Statistics is involved in every physical measurement.The weighted mean appears when a physical quantity is measured by different methods in different laboratories, producing different results. The formula is: The formula is:\r\n (1)\r\nwith - absolute weights and - relative weights.\r\n\r\nA relation exists:\r\n (2)\r\nwith - the individual standard deviations.\r\n\r\nTo , two different dispersions are associated, D1 (internal) and D2 (external).\r\nThe ratio D2/D1 is:\r\n (3)\r\n\r\nIn practice (D2/D1)>1. So D2 is the confident one.\r\nIn principle should produce as great deviations , as the associated are great.\r\nFor equal treatment of these deviations, relative deviations may be considered:\r\n (4)\r\nwhich express the deviations in units .\r\nThe n values from (4) must have same near (equivalent) values.Their arithmetical mean tends to zero.Thus, as in the case of a unique , we may reach the minimum of the expression:\r\n (5)\r\nThe annulation of the derivative for , gives:\r\n (6)\r\n\r\nand finally formulas (1) and (2) are obtained.\r\nThis calculus accepts great even with systematic uncertainties.In practice the theoretical unknown values , are replaced in formulae (1), (2) by the experimental value , which fluctuate. \r\n\r\nA dispersion D3 is obtained, by error propagation, and added to D1, D2.\r\nFirst are calculates:\r\n (7)\r\nThen:\r\n (8)\r\nwith - the relative standard deviation for and is the values of measurements which provided .For (8) it was used the equality:\r\n (9)\r\n\r\n express the fluctuation of the only statistical part of (systematic components do not fluctuate) and so formulae (9) may be applied. Adding D2 with D3, a correct D2c results. To neglect D3, ni must have same values (rarely the cases).\r\n