $ x_ {50, r} = x_ {50, k} \ cdot e ^ {(r - k) \ cdot s_k ^ 2} $

Characteristic values: $ s_k $ = Standard deviation, as a measure of the width of the distribution, $ r = 0, 1, 2, 3 $, $ k = 0, 1, 2, 3 $

Click here to expandWith $ x_ {81}, x_ {50}, x_ {33} $ the Features of fineness below which $ 81%, 50%, $ 33% of the respective quantities are. The standard deviation is given only by the slope of the straight line and is completely independent of the type of quantity. If the type of quantity is to be changed, the distribution line must be shifted in parallel from the old median value to the new median value.
Click here to expand

In the next section of the course, we will look at measurement methods for particle size analysis.