# Hyperlink 18. Calculating the Mass of Sample Needed to Achieve a Specific Fundamental Error

One can estimate the mass of a sampled needed to achieve a specific fundamental error (FE) by applying Gy theory equations that relate the mass of the sample, diameter of the largest particle, and the variance of the fundamental error.

A simplified Gy theory equation useful for this calculation is as follows (Eq.1 ):

s^{2}_{FE} = Cd^{3}/M_{s} (1)

where

M_{s} |
= mass of the sample |
---|---|

d | = diameter of the mesh opening that retains no more than 5% of the sample |

s^{2}_{FE} |
= variance of the fundamental error |

C | = sampling constant |

Rearranging to solve for the mass of the sample one obtains the following (Eq. 2):

M_{s} = Cd^{3}/s^{2}FE (2)

A common value for d is 0.2 cm, as the media of interest usually is soil, which is generally defined as those particles <2 mm in diameter. The desired FE is often selected as 15%, as one would like to keep the overall error fairly low. If there are several sampling stages, then each stage contributes an FE to the overall error. Note that each sampling stage also contributes to the other six sampling errors that compose the overall error, so it is desirable to keep these errors to a minimum also.

The sampling constant is often given a value of 22.5. However, one should be aware of the assumptions used in obtaining this value as well as the assumptions used to generate equation (1).

Using these values, the calculated mass of the sample is 8 g.

The most problematic assumption from an environmental perspective in using a sampling constant of 22.5 is that this assumes that the concentration of the analyte of interest is in the percent range. For environmental sampling where the concentrations of interest are on the order of parts per million (ppm), then the sampling constant will be about 200,000.

Using this sampling constant (2 × 105), the calculated mass of the sample would be 71 kg with the same FE and particle diameter as above.

The equation with the sampling constant broken out into its components is as follows (Eq. 3):

s^{2}FE = cβfgd^{3}/M_{s} (3)

where

M_{s} |
= mass of the sample |
---|---|

c | = constitution parameter |

β | = dimensionless liberation factor |

f | = dimensionless shape factor |

g | = dimensionless size range factor |

d | = diameter of the mesh opening that retains no more than 5% of the sample |

Guidelines for the values of these parameters are given by Gy (1998):

The constitution parameter, c, depends upon the amount of the analyte of interest, a, in the lot and the mean density of the lot. If the amount of a in the lot is small, a <<1, then an approximation for c is given by c =δ

_{M}/a_{L}, where dM is the mean density of the lot and aL is the decimal fraction of a in the lot.The dimensionless liberation parameter, β, can have values from 0 to 1. The parameter is 0 when the components are completely homogenized (an impossible situation) and is 1 when the components are completely liberated. It is best to set β = 1 if one is not certain of the state of liberation.

The dimensionless shape parameter, f, also can have values from 0 to 0. For a sphere f = 0.52. For most compact particles f has values near 0.5.

The dimensionless size range parameter, g, also can have values from 0 to 1. Some values used in practice are:

- Undifferentiated, unsized materials, mean value g = 0.25
- Undersized material passing through a screen g = 0.40
- Oversize material retained by a screen g = 0.50
- Material sized between two screens g = 0.6/0.75
- Naturally sized materials, e.g., cereal grains g = 0.75
- Uniformly sized objects, e.g., bearing balls g = 1.0

Rearranging Eq. 3 to solve for the mass of the sample and substituting dM/aL for c yields the following (Eq. 4):

M_{s} = (δ_{M}/a_{L})βfgd^{3}/s^{2}_{FE} (4)

For this example, suppose that a DQO has been established limiting FE to 15%. By using the following values for the parameters in the equation for the variance of the fundamental error: δ_{M} = 1.6 g/cm^{3} (a typically density for soil), β = 1 (as suggest above), f = 0.5 (also as suggested above), g = 0.25 (for unsieved soils), and d = 0.2 cm (from the definition of soil), one can solve for the mass of the sample for anticipated situations (Eqs. 5/6):

M_{s} = (1.6/a_{L})(1)(0.5)(0.25)(0.2)^{3}/s^{2}_{FE} (5)

or

M_{s} = (1.6 × 10^{–3})/(a_{L}s^{2}_{FE}) (6)

Therefore, for a desired FE of 15% and a_{L} = 1 × 10^{–6}, the mass of the sample needs to be 71 kg.

The assumption made in both Eqs. 1 and 3 is that the mass of the sample is much less than the mass of the population or lot, ML. The equation both are derived from is as follows (Eq. 7):

(7)

where

M_{s} |
= mass of the sample |
---|---|

M_{L} |
= mass of the lot |

c | = constitution parameter |

β | = dimensionless liberation parameter |

f | = dimensionless shape parameter |

g | = dimensionless size range parameter |

d | = diameter of the largest particle |

This assumption is probably true for field sampling but may not be true for laboratory subsampling.

When using any of these equations to determine the mass of the sample, it is important to note that the values obtained are approximate (on the order of magnitude of the mass needed). The actual mass of sample needed to achieve a specific fundamental error may be greater than that calculated.

One can experimentally determine whether the mass calculated is sufficient by analyzing at least 10 replicate samples. If the variance of the results is less than the variance of the desired fundamental error, then the mass of the sample is sufficient. If the variance of the results is greater than desired, then overall sampling and analysis process must be reexamined. Some part of the process—field sampling, laboratory subsampling, processing, analysis, etc.—is not in control and must be corrected to achieve the desired variance.