This is a description of how we usually calculate the uncertainties of concentrations and ages and some practical examples on cosmonuclide analysis.

##### Theory

Considering that

- All operators (e.g. analytical data) are well represented by
**normal distributions**. - We are considering uncertainties of
**independent variables**(no covariance). - The
**uncertainty of the result**is small enough to be well represented by a normal distribution. This means that the result is roughly linear in the area of the uncertainty and therefore the resulting distribution is**not asymmetrical**.

the error propagation should be performed by considering the partial derivatives of the result respect the operators:

where and are the operators within a standard deviation (one sigma) uncertainties.

**Examples of common operations**

Operation | Formula | Uncertainty |

General | ||

Addition | ||

Subtraction | ||

Multiplication | ||

Division | ||

Power | ||

Natural logarithm | ||

Logarithm to base 10 |

**Examples applied to cosmogenic isotope data**

###### Calculating a ^{10}Be age

The apparent ^{10}Be age* t* of a surface sample with a ^{10}Be concentration of *C* is

where *P* is the production rate and λ is the radioactive decay constant. According to the general equation, the uncertainty of the apparent age σ* _{t}* should be calculated as

where σ* _{C}* and σ

*are the uncertainties of the concentration and the production rate, respectively. However, as for , the following approximation is correct for most of the typically calculated exposure ages:*

_{P}^{26}Al/^{10}Be burial age

If we calculate a ^{26}Al/^{10}Be burial age *t* as

where *R* is the ^{26}Al/^{10}Be ratio, *R _{i}* is the initial ratio, λ

_{10}and λ

_{26}are the radioactive decay constants, and

*σ*and

_{R}*σ*

_{R}_{i}are the considered independent one-sigma uncertainties, the error of the burial age should be:

In this case, this equation is often not realistic because the previous equation does not fit the third assumption (asymmetrical error) for typical *R* uncertainties from too young (*R/R _{i}*>0.8) or too old samples (

*R/R*<0.2). Therefore, in this case, it is highly recommendable to calculate the extreme values of

_{i}*t*(

*t*+

*σ*and

_{t}*t*–

*σ*) by varying the burial equation as follows:

_{t}to check the asymmetry and, if necessary, calculate the positive and negative errors of *t* (*t*+*σ _{t}* and

*t*–

*σ*) from the maximum and minimum values obtained from the last equations.

_{t}###### Calculating a ^{21}Ne concentration

If we consider that a cosmogenic ^{21}Ne concentration is calculated as:

where *R* are the measured ^{21}Ne/^{20}Ne ratios, *N _{20}* is the total number of

^{20}Ne atoms,

*M*is the mass of the sample and we are considering

*σ*

_{RSample}_{ }and

*σ*as independent errors, the uncertainty of the

_{N20}^{21}Ne concentration should be

that can be also expressed as

and it is always larger than

which is not a realistic way of calculating the uncertainty of [^{21}Ne] unless *R _{Sample}*>>

*R*Also, for samples where

_{Air.}*R*values are close to

_{Sample}*R*, the uncertainty of [

_{Air}^{21}Ne] might be asymmetrical, and it is highly recommendable to check the shape of the [

^{21}Ne] uncertainty as explained in the burial section.