* If it is required to use a single number X as an estimate for the value of numbers, then the arithmetic mean does this best, in the sense of minimizing the sum of squares ( x < sub > i </ sub > − X )< sup > 2 </ sup > of the residuals.

The fundamental technique is a partitioning of the total sum of squares SS into components related to the effects used in the model.

The number of degrees of freedom DF can be partitioned in a similar way: one of these components ( that for error ) specifies a chi-squared distribution which describes the associated sum of squares, while the same is true for " treatments " if there is no treatment effect.

See also Lack-of-fit sum of squares.

and then computes the sum of the squares of the differences from the mean,

It turns out that a more suitable quantity for updating is the sum of squares of differences from the ( current ) mean,, here denoted:

Image: Pythagorean. svg | Pythagoras ' theorem: The sum of the areas of the two squares on the legs ( a and b ) of a right triangle equals the area of the square on the hypotenuse ( c ).

The celebrated Pythagorean theorem ( book I, proposition 47 ) states that in any right triangle, the area of the square whose side is the hypotenuse ( the side opposite the right angle ) is equal to the sum of the areas of the squares whose sides are the two legs ( the two sides that meet at a right angle ).

A proposition that says: " The product of the sum and the difference of a and b should give us the difference of the squares of a and b " does express a normative proposition, but this normative statement is based on the theoretical statement "( a + b )( a-b )= a²-b² ".

The energy of formation of a molecule containing only single bonds then can be approximated from an electronegativity table, and depends on the constituents and sum of squares of differences of electronegativities of all pairs of bonded atoms.

Mathematical and logical propositions ( e. g. " that the square of the hypotenuse is equal to the sum of the squares of the two sides ") are examples of the first, while propositions involving some contingent observation of the world ( e. g. " the sun rises in the East ") are examples of the second.

Given that estimation is undertaken on the basis of a least squares analysis, estimates of the unknown parameters β < sub > j </ sub > are determined by minimising a sum of squares function

takes a pair of inputs, and and returns the sum of their squares,.

Another, more general, approach is to minimize the sum of squares of the errors defined in the form

An example of a non-multiplicative function is the arithmetic function r < sub > 2 </ sub >( n )-the number of representations of n as a sum of squares of two integers, positive, negative, or zero, where in counting the number of ways, reversal of order is allowed.

While Diophantus is concerned largely with rational solutions, he assumes some results on integer numbers ; in particular, he seems to assume that every integer is the sum of four squares, though he never states as much explicitly.

Every ordered field is a formally real field, i. e., 0 cannot be written as a sum of nonzero squares.

Finite fields and more generally fields of finite charactristic cannot be turned into ordered fields, because in characteristic p, the element-1 can be written as a sum of ( p-1 ) squares 1 < sup > 2 </ sup >.

However, instead of adding to one, the sum of the squares of the coefficient magnitudes,, must equal one.

In quantum computation, on the other hand, allowed operations are unitary matrices, which are effectively rotations ( they preserve that the sum of the squares add up to one, the Euclidean or L2 norm ).

For example, in Z < sup > n </ sup > with Euclidean metric, a sphere of radius r is nonempty only if r < sup > 2 </ sup > can be written as sum of n squares of integers.

In number theory, Waring's problem, proposed in 1770 by Edward Waring, asks whether for every natural number k there exists an associated positive integer s such that every natural number is the sum of at most s k < sup > th </ sup > powers of natural numbers ( for example, every number is the sum of at most 4 squares, or 9 cubes, or 19 fourth powers, etc .).

where is the vector sum of the physical forces applied to the particle and is the absolute acceleration ( that is, acceleration in a stationary frame ) of the particle, given by:

If the two input signals are both sinusoids of specified frequencies f < sub > 1 </ sub > and f < sub > 2 </ sub >, then the output of the mixer will contain two new sinsoids that have the sum f < sub > 1 </ sub > + f < sub > 2 </ sub > frequency and the difference frequency absolute value | f < sub > 1 </ sub >-f < sub > 2 </ sub >|.

In mathematics, an infinite series of numbers is said to converge absolutely ( or to be absolutely convergent ) if the sum of the absolute value of the summand is finite of convergence.

It is usual to represent the state so that the sum of the absolute squares of the amplitudes add up to one:

Since the sum of the absolute squares of the amplitudes must be constant, must be unitary:

The absolute vorticity at a point can also be expressed as the sum of the relative vorticity at that point and the Coriolis parameter at that latitude ( i. e., it is the sum of the Earth's vorticity and the vorticity of the air relative to the Earth ).

The total energy of the Fermi gas at absolute zero is larger than the sum of the single-particle ground states because the Pauli principle implies a sort of interaction or pressure that keeps fermions separated and moving.

The multiplication of two complex numbers can be expressed most easily in polar coordinates – the magnitude or modulus of the product is the product of the two absolute values, or moduli, and the angle or argument of the product is the sum of the two angles, or arguments.

The adder is designed to overflow when the sum of the absolute value of its operands exceeds its capacity ( 2 < sup > N </ sup >− 1 ).

Addition is also commutative, so permuting the terms of a finite sequence does not change its sum ( for infinite summations this property may fail ; see absolute convergence for conditions under which it still holds ).

#* Manhattan distance-the sum of the absolute differences in value for any variable

At any temperature greater than absolute zero, potential energy and kinetic energy constantly converted into one another, but the sum remains constant in an isolated system ( cf.

Taxicab geometry, considered by Hermann Minkowski in the 19th century, is a form of geometry in which the usual distance function or metric of Euclidean geometry is replaced by a new metric in which the distance between two points is the sum of the absolute differences of their coordinates.

Given the rapid growth in absolute value of Γ ( z + k ) when k → ∞, and the fact that the reciprocal of Γ ( z ) is an entire function, the coefficients in the rightmost sum are well-defined, and locally the sum converges uniformly for all complex s and x.

Likewise, the sum of absolute errors ( SAE ) refers to the sum of the absolute values of the residuals, which is minimized in the least absolute deviations approach to regression.

Debts continued to grow in the course of the 1980s to over 40 billion Deutsche Marks owed to western institutions, a sum not astronomical in absolute terms ( the GDR's GDP was perhaps 250 billion DM ) but large in relation to the GDR's capacity to export sufficient goods to the west to provide the hard currency to service these debts.

The order of a given intermodulation product is the sum of the absolute values of the coefficients,

The Bernoulli numbers appear in the Taylor series expansions of the tangent and hyperbolic tangent functions, in formulas for the sum of powers of the first positive integers, in the Euler – Maclaurin formula, and in expressions for certain values of the Riemann zeta function.

American media critic Herbert Schiller wrote: " The concept of cultural imperialism today best describes the sum of the processes by which a society is brought into the modern world system and how its dominating stratum is attracted, pressured, forced, and sometimes bribed into shaping social institutions to correspond to, or even promote, the values and structures of the dominating centre of the system.

The sum of the two values, i. e. the total number of pips, may be referred to as the rank or weight of a tile, and a tile with more pips may be called heavier than a lighter tile with fewer pips.

All future cash flows are estimated and discounted to give their present values ( PVs ) — the sum of all future cash flows, both incoming and outgoing, is the net present value ( NPV ), which is taken as the value or price of the cash flows in question.

: The value of a whole must not be assumed to be the same as the sum of the values of its parts ( Principia, § 18 ).

They might have some value, but when we consider the total value of a consciousness experiencing a beautiful object, it seems to exceed the simple sum of these values ( Principia 18: 2 ).

Note that we have to restrict the sum to ordered values of m < sub > 1 </ sub >, ..., m < sub > N </ sub > to ensure that we do not count each multi-particle state more than once.

Maximize the sum of the values of the items in the knapsack so that the sum of the weights must be less than the knapsack's capacity.

For a data set, the arithmetic mean is equal to the sum of the values divided by the number of values.

The output byte is selected by looking up the values of S ( i ) and S ( j ), adding them together modulo 256, and then looking up the sum in S ; S ( S ( i ) + S ( j )) is used as a byte of the key stream, K. For as many iterations as are needed, the PRGA modifies the state and outputs a byte of the keystream.

In each iteration, the PRGA increments i, looks up the ith element of S, S, and adds that to j, exchanges the values of S and S, and then uses the sum S + S ( modulo 256 ) as an index to fetch a third element of S, ( the keystream value K below ) which is XORed with the next byte of the message to produce the next byte of either ciphertext or plaintext.

In calculating the arithmetic mean of a sample, for example, the algorithm works by summing all the data values observed in the sample then divides this sum by the number of data items.

In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point.

* Summed area table, an algorithm for generating the sum of values in a rectangular subset of a grid

The fitness of the solution is the sum of values of all objects in the knapsack if the representation is valid, or 0 otherwise.

Intuitively, the additivity property says that the probability assigned to the union of two disjoint events by the measure should be the sum of the probabilities of the events, e. g. the value assigned to " 1 or 2 " in a throw of a die should be the sum of the values assigned to " 1 " and " 2 ".

* the sum of all the function values at x is 1, i. e.,.

Sometimes, the requirement is not as strict: the sum of all the function values at a particular point is only required to be positive rather than a fixed number for all points in the space

The sum of all these values will give the standard enthalpy of formation of sodium chloride.

In numerical analysis, a quadrature rule is an approximation of the definite integral of a function, usually stated as a weighted sum of function values at specified points within the domain of integration.

An integer sequence is called a complete sequence if every positive integer can be expressed as a sum of values in the sequence, using each value at most once.