Significant figures values, uncertainty

We deal with two types of numbers in chemistry—exact and measured. Exact values are just that—exact, by definition. There is no uncertainty associated with them. There are exactly 12 items in a dozen and 144 in a gross. Measured values, like the ones you deal with in the lab, have uncertainty associated with them because of the limitations of our measuring instruments. When those measured values are used in calculations, the answer must reflect that combined uncertainty by the number of significant figures that are reported in the final answer. The more significant figures reported, the greater the certainty in the answer. 

The half-life and the abundance (in bold face from [7]) are shown followed by their units "%" symbol in the case of abundance) which are followed by the uncertainty, in italics,in the last significant figures. For example, 8.1 s 10 means 8.1 1,0 s. For some very short-lived nuclei, level widths rather than half-lives are given, There also, the width is followed by unite e.g., eV, keV, or MeV) which are followed by the uncertainty in italics, if known, As stated above when a limit or an approximate value is given it is based on systematics (sy), mostly from [5]. A in this field indicates that T is not known, For 2p and 2e decay only the lowest value of their several limits e.g., for Ov or 2v, etc.) is given,... 

The atomic weights of many elements are not invariant but depend on the origin and treatment of the material.The standard values of Ar(E) and the uncertainties (in parentheses following the last significant figure to which they are attributed) apply to elements of natural terrestrial origin.The footnotes to this table elaborate the types of variation that may occur for individual elements and that may be larger than the listed uncertainties of values of Ar(E). Names of elements with atomic numbers 110 to 116 are provisional. 

According to Eq. (5) the uncertainty in p will depend on the uncertainty in each of five variables however, the value oif Pl is known to five significant figures and its uncertainty may be neglected in comparison to those in the other variables. We may take as reasonable 95% confidence limits for the weighings AWi = AW2 = 0.001 g and AW3 = 0.002 g. The higher value for AW3 includes the possible failure to attain an exact filling of the pycnometer with water. 

It is very important to report the results of analyses in ways that are readily and unambiguously understood. The methods used to determine average values, standard deviations, confidence intervals, uncertainties etc. must be clearly defined. In some application areas, the concept of a reporting limit is useful it is stated that the concentration of the analyte is not less than a particular value. Prudence indicates that this limit should be greater than the limit of detection (see Section 2.3). Use of an appropriate number of significant figures is also critical in reporting results, as this indicates the precision of the measurements. 

When the value of a number is given, the significant figures provide an indication of the uncertainty in the value for example, a value of 2.7 indicates that the number lies between 2.65 and 2.75. Give ranges within which each of the following values lie. 

Another value that can have an unlimited number of significant figures is a conversion factor. There is no uncertainty in the values that make up this conversion factor, such as I m = 1000 mm, because a millimeter is defined as exactly one-thousandth of a meter. 

Compiled from Rossini [4], Kirk-Othmer [12], and Guthrie [13]. The values given are approximate. Uncertainty is indicated by significant figures. Trends seen are realistic. 

Figure 8.2 Electron affinities of the elements in the form of a Periodic Table. The values in parentheses are the uncertainties in the last figure. The other statistics are given with their proper number of significant figures. The dates are those of the determination or selection as the evaluated values [10-17].

When dissolving the substance, the analyst must have a properly calibrated balance. Preferably it should be calibrated just before use. In that case the uncertainty due to weighing will be minimal and negligible compared to the purity uncertainty. If several substances are mixed in the solution, contaminants present in one substance may affect the exact mass of another substance (e.g. isomers of compounds such as dioxins, mixtures of metals, etc.). The certified value of the concentration must be corrected for the purity figure and for the presence of cross-contaminants. The uncertainty of the value, possibly as a 95 Mi confidence interval, and the significance of the uncertainty must be given. 

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