On the Stability Assessment of Reference Materials

Abstract

As well as presenting comparative characteristics of domestic and international approaches to assessing the instability of reference materials, the paper describes proposed changes in R 50.2.031–2003. A proposed mathematical apparatus is based on an algorithm of actions for estimating the standard uncertainty from instability and the validity period of a reference material (RM). Approaches to estimating uncertainty from instability in cases of the absence or presence of a significant trend in a RM’s certified characteristic over time are considered. The minimum number of measurements necessary to study the stability of an RM is determined and justified. Smoothing of measurement results when assessing the stability of an RM is shown to lead to an overestimation of the period of validity.

Appendices

Appendix 1: Determination of Inequality (2)

The target value of the error from RM instability $\hat{\Delta }_{T}$ is assumed to be set. Then from the formulas (3), (5) and (7), the equality follows

$$\left| {\hat{a}} \right| \cdot \tau_{\Gamma } + t_{p,N - 2} \cdot S\left( \varepsilon \right) \cdot \sqrt {\frac{1}{N} + \frac{{(\tau_{\Gamma } - \overline{\tau })^{2} }}{{\mathop \sum \nolimits_{n = 0}^{N - 1} (\tau_{n} - \overline{\tau })^{2} }}} = \hat{\Delta }_{T} ,$$

(15)

as well as the inequality

$$t_{p,N - 2} \cdot S\left( \varepsilon \right) \cdot \sqrt {\frac{1}{N} + \frac{{(\tau_{\Gamma } - \overline{\tau })^{2} }}{{\mathop \sum \nolimits_{n = 0}^{N - 1} (\tau_{n} - \overline{\tau })^{2} }}} \le \hat{\Delta }_{T} .$$

(16)

Squaring both parts (16), we get

$$t^{2}_{p,N - 2} \cdot \frac{{S^{2} \left( \varepsilon \right)}}{N} \cdot \left( {1 + \frac{{(\tau_{N - 1} - \overline{\tau })^{2} }}{{S_{\tau }^{2} }}} \right) \le \hat{\Delta }_{T}^{2} ,$$

(17)

where

$$S_{\tau }^{2} = \frac{1}{N}\mathop \sum \limits_{n = 0}^{N - 1} (\tau_{n} - \overline{\tau })^{2} .$$

(18)

Given that the inequality $\tau_{\Gamma } > \tau_{N - 1}$ is satisfied for the expiration date from (17), we obtain

$$t^{2}_{p,N - 2} \cdot \frac{{S^{2} \left( \varepsilon \right)}}{N} \cdot \left( {1 + \frac{{(\tau_{N - 1} - \overline{\tau })^{2} }}{{S_{\tau }^{2} }}} \right) < \hat{\Delta }_{T}^{2} .$$

(19)

If measurements are made at regular intervals, that is

$$\tau_{n} = \Delta \tau \cdot n, n = \underline{0,N - 1} ,$$

(20)

then the values $\overline{\tau }$ and $S_{\tau }^{2}$ can be calculated analytically

$$\overline{\tau } = \frac{1}{N}\mathop \sum \limits_{n = 1}^{N - 1} \tau_{n} = \Delta \tau \cdot \frac{1}{N}\mathop \sum \limits_{n = 1}^{N - 1} n = \Delta \tau \cdot \frac{N - 1}{2}.$$

(21)

It follows that

$$\overline{\tau } = \Delta \tau \cdot \frac{N - 1}{2}\quad {\text{and}}\quad \tau_{N - 1} - \overline{\tau } = \Delta \tau \cdot \frac{N + 1}{2}.$$

(22)

For the value (18), we obtain

$$S_{\tau }^{2} = \frac{1}{N}\mathop \sum \limits_{n = 0}^{N - 1} (\tau_{n} - \overline{\tau })^{2} = \frac{1}{N} \cdot \mathop \sum \limits_{n = 1}^{N - 1} \tau_{n}^{2} - \overline{\tau }^{2} = (\Delta \tau )^{2} \cdot \frac{1}{N} \cdot \mathop \sum \limits_{n = 1}^{N - 1} n^{2} - (\Delta \tau )^{2} \cdot \left( {\frac{N - 1}{2}} \right)^{2} .$$

(23)

Taking into account the formula for summing squares of natural numbers [10], we obtain

$$\frac{1}{N} \cdot \mathop \sum \limits_{n = 1}^{N - 1} n^{2} = \frac{{\left( {N - 1} \right) \cdot \left( {2 \cdot N - 1} \right)}}{6},$$

(24)

and for magnitude (23)

$$S_{\tau }^{2} = (\Delta \tau )^{2} \cdot \frac{{N^{2} - 1}}{12}.$$

(25)

Taking into account formulas (22) and (25), we obtain

$$\frac{{(\tau_{N - 1} - \overline{\tau })^{2} }}{{S_{\tau }^{2} }} = 3 \cdot \frac{N - 1}{{N + 1}}.$$

(26)

Thus, for measurements carried out at regular intervals, the inequality (19) takes the form

$$t^{2}_{p,N - 2} \cdot \frac{{S^{2} \left( \varepsilon \right)}}{N} \cdot \left( {1 + 3 \cdot \frac{N - 1}{{N + 1}}} \right) < \hat{\Delta }_{T}^{2} .$$

(27)

It follows from (27) that the number of dimensions N should be chosen as the minimum for which condition (2) is satisfied.

Appendix 2: Quadratic Equation for Determining the RM Period of Validity

It is considered that the maximum allowable error value from instability $\hat{\Delta }_{T}$ is known. Then, during the expiration date, the error from instability should not exceed the maximum allowable value of $\hat{\Delta }_{T} \left( \tau \right) \le \hat{\Delta }_{T}$; as a consequence of the monotonous increase of the function describing the error from RM instability for the moments of time $\tau > \tau_{N - 1}$, the expiration date is defined as the positive root of the quadratic equation relative to $\tau_{\Gamma }$

$$\hat{\Delta }_{T} \left( {\hat{\tau }_{\Gamma } } \right) = \hat{\Delta }_{T} .$$

(28)

Taking into account formulas (3) and (6), Eq. (8) takes the form

$$\left| {\hat{a}} \right| \cdot \hat{\tau }_{\Gamma } + t_{p,N - 2} \cdot S\left( \varepsilon \right) \cdot \sqrt {\frac{1}{N} + \frac{{(\hat{\tau }_{\Gamma } - \overline{\tau })^{2} }}{{\mathop \sum \nolimits_{n = 0}^{N - 1} (\tau_{n} - \overline{\tau })^{2} }}} = \hat{\Delta }_{T} .$$

(29)

By transferring the first of the terms from the left side to the right and squaring both parts of the equation, the following quadratic equation is obtained

$$\frac{1}{N} + \frac{{(\hat{\tau }_{\Gamma } - \overline{\tau })^{2} }}{{\mathop \sum \nolimits_{n = 0}^{N - 1} (\tau_{n} - \overline{\tau })^{2} }} = \frac{{(\hat{\Delta }_{T} - \left| {\hat{a}} \right| \cdot \hat{\tau }_{\Gamma } )^{2} }}{{(t_{p,N - 2} \cdot S\left( \varepsilon \right))^{2} }} ,$$

(30)

then $\tau_{\Gamma }$ can be expressed as

$$\hat{\tau }_{\Gamma } = \frac{q + w}{e} ,$$

(31)

where

$$q = \sqrt {\begin{array}{*{20}l} {S^{2} \left( \varepsilon \right) \cdot \mathop \sum \limits_{{n = 0}}^{{N - 1}} (\tau _{n} - \bar{\tau })^{2} \cdot N \cdot t_{{p,N - 2}}^{2} \cdot \left( {N \cdot \hat{\Delta }_{T} \cdot \left[ {\hat{\Delta }_{T} - 2 \cdot \bar{\tau } \cdot \left| {\hat{a}} \right|} \right]} \right.} \hfill \\ {\left. {~ + \left[ {\left( {\bar{\tau }^{2} \cdot N + \mathop \sum \limits_{{n = 0}}^{{N - 1}} (\tau _{n} - \bar{\tau })^{2} } \right) \cdot \hat{a}^{2} - S^{2} \left( \varepsilon \right) \cdot t_{{p,N - 2}}^{2} } \right]~} \right)} \hfill \\ \end{array} } ,$$

(32)

$$w = S^{2} \left( \varepsilon \right) \cdot \overline{\tau } \cdot N \cdot t_{p,N - 2}^{2} - \mathop \sum \limits_{n = 0}^{N - 1} (\tau_{n} - \overline{\tau })^{2} \cdot N \cdot \left| {\hat{a}} \right| \cdot \hat{\Delta }_{T} ,$$

(33)

$$e = S^{2} \left( \varepsilon \right) \cdot N \cdot t_{p,N - 2}^{2} - \mathop \sum \limits_{n = 0}^{N - 1} (\tau_{n} - \overline{\tau })^{2} \cdot N \cdot \hat{a}^{2} .$$

(34)

Appendix 3

Microsoft® Excel® “Calc.xlsm” file with calculation example is available via: https://uniim.ru/calculations/.