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Both the "compatibility" function STDEVP and the "consistency" function STDEV.P in Excel 2010 return the 0.5 population standard deviation for the given set of values. However, numerical inaccuracy still can be shown using this example by extending the existing figure to include 10 15 , whereupon the erroneous standard deviation found by Excel ...
Under zero-based numbering, the initial element is sometimes termed the zeroth element, [1] rather than the first element; zeroth is a coined ordinal number corresponding to the number zero. In some cases, an object or value that does not (originally) belong to a given sequence, but which could be naturally placed before its initial element ...
The first ordinal number that is not a natural number is expressed as ω; this is also the ordinal number of the set of natural numbers itself. The least ordinal of cardinality ℵ 0 (that is, the initial ordinal of ℵ 0) is ω but many well-ordered sets with cardinal number ℵ 0 have an ordinal number greater than ω.
Including 0, the set has a semiring structure (0 being the additive identity), known as the probability semiring; taking logarithms (with a choice of base giving a logarithmic unit) gives an isomorphism with the log semiring (with 0 corresponding to ), and its units (the finite numbers, excluding ) correspond to the positive real numbers.
By using S as the set of all functions from A to B, and defining, for each i in B, the property P i as "the function misses the element i in B" (i is not in the image of the function), the principle of inclusion–exclusion gives the number of onto functions between A and B as: [14]
2008 Excel 12.0 (part of Office 2008) 2010 Excel 14.0 (part of Office 2011) 2015 Excel 15.0 (part of Office 2016—Office 2016 for Mac brings the Mac version much closer to parity with its Windows cousin, harmonizing many of the reporting and high-level developer functions, while bringing the ribbon and styling into line with its PC counterpart ...
The arrays V and COL_INDEX are of length NNZ, and contain the non-zero values and the column indices of those values respectively; COL_INDEX contains the column in which the corresponding entry V is located. The array ROW_INDEX is of length m + 1 and encodes the index in V and COL_INDEX where the given row starts.
Leading zeros are also present whenever the number of digits is fixed by the technical system (such as in a memory register), but the stored value is not large enough to result in a non-zero most significant digit. [7] The count leading zeros operation efficiently determines the number of leading zero bits in a machine word. [8]