Name

Error equation

Note

Ref.

SSE/ERRSQ

${\displaystyle \sum }_{i=1}^n{\left(q_{e,cal}-q_{e,exp}\right)}^2$

It is indicator for accuracy, in which the best fit of the data can be assessed from the sum-of-squares value. The smallest value for SSE indicates the best fit data for the model.

[82]

HYBRID

${\displaystyle \sum }_{i=1}^n\frac{100}{n-p}\left[\frac{\left(q_{e,meas}-q_{e,cal}\right)}{q_{e,meas}}\right]$

The error function was developed to improve ERRSQ fit at low concentrations.

[83]

ARE

$\frac{100}{n}{\displaystyle \sum }_{i=1}^n\left|\frac{\left(q_{e,meas}-q_{e,cal}\right)}{q_{e,meas}}\right|$

which indicates a tendency to under or overestimate the experimental data, attempts to minimize the fractional error distribution across the entire studied concentration range

[84]

χ²

${\displaystyle \sum }_{i=1}^n\frac{{\left(q_{e,cal}-q_{e,\exp }\right)}^2}{q_{e,cal}}$

χ² is also similar to SSE. Smaller values of χ² also indicate a better fit of the model.

[69]

SE

$\sqrt{\frac{1}{n}{\displaystyle \sum }_{i=1}^n{\left(q_{e,cal}-q_{e,exp}\right)}^2}$

It is also used to judge the equilibrium model. A smaller value for SE indicates a better fit of the model

[84]

∆q (%)

$100\sqrt{\frac{1}{n-1}{\displaystyle \sum }_{i=1}^n{\left(\frac{q_{e,meas}-q_{e,cal}}{q_{e,meas}}\right)}^2}$

According to the number of degrees of freedom in the system, it is similar to some respects of a modified geometric mean error distribution

R²

$\frac{{{\displaystyle \sum }}_{i=1}^n{\left(q_{cal}-\overline{q_{exp}}\right)}^2}{{{\displaystyle \sum }}_{i=1}^n{\left(q_{e,cal}-\overline{q_{e,exp}}\right)}^2+{{\displaystyle \sum }}_{i=1}^n{\left(q_{e,cal}-q_{e,exp}\right)}^2}$

The correlation coefficient (R²) is the common measure of analytical accuracy. Its value is within the range of 0 < R² ≤ 1, where a high value reflects an accurate analysis.

[85]

SAE

${\displaystyle \sum }_{i=1}^n\left|q_{e,mass}-q_{e,cal}\right|$

with an increase in the errors will provide a better fit, leading to the bias towards the high concentration data

[84]

S_RE

$\sqrt{\frac{\left[{{\displaystyle \sum }}_{i=1}^n{\left(q_{e,mass}-q_{e,cal})-\text{ARE}\right)}^2\right]}{n-1}}$

[84]