# What is log normal distribution

## Log normal distribution, LNVT distribution

The **logarithmic normal distribution** (LNVT for short) can be derived from the **Gaussian****Error 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

**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:

$ 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 function****the 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:

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 $.

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.

**Particle 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:

**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 $

**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.

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

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