_{i}} of “n” independent determinations of radiative line strength. Each ensemble has a mean “$\overline{x}$” and unbiased standard deviation “s”. The Coefficient of Variation, ${C}_{V}$, is defined as:

_{i}}. Say we take different samples “j” from this population. The central limit theorem holds that the mean of such samples {${\overline{x}}_{\text{j}}$} will have a standard deviation of approximately $s/\sqrt{n}$, the standard deviation of the mean. Here, ${\overline{x}}_{\text{j}}={{\displaystyle \sum}}_{i}{x}_{\text{ij}}/{\text{n}}_{\mathrm{j}}$ is the jth sample mean and ${s}_{\text{j}}^{2}={{\displaystyle \sum}}_{\text{i}}{({x}_{\text{ij}}-{\overline{x}}_{j})}^{2}/({\text{n}}_{\mathrm{j}}-1)$. Therefore, as a matter of nomenclature, we refer to:

_{i}} of size n, ${C}_{V}^{n}$ has the useful limit (with condition):

_{>}is positive.

_{j}independent determinations, and mean ${\overline{x}}_{\text{j}}$, the Coefficient of Variation of the mean is:

_{p}

_{+}to indicate the upper confidence bound of the Coefficient of Variation, and ${U}_{\text{p}+}^{\text{rel}}$ to designate the expanded relative uncertainty of the mean. Such plots can also illuminate systematic trends and outliers.

^{−1}are included in this figure.

_{0}must have asymptotic bounds of one and zero. We have chosen:

_{0}($\overline{x}$) has asymptotic values of one and zero, consistent with Equation (3). This ad hoc fit curve has the equivalent functional form to the cumulative distribution function of the log-normal distribution.

_{j}. In Equation (6), the “$\approx $” symbol is meant to indicate a LS “best-fit” for the two LS fit parameters).

**Figure 1.**The mean line strength $(\overline{S})$ vs. the Coefficient of Variation of the mean, ${C}_{V}^{n}$, for radiative transition rates in Na III for which the energy of the upper level is greater than 415, 000 cm

^{−1}. Seven outliers, three of which are off the scale, were given very small weights. Because only two data sources are used, the ${C}_{V}^{n}$ for each transition is given by Equation (4a).

^{−1}. These levels belong to the same Na III spectrum, but are more widely spaced than the upper levels of the Figure 2 transitions. Thus level “mixing” is less for transitions from lower-lying levels. This results in smaller computational discrepancies between the two methods. If we had not separated out the data for Figure 3, we would have overestimated the relative uncertainty for this data.

**Figure 2.**The ${C}_{V}^{n}$ data points are the same as Figure 1, to which two curves have been added. Seven outliers, three of which are off the scale, were given small weights. Because only two data sources were used, the ${C}_{V}^{n}$ for each transition is given by Equation (4a). For the lower curve, parameters β[LS] and ${\overline{x}}_{1/2}$[LS] in Equation (6) were evaluated by a LS fit, and β[LS] was found to be 4.2 (“slope”). Keeping the same β, ${\overline{x}}_{1/2}$[p] was then adjusted in Equation (7) until 95% of the points (excluding outliers) lie under the upper curve, for which ${\overline{x}}_{1/2}$[p] = 5.1 × 10

^{−5}(“intercept”). Compare to Figure 3.

**Figure 3.**Coefficient of Variation of the mean (${C}_{V}^{n}$) for radiative transitions in Na III in which the energy of the upper level is less than 415,000 cm

^{−1}, in contrast to Figure 1 and Figure 2. One value was weighted as an outlier. Each point is the ${C}_{V}^{n}$ for a different radiative transition. These data were taken from the same sources as in Figure 2, and the scales are the same. Ninety-five percent of the points lie beneath the upper curve, for which β = 13.8 and ${\overline{x}}_{1/2}$ = 2.3 × 10

^{−19}.

## Acknowledgments

## Appendix A:

_{0}($\overline{x}$) prevail (i.e., ${C}_{V}^{n}$ <≈ 0.3). Φ

_{0}has proven robust in fitting a wide range of different data types, including those that span essentially the entire ${C}_{V}^{n}$ range of 0 to 1. Other functional forms may work better for different data sets. The whole procedure could also be performed graphically, without the explicit use of a parametric fitting function.

**Figure A1.**The fitting function Φ

_{0}($\overline{x}$), Equation (5), for the Coefficient of Variation of the mean ${C}_{V}^{n}$ as a function of $\overline{x}/{\overline{x}}_{1/2}$ for seven values of β. The mean is represented by $\overline{x}$ and ${\overline{x}}_{1/2}$ is the value of the mean for which ${C}_{V}^{n}$ = 0.5.

_{0}($\overline{x}$) approaches one, such as:

## Appendix B:

^{−7}, is given by Hastings [8] (slightly modified here):

^{2}),

## Conflicts of Interest

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