The arithmetic mean of a variable is often denoted by a bar, for example ( read " x bar ") would be the mean of some sample space.

* 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.

model includes terms for the main effects ( x, y, z ) and

In abstract algebra, an algebraically closed field F contains a root for every non-constant polynomial in F, the ring of polynomials in the variable x with coefficients in F.

Since p ( x ) is irreducible, this means that p ( x ) = k ( x − a ), for some k ∈ F

Aspect ratios are mathematically expressed as x: y ( pronounced " x-to-y ") and x × y ( pronounced " x-by-y "), with the latter particularly used for pixel dimensions, such as 640 × 480.

for x in data:

for x in data:

for x in data:

for x in data:

for x in data:

for x in data:

for x, weight in dataWeightPairs: # Alternatively " for x, weight in zip ( data, weights ):"

for x in data:

The usual order relation ≤ on the real numbers is antisymmetric: if for two real numbers x and y both inequalities x ≤ y and y ≤ x hold then x and y must be equal.

The coordinate systems chosen for atomic orbitals are usually spherical coordinates ( r, θ, φ ) in atoms and cartesians ( x, y, z ) in poly-atomic molecules.

In German, words starting with sch-( constituting the German phoneme ) would be intercalated between words with initial sca-and sci-( all incidentally loanwords ) instead of this graphic cluster appearing after the letter s, as though it were a single letter — a lexicographical policy which would be de rigueur in a dictionary of Albanian, i. e. dh -, ë -, gj -, ll -, rr -, th -, xh-and zh-( all representing phonemes and considered separate single letters ) would follow the letters d, e, g, l, n, r, t, x and z respectively.

As an example, the field of real numbers is not algebraically closed, because the polynomial equation x < sup > 2 </ sup > + 1 = 0 has no solution in real numbers, even though all its coefficients ( 1 and 0 ) are real.

for all r ∈ R and x, y ∈ A.

for all x ∈ A.

for all x, y, and z in A.

for all r ∈ R and x ∈ A.

* An example of a non-unitary associative algebra is given by the set of all functions f: R → R whose limit as x nears infinity is zero.

For example, the equation y = x corresponds to the set of all the points on the plane whose x-coordinate and y-coordinate are equal.

For example, the parent function y = 1 / x has a horizontal and a vertical asymptote, and occupies the first and third quadrant, and all of its transformed forms have one horizontal and vertical asymptote, and occupies either the 1st and 3rd or 2nd and 4th quadrant.

For two geometric objects P and Q represented by the relations P ( x, y ) and Q ( x, y ) the intersection is the collection of all points ( x, y ) which are in both relations.

Thus, all the antiderivatives of x < sup > 2 </ sup > can be obtained by changing the value of C in F ( x ) = ( x < sup > 3 </ sup >/ 3 ) + C ; where C is an arbitrary constant known as the constant of integration.

If F is an antiderivative of f, and the function f is defined on some interval, then every other antiderivative G of f differs from F by a constant: there exists a number C such that G ( x ) = F ( x ) + C for all x.

for all x and y in the algebra.

for all x and y.

In linear algebra, a bilinear transformation is a binary function where the sets X, Y, and Z are all vector spaces and the derived functions f < sup > x </ sup > and f < sub > y </ sub > are all linear transformations.

An ideal of the Boolean algebra A is a subset I such that for all x, y in I we have x ∨ y in I and for all a in A we have a ∧ x in I.

The latter notation corresponds to viewing R as the characteristic function on " X " x " Y " for the set of pairs of G.

Set-theoretically, one may represent a binary function as a subset of the Cartesian product X × Y × Z, where ( x, y, z ) belongs to the subset if and only if f ( x, y ) = z.

Conversely, a subset R defines a binary function if and only if, for any x in X and y in Y, there exists a unique z in Z such that ( x, y, z ) belongs to R.

Given any element x of X, there is a function f < sup > x </ sup >, or f ( x ,·), from Y to Z, given by f < sup > x </ sup >( y ) := f ( x, y ).

Similarly, given any element y of Y, there is a function f < sub > y </ sub >, or f (·, y ), from X to Z, given by f < sub > y </ sub >( x ) := f ( x, y ).

f ( x ) for all x in X and f in X ′.

Because F ( x ) is a map from X ′ to K, it is an element of X ′′.

The map F: x → F ( x ) is thus a map X → X ′′.

The mean temperature of the surface was then computed according to the following relation: Af where x is the fraction of the plug area covered by the hot spot.

hence Af and the minimal polynomial is simply x, which says the operator is 0.

Some adaptations of the Latin alphabet are augmented with ligatures, such as æ in Old English and Icelandic and Ȣ in Algonquian ; by borrowings from other alphabets, such as the thorn þ in Old English and Icelandic, which came from the Futhark runes ; and by modifying existing letters, such as the eth ð of Old English and Icelandic, which is a modified d. Other alphabets only use a subset of the Latin alphabet, such as Hawaiian, and Italian, which uses the letters j, k, x, y and w only in foreign words.

Flowers of x Brassolaeliocattleya ' Turanbeat ', a hybrid between the genera Brassavola, Laelia and Cattleya.

For example, the open interval ( 0, 1 ) does not have a least element: if x is in ( 0, 1 ), then so is x / 2, and x / 2 is always strictly smaller than x.

If F is algebraically closed and p ( x ) is an irreducible polynomial of F, then it has some root a and therefore p ( x ) is a multiple of x − a.

A common misunderstanding is that x and y represent actual length and height.

: When comparing the above illustration to the below text, please note that the above x: y aspect ratio values are shown as vertical orientation rectangles to better demonstrate visual differences, whereas the aspect ratio values of the text below are written as rotated horizontal orientation rectangles ( e. g. compare 3: 4 vertical orientation illustration to 4: 3 horizontal orientation text ).

Here, < sub > n </ sub > denotes the sample mean of the first n samples ( x < sub > 1 </ sub >, ..., x < sub > n </ sub >), s < sup > 2 </ sup >< sub > n </ sub > their sample variance, and σ < sup > 2 </ sup >< sub > n </ sub > their population variance.