A Practical Guide to Isotherms of ADSORPTION on Heterogeneous Surfaces

### Statistical deviations in experimental data (Numerical simulation)

Simulation of quasi-gaussian errors

Quite often, the distribution of errors depends on variable magnitude. Then you may fit by using statistical weights or by application of data transformation that depends on the error vs. variable dependence.

Two types of deviations are shown:

• deviations locally proportional to the magnitude of y, err(y) ~ ky (blue points and fitted line)
• deviations locally proportional to the inverse of y, err(y) ~ m/y (red points and fitted line)
The actual deviations were calculated by multiplication of "local magnitude" by a result of quasi-gaussian random sampling (shown below).

Quasi-gaussian distribution of errors was calculated by superposition of 5 random() functions (continuous distribution with total width w = 1, min = 0, max = 1, mean = 0.5) and scaling it in such a way, that the minimum and maximum were -0.5 and 0.5, respectively (total width w = 1, mean = 0):

```  random5() = [random()+random()+random()+random()+random()]/5 - 0.5
```
or generally for superposition of n random() functions:
```  random_n() = [random()+ ... +random()]/n - 0.5
```
Continuous random() distribution of width w has standard deviation
σ( random() ) = 0.5w/√(3)
whereas the superposition of n random() functions has standard deviation of:
σ( random_n() ) = 0.5w/√(3n)

Quasi-gaussian distribution of errors

Simulation of quasi-gaussian errors with outliers (description)

Quite often, the distribution of errors depends partially on variable magnitude. Then you may fit by using statistical weights or by application of data transformation that depends on the error vs. variable dependence.

Two types of deviations are shown:

• deviations being a combination of a constant statistical deviation (e.g backgound noise) and a term locally proportional to the magnitude of y, err(y) ~ ky+a (blue points and fitted line)
• deviations locally proportional to the inverse of y, err(y) ~ m/y (red points and fitted line)
The actual deviations were calculated by multiplication of "local magnitude" by a result of quasi-gaussian random sampling (shown below).

Quasi-gaussian distribution of errors was calculated by superposition of 5 random() functions (continuous distribution with total width w = 1, min = 0, max = 1, mean = 0.5) and scaling it in such a way, that the minimum and maximum were -0.5 and 0.5, respectively (total width w = 1, mean = 0) - see above. The actual random error/deviation was calculated as a product of a such simulated standarized error (w=1) by a factor dependent on magnitude of variable.

In fact two distributions were used. One - as above, i.e. without outliers - for errors err(y) ~ m/y and err(x) ~ m/x. However, the other one - distribution for errors err(y) ~ (ky+a) - was created by random switching between the original error distribution (width w1) with probability 80% and one (identical in shape/character) with probability of 20% and triple width (w2 = 3 w1), thus producing "outliers".

Quasi-gaussian distribution of errors with outliers

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