Weighted arithmetic mean in the context of "List of atomic weights"

In chemistry, the molar mass (M) (sometimes called molecular weight or formula weight, but see related quantities for usage) of a chemical substance (element or compound) is defined as the ratio between the mass (m) and the amount of substance (n, measured in moles) of any sample of the substance: M = m/n. The molar mass is a bulk, not molecular, property of a substance. The molar mass is a weighted average of many instances of the element or compound, which often vary in mass due to the presence of isotopes. Most commonly, the molar mass is computed from the standard atomic weights and is thus a terrestrial average and a function of the relative abundance of the isotopes of the constituent atoms on Earth.

The molecular mass (for molecular compounds) and formula mass (for non-molecular compounds, such as ionic salts) are commonly used as synonyms of molar mass, as the numerical values are identical (for all practical purposes), differing only in units (dalton vs. g/mol or kg/kmol). However, the most authoritative sources define it differently. The difference is that molecular mass is the mass of one specific particle or molecule (a microscopic quantity), while the molar mass is an average over many particles or molecules (a macroscopic quantity).

Relative atomic mass (symbol: A_r; sometimes abbreviated RAM or r.a.m.), also known by the deprecated synonym atomic weight, is a dimensionless physical quantity defined as the ratio of the average mass of atoms of a chemical element in a given sample to the atomic mass constant. The atomic mass constant (symbol: m_u) is defined as being ⁠1/12⁠ of the mass of a carbon-12 atom. Since both quantities in the ratio are masses, the resulting value is dimensionless. These definitions remain valid even after the 2019 revision of the SI.

For a single given sample, the relative atomic mass of a given element is the weighted arithmetic mean of the masses of the individual atoms (including all its isotopes) that are present in the sample. This quantity can vary significantly between samples because the sample's origin (and therefore its radioactive history or diffusion history) may have produced combinations of isotopic abundances in varying ratios. For example, due to a different mixture of stable carbon-12 and carbon-13 isotopes, a sample of elemental carbon from volcanic methane will have a different relative atomic mass than one collected from plant or animal tissues.

In European football, the UEFA coefficients are statistics based in weighted arithmetic means used for ranking and seeding teams in club and international competitions. Introduced in 1979 for men's football tournaments (country rankings only), and after applied in women's football and futsal, the coefficients are calculated by UEFA, who administer football within Europe, and the Asian parts of some transcontinental countries.

The confederation publishes three types of rankings: one analysing a single season, a five-year span, and a ten-year span. For men's competitions, three sets of coefficients are calculated:

Weighted average is a single number or value that best represents a set of data, when each data point is assigned different "weights" or importance. The most common weighted average is the weighted arithmetic mean, which is similar to an ordinary arithmetic mean except some data points contribute more than others. Other cases include the weighted geometric mean and weighted harmonic mean.