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If and arithmetic
* 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.
If elements in the sample space increase arithmetically, when placed in some order, then the median and arithmetic average are equal.
If, for instance, an addition operation was requested, the arithmetic logic unit ( ALU ) will be connected to a set of inputs and a set of outputs.
If the addition operation produces a result too large for the CPU to handle, an arithmetic overflow flag in a flags register may also be set.
If a set of non-identical numbers is subjected to a mean-preserving spread — that is, two or more elements of the set are " spread apart " from each other while leaving the arithmetic mean unchanged — then the geometric mean always decreases.
If a set of non-identical numbers is subjected to a mean-preserving spread — that is, two or more elements of the set are " spread apart " from each other while leaving the arithmetic mean unchanged — then the harmonic mean always decreases.
( If neither is the case, then a weighted harmonic mean or weighted arithmetic mean is needed.
If these ratios are averaged using an arithmetic mean ( a common error ), high data points are given greater weights than low data points.
In arithmetic and number theory, the least common multiple ( also called the lowest common multiple or smallest common multiple ) of two integers a and b, usually denoted by LCM ( a, b ), is the smallest positive integer that is divisible by both a and b. If either a or b is 0, LCM ( a, b ) is defined to be zero.
If the data set were based on a series of observations obtained by sampling from a statistical population, the arithmetic mean is termed the " sample mean " () to distinguish it from the " population mean " ( or < sub > x </ sub >).
An important example is the ring Z / nZ of integers modulo n. If n is written as a product of prime powers ( see fundamental theorem of arithmetic ):
If the Commercial Instruction Set option was installed, packed decimal arithmetic could be performed as memory-to-memory with some memory-to-register operations.
If all the weights are equal, then the weighted mean is the same as the arithmetic mean.
Referring to Gödel's incompleteness theory, he said: " If we cannot prove the consistency of arithmetic it seems a bit much to hope that God's existence is easier to deal with ," concluding that God is " ontologically necessary, but not logically necessary.
If n is non-zero, and is neither 1 nor − 1, by the fundamental theorem of arithmetic, there exist primes p < sub > i </ sub >, and positive integers e < sub > i </ sub >, with.
If n numbers are given, each number denoted by a < sub > i </ sub >, where i = 1, ..., n, the arithmetic mean is the of the a < sub > i </ sub >' s divided by n or
If exactly one value is left, it is the median ; if two values, the median is the arithmetic mean of these two.
If the matrix being factorized is positive definite as required, the numbers under the square roots are always positive in exact arithmetic.
If absolute indexes are used in the generated ( or manually created ) unwound code, rather than variables, the code created may actually be able to avoid generated pointer arithmetic instructions altogether, using offsets instead ).
If the initial term of an arithmetic progression is and the common difference of successive members is d, then the nth term of the sequence () is given by:
The behavior of the arithmetic progression depends on the common difference d. If the common difference is:
If ƒ and g are two arithmetic functions ( i. e. functions from the positive integers to the complex numbers ), one defines a new arithmetic function ƒ * g, the Dirichlet convolution of ƒ and g, by

If and mean
If he sees the heroic in a Sartoris or a Sutpen, he sees also -- and he shows -- the blind and the mean, and he sees the Compson family disintegrating from within.
If that's done, the house can be designed and oriented for best operation, and this can mean savings both in the size of equipment and in the cost of the house itself.
`` If I have seemed to emphasize the structure of the composition, I mean to project equal concern for color.
If we did not mean to say this, why should we be so relieved on finding that the suffering had not occurred??
If the list is a statistical population, then the mean of that population is called a population mean.
If the list is a statistical sample, we call the resulting statistic a sample mean.
* If numbers have mean X, then.
In the biography, Paul Allen wrote, regarding a suggestion in Cosmopolitan that his plays were becoming autobiographical: " If we take that to mean that his plays tell his own life story, he still hasn't started.
If are independent and identically distributed random variables, each with a standard Cauchy distribution, then the sample mean has the same standard Cauchy distribution ( the sample median, which is not affected by extreme values, can be used as a measure of central tendency ).
If a probability distribution has a density function f ( x ), then the mean is
If liberty is taken to mean " the ability to exercise one's rights as provided for by the law and nature " then this is true, but if it means " the state when one is not held to nor required to perform anything against their will " then this is clearly false.
If true this would mean John's chronology is much closer to Synoptic chronology than often assumed.
If p is a non-zero real number, we can define the generalized mean with exponent p ( or power mean with exponent p ) of the positive real numbers as:
If " hate speech " is taken to mean ethnic agitation, it is prohibited in Finland and defined in the section 11 of the penal code, War crimes and crimes against humanity, as publishing data, an opinion or other statement that threatens or insults a group on basis of race, nationality, ethnicity, religion or conviction, sexual orientation, disability, or any comparable basis.
If a set of weights, ..., is associated to the dataset, ...,, the weighted harmonic mean is defined by

If and was
If, when this was all over, she found the words to tell him about it, she wondered if he would ever understand.
If you don't leave this country within 3 days, your life will be taken the same as Powell's was.
If we was both armed, you wouldn't talk so tough ''.
If it were the enemy, tactically his position was correct.
If the turn was too tight, a barrel roll would bring them out.
If it were not that I knew who it was I could have mistaken it for my Aunt so well did her clothes fit him.
If Franklin was an authentic genius, then Alexander Hamilton, with his exceptional precocity, consuming energy, and high ambition, was a political prodigy.
`` If you can conveniently let me have twenty dollars '', he wrote one friend in 1791 when he was Secretary of the Treasury.
If an automobile were approaching him, he would know what was required of him, even though he might not be able to act quickly enough.
If living Jews were unavailable for study, the Bible was at hand.
If there was ever a thought in her mind she might devote her life to religion, it was now dispelled.
If his scholarship and formal musicianship were not all they might have been, Mercer demonstrated at an early age that he was gifted with a remarkable ear for rhythm and dialect.
If she were not at home, Mama would see to it that a fresh white rose was there.
If, as Reid says, `` nearly all his poetry was produced when he was not taking opium '', there may be some reason to doubt that he was under its influence in the period from 1896 to 1900 when he was writing the poems to Katie King and making plans for another book of verse.
If Robinson was a liar and a slanderer, he was also a very canny gentleman, for nothing that Pike could do would pry so much as a single word out of him.
If his circumspection in regard to Philip's sensibilities went so far that he even refused to grant a dispensation for the marriage of Amadee's daughter, Agnes, to the son of the dauphin of Vienne -- a truly peacemaking move according to thirteenth-century ideas, for Savoy and Dauphine were as usual fighting on opposite sides -- for fear that he might seem to be favoring the anti-French coalition, he would certainly never take the far more drastic step of ordering the return of Gascony to Edward, even though, as he admitted to the English ambassadors, he had been advised that the original cession was invalid.
If the historian was convinced of his own correctness, then he should not allow his vision to become fogged by disturbing facts.
If their schedules were to synchronize, there was no point in wasting time.
If he had any worries, it was only the small ones, about Mother in New York, and his daughter Edwina and what she might be doing at this hour, with her Aunt Asia, in Philadelphia.

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