ON THE DISTRIBUTION LAW OF THE ANODIC CURRENT MAGNITUDE IN THE METHOD OF AMALGAM POLAROGRAPHY

B. A. KUBRAK, A. A. KAPLIN, A. N. POKROVSKAYA, V. N. POLYAKOVA

(Presented to the scientific-methodological seminar
of the Department of Physical Chemistry)

Knowledge of the distribution law of the anodic current magnitude in the APN method under repeated measurements is necessary as an initial prerequisite in deriving theoretical relationships for the minimum determinable concentration, resolving power, calculation of confidence intervals in processing analytical results, solution of certain physicochemical questions, etc.

A feature of the APN method when there is a large number of repeated measurements is the influence of a number of processes occurring over time in the solution and on the electrode surface (for example, adsorption of surfactants on the electrode, leaching of impurities from the vessel walls, etc.) on the magnitude of the mean and the variance. This does not allow one to assert that such a measurement from a relatively large sample (more than 20), especially at long accumulation times, determines an estimate of the mean and variance of the population.

We have assumed that the measurement of 3–5 successive peaks is a sample from a normally distributed population. A sample of 3–5 measurements also corresponds to the analytical practice of the APN method. To test this hypothesis, we used groups of several successive observations made at different times. Combining all observations into one population is possible upon transition to the variable

\[ u=\frac{x_v-x_1}{S_1}. \tag{1} \]

If the measurements of peak heights obey the normal distribution law, then the variable

\[ t=\frac{u\sqrt{n-2}}{\sqrt{\,n-1-u^2\,}} \tag{2} \]

follows Student’s distribution with \((n-2)\) degrees of freedom for sample size \(n\).

To test the hypothesis that the variable \(u\) follows Student’s distribution, we apply the Kolmogorov criterion [1] as follows.

For one measurement from the sample with any permanently fixed number \(v\), the variable is calculated by formula (1) for all samples; i.e., for \(N\) samples, \(N\) values of \(u\) are calculated. From the magnitude \(u\)

\(N\) values of \(t\) are computed by formula (2), and are arranged in increasing order. Then, depending on \(t\) and on the number of degrees of freedom \(f=n-2\), the values of the distribution function are found, with \(F(-t)=1-F(t)\).

The values of \(t\) are recorded in increasing order and each has frequency of occurrence equal to 1; therefore the values of the empirical distribution function \(F_j^*\) are determined from the expression:

\[ F_j^*=\frac{j-\frac{1}{2}}{N}, \tag{3} \]

where

\(j\) is the ordinal number of the value \(t\) from the table, and
\(N\) is the number of samples. The values \(F_j^*\) are entered in the table. Then the differences \(F(t_j)-F_j^*=d_n\) are found, and the larger of the differences \(d_n\) is compared with the quantity

\[ \frac{\lambda}{\sqrt{N}}, \]

where \(\lambda\) is found from the table of values of Kolmogorov’s function for \(N(x)\), equal to the specified probability, usually \(0.90\text{--}0.95\) [1].

If

\[ d_n<\frac{\lambda}{\sqrt{N}}, \]

then it may be concluded that the variable \(t\) with empirical distribution \(F_t^*\) follows Student’s distribution law, and that the samples \(n\) have been taken from a normally distributed population.

In the present work, the normality of the distribution of peak heights of the elements Cu, Pb, Bi, and Cd was tested in the analysis of a number of substances by the AAP method on small samples. At different concentrations of the elements, 10 samples were taken (\(N=10\)), with 5 measurements in each sample (\(n=5\)).

The calculation of \(t\) was performed for the third measurement in each sample.

The values of the empirical distribution \(F_j^*\) are given in Table 1; the columns of the table give the values \(d_n=F(t_j)-F_j^*\), respectively, for the following concentrations of the elements (g/ml):

1. \(C(\mathrm{Cu})=1.6\cdot10^{-7}\)
2. \(C(\mathrm{Cu})=4\cdot10^{-7}\)
3. \(C(\mathrm{Cu})=8\cdot10^{-7}\)
4. \(C(\mathrm{Pb})=1.6\cdot10^{-7}\)
5. \(C(\mathrm{Pb})=4\cdot10^{-7}\)
6. \(C(\mathrm{Pb})=8\cdot10^{-7}\)
7. \(C(\mathrm{Bi})=1\cdot10^{-6}\)
8. \(C(\mathrm{Bi})=4\cdot10^{-6}\)
9. \(C(\mathrm{Bi})=8\cdot10^{-6}\)

10. \(C(\mathrm{Bi})=1\cdot10^{-7}\)

11. \(C(\mathrm{Bi})=1.5\cdot10^{-7}\)

12. \(C(\mathrm{Bi})=2\cdot10^{-7}\)
13. \(C(\mathrm{Bi})=3\cdot10^{-7}\)
14. \(C(\mathrm{Bi})=4.8\cdot10^{-7}\)
15. \(C(\mathrm{Bi})=1.2\cdot10^{-8}\)
16. \(C(\mathrm{Bi})=4.4\cdot10^{-8}\)
17. \(C(\mathrm{Bi})=8\cdot10^{-8}\)
18. \(C(\mathrm{Cd})=1.2\cdot10^{-7}\)
19. \(C(\mathrm{Cd})=4\cdot10^{-7}\)
20. \(C(\mathrm{Cd})=8\cdot10^{-7}\)

As follows from Table 1, the largest difference \(d_{\max}=0.407\) is less than the quantity

\[ d_{\mathrm{cr}}^{0.95}=\frac{\lambda(0.95)}{\sqrt{N}}=\frac{1.36}{\sqrt{N}}\simeq0.429, \]

or

\[ d_{\mathrm{cr}}^{0.90}=\frac{\lambda(0.90)}{\sqrt{N}}=\frac{1.22}{\sqrt{10}}=0.386. \]

On the basis of the calculations performed, it may be concluded that the variable \(t\) follows Student’s distribution law, and that the small samples have been taken from a normally distributed population. The conclusions are not of a general nature and imply additional quantitative measurements for different variants of the method, elements, and concentration ranges.

Table 1

Table 2

Table 3

Conclusions

1. A method has been considered for the quantitative calculation of the agreement between an empirical distribution and the normal law for small samples.

2. Using a series of elements as an example, it has been shown that the hypothesis of a normal distribution of anodic peaks in the APM method is not rejected.

References

1. I. Marinescu, Ch. Moineagu et al. Fundamentals of Mathematical Statistics and Its Application. Moscow, “Statistika,” 1970.