Where x·y, x, y are the average values of the samples; σ(x), σ(y) - standard deviations.
In addition, the linear pair correlation coefficient can be determined through the regression coefficient b: , where σ(x)=S(x), σ(y)=S(y) - standard deviations, b - coefficient before x in the regression equation y= a+bx .
Other formula options:
or 
K xy - correlation moment (covariance coefficient) 
The linear correlation coefficient takes values from –1 to +1 (see Chaddock scale). For example, when analyzing the closeness of the linear correlation between two variables, a paired linear correlation coefficient equal to –1 was obtained. This means that there is an exact inverse linear relationship between the variables.
Geometric meaning of the correlation coefficient: r xy shows how different the slope of two regression lines: y(x) and x(y) is, and how much the results of minimizing deviations in x and y differ. The greater the angle between the lines, the greater r xy.Properties of the correlation coefficient
Instructions. Specify the amount of input data. The resulting solution is saved in a Word file (see Example of finding a regression equation). A solution template is also automatically created in Excel. .
In mathematical descriptions the term " numerical coefficient ", in particular, when working with literal expressions and expressions with variables, it is convenient to use the concept of a numerical coefficient of an expression. In this article we will give a definition of the numerical coefficient of an expression and analyze examples of finding it.
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In N. Ya. Vilenkin’s textbook mathematics for 6th grade the following is given determination of the numerical coefficient of an expression.
Definition.
If a letter expression is the product of one or more letters and one number, then this number is called numerical coefficient of expression.
By the way, the numerical coefficient is often called simply the coefficient.
The voiced definition allows us to give examples of numeric coefficients of expressions. First, consider the product of the number 3 and the letter a of the form 3·a. The number 3 is the numerical coefficient of this expression by definition. Another example: in the product x·y·0.2·x·x·z the only numerical factor is 0.2, which is the numerical coefficient of this expression.
Now let's give a counter example. The number 3 is not a numerical coefficient of the expression 3·x+y, since the original expression is not a product. But this number 3 is the numerical coefficient of the first of the terms in the original expression.
And the product 5·a·2·b·3·c contains not one, but three numbers. To determine the numerical coefficient of this expression, it must be converted into a product containing a single numerical factor. We will figure out how this is done in the next paragraph of this article; this is the process.
It is worth noting that products of identical letters can be written in the form , so the definition of a numerical coefficient is also suitable for expressions with powers. For example, the expression 5 x 3 y z 2 is essentially an expression of the form 5 x x x x y z z, its coefficient, by definition, is the number 5.
You also need to focus on the numerical coefficients 1 and −1. Their peculiarity is that they are almost never written down explicitly. If an expression is a product of several letters (without a numerical factor) and there is a plus sign in front, or there is no sign, then the numerical coefficient of such an expression is considered to be the number 1. If a product of several letters is preceded by a minus sign, then the coefficient of such an expression is considered to be the number −1. For example, the numerical coefficient of the expression a b is equal to one (since a b can be written as 1 a b ), and the numerical coefficient of the expression −x is equal to minus one (since −x is identically equal to the expression (−1) x ) .
Subsequently, the definition of a numerical coefficient is expanded from the product of a number and several letters to the product of one number and several letter expressions. So, for example, in a product the number −5 can be considered a numerical coefficient. Similarly, the number 3 is the coefficient of the expression 3·(1+1/x)·x, and is the coefficient of the expression 
.
When an expression is a product with one numeric factor, that factor is the numeric coefficient. When an expression has a different form, then finding its numerical coefficient implies preliminary performance of some identical transformations, with the help of which the original expression is reduced to a product with one numerical factor.
Example.
Find the numerical coefficient of the expression −4·x·(−2) .
Solution.
Let's group the factors, which are numbers, and then multiply them: −4 x (−2)=((−4) (−2)) x=8 x. Now the required coefficient is clearly visible; it is equal to 8.
COEFFICIENT
COEFFICIENT
in algebra: a constant indicating how many times a term is taken standing nearby with her expression; in physics: a number that measures the strength of a Ph.D. phenomena, for example, elasticity.
Complete dictionary foreign words, which have come into use in the Russian language. - Popov M., 1907 .
COEFFICIENT
in mathematics there is a constant quantity for which. an unknown or variable quantity is multiplied; eg in expressions 2x - the number 2 is k. If there is no coefficient for a variable value, then the coefficient 1 is implied. In physics, k is a number that is used to measure various specific actions of a substance and which is constant for the same substances; eg expansion of bodies - the ratio of the increase in the length or volume of a body from an increase in temperature by 1° to the original length or volume of the body.
Dictionary of foreign words included in the Russian language. - Pavlenkov F., 1907 .
COEFFICIENT
novolatinsk coefficiens, from cum, with, and efficere, to promote. In algebra, a quantity that appears before a quantity and indicates how many times that quantity is taken.
Explanation of 25,000 foreign words that have come into use in the Russian language, with the meaning of their roots. - Mikhelson A.D., 1865 .
ODDS or UPCOMING
(new Latin coefficiens, from cum - with, and efficere - to promote). In algebra, a quantity that appears before a quantity and indicates how many times that quantity is taken.
Dictionary of foreign words included in the Russian language. - Chudinov A.N., 1910 .
Coefficient
(lat. coefficiens (coelfi-cientis) facilitating) mat. a usually constant or known quantity that is a factor of another, usually variable or unknown quantity; k. proportionality - a constant number which, when multiplied by any value of one quantity, gives a product equal to the corresponding value of another quantity proportional to the first; useful action - a value showing what part of the expended energy is converted into useful work; usually expressed as a percentage.
New dictionary foreign words.- by EdwART,, 2009 .
Coefficient
coefficient, m. [ New Latin coefficiens – facilitating]. 1. Numerical factor in an algebraic expression (mat.). || The number by which to multiply something. value (price, size, etc.) in order to obtain the required one under given conditions (special). Set a coefficient for converting old prices to new ones. 2. A number that quantifies something. property physical body(physical). Efficiency factor (ratio of quantity useful work, produced by someone. mechanism, to the amount of energy it absorbs).
Big dictionary foreign words.- Publishing house "IDDK", 2007 .
Coefficient (Ian), A, m. (German Koeffizient lat. coeffîciens (coefficiēntis) facilitating).
1.
mat. Numerical factor in an algebraic expression.
2.
physical A quantity that determines something. property of a physical body or mechanism. TO. useful action(efficiency).
3.
The number by which to multiply something. value to get what you are looking for. You can calculate your salary by multiplying the minimum wage by k. , corresponding to your rank.
4.
decomposition Supplement to wages, compensating for difficult or abnormal working conditions. They are paid north k.
Coefficient- relating to coefficients 1-4, coefficients.
Explanatory dictionary of foreign words by L. P. Krysin. - M: Russian language, 1998 .
In statistics, an indicator expressed as relative values. Reflects: the rate of development of any phenomenon (the so-called dynamics coefficient), the frequency of occurrence of the phenomenon (for example, the birth rate), the relationship of qualitatively different phenomena...
COEFFICIENT, a number by which some unknown quantity is multiplied in an algebraic expression. In the expression 1 + 5x + 2x2, the numbers 5 and 2 are the coefficients of x and x2, respectively. In physics, a coefficient is a number characterizing a certain... ... Scientific and technical encyclopedic dictionary
Component, component, term, multiplier, factor, ratio, proportion, ratio, degree, percentage, indicator, index, parameter, characteristic; efficiency Dictionary of Russian synonyms. coefficient noun, number of synonyms: 9 gross coefficient ... Synonym dictionary
coefficient- a, m. coefficient, n. lat. coefficients, ntis. 1. Mat. A multiplier (numeric or alphabetic) in an algebraic expression. Sl. 18. It should not be left to young men to make notes on algebraic multiplication and elevation of powers. As members... ... Historical Dictionary of Gallicisms of the Russian Language
- (from Latin co together and efficiens producing) a multiplier, usually expressed in numbers. If the product contains one or more variables (or unknown) quantities, then the coefficient of them is also called the product of all constants, including ... Big Encyclopedic Dictionary
Coefficient K1, proposed by V.S. Ivlev (1938) is a trophic coefficient determined by the equation: , where Q1 is the energy of the substance newly formed in the body (growth energy); Q energy of food consumed. Ecological encyclopedic... ... Ecological dictionary
coefficient J- heel deviation coefficient The change in compass deviation for each degree of list of the ship to starboard, if the ship is heading north according to the compass. [GOST R 52682 2006] Topics of navigation, observation, control means Synonyms coefficient... ... Technical Translator's Guide
- (from Latin co together and efficiens producing), a multiplier, usually expressed in numbers. If the product contains one or more variables (or unknowns), then the coefficient for them is also called the product of all constants, in ... Modern encyclopedia
- (coefficient) Numbers or algebraic expressions that define the structure of a mathematical expression or equation. For example, in the equation y = ax2+bx+c, a is the coefficient of x2, b is the coefficient of x, and c is the constant term. Economy. Intelligent... ... Economic dictionary
See Industrial Discovery Efficiency Ratio. Geological Dictionary: in 2 volumes. M.: Nedra. Edited by K. N. Paffengoltz et al. 1978 ... Geological encyclopedia
« Numerical coefficient", or simply " coefficient" is a term that implies the same mathematical concept. It is very simple to understand the meaning of the term, and to find the numerical coefficient on specific example even easier. But first, let's look at the official definition.
According to a mathematics textbook, if an expression consists of one number and several letter symbols multiplied by each other, then this number will be the coefficient of the entire expression. In this case, the number of letters does not matter - the number can be multiplied by one letter, by two or by five at once, it still remains a coefficient.
For example, consider the following expressions:
It must be recalled that the last rule also applies to expressions where numerical designations are not next to each other, but are separated by letters. For example, 2*c*4*a - we can safely rewrite this expression as 2*4*c*a, because when multiplying, it does not matter in what order the factors are. And thus, the coefficient is still found easily and simply - it will be the number “8”.
Don’t get lost if the problem asks you to find the coefficient for literal expression without numbers - for example, y*z. In this case, the number “1” is always used - since the expression from the example can be written as 1*y*z. The coefficient is found in expressions with both positive and negative factors.
The overall coefficient cannot be found if operations other than multiplication are provided. For example, if you take 3*c + a, then the number “3” will be a coefficient for only one of the terms, but not for the entire expression.
The equation of a reaction in chemistry is called the notation chemical process using chemical formulas and mathematical symbols.
This entry is a schema chemical reaction. When the "=" sign appears, it is called an "equation". Let's try to solve it.
There is one atom in calcium, since the coefficient is not worth it. The index is also not written here, which means one. On the right side of the equation, Ca is also one. We don't need to work on calcium.
Let's look at the next element - oxygen. Index 2 indicates that there are 2 oxygen ions. There are no indices on the right side, that is, one particle of oxygen, and on the left there are 2 particles. What are we doing? No additional indexes or fixes in chemical formula You cannot enter it because it is written correctly.
The coefficients are what is written before the smallest part. They have the right to change. For convenience, we do not rewrite the formula itself. On the right side, we multiply one by 2 to get 2 oxygen ions there.
After we set the coefficient, we got 2 calcium atoms. There is only one on the left side. This means that now we must put 2 in front of calcium.
Now let's check the result. If the number of atoms of an element is equal on both sides, then we can put the “equal” sign.
Another clear example: two hydrogens on the left, and after the arrow we also have two hydrogens.
We multiplied the entire formula by 2, and now the amount of hydrogen has changed. We multiply the index by the coefficient, and we get 4. And on the left side there are two hydrogen atoms left. And to get 4, we have to multiply hydrogen by two.
This is the case when the element in one and the other formula is on the same side, up to the arrow.
One sulfur ion on the left, and one ion on the right. Two oxygen particles, plus two more oxygen particles. This means that there are 4 oxygens on the left side. On the right there are 3 oxygens. That is, on one side there is an even number of atoms, and on the other, an odd number. If we multiply the odd number by two times, we get an even number. First we bring it to an even value. To do this, multiply the entire formula after the arrow by two. After multiplication, we get six oxygen ions, and also 2 sulfur atoms. On the left we have one microparticle of sulfur. Now let's equalize it. We put the equations on the left before gray 2.
Called.
This example is more complex because there are more elements of matter.
This is called a neutralization reaction. What needs to be equalized here first:
The conclusion suggests itself is that you need to multiply the entire formula by two.
Now let's see how much sulfur there is. One on the left and right sides. Let's pay attention to oxygen. On the left side we have 6 oxygen atoms. On the other hand - 5. Less on the right, more on the left. An odd number must be brought to an even number. To do this, we multiply the formula of water by 2, that is, from one oxygen atom we make 2.
Now there are already 6 oxygen atoms on the right side. There are also 6 atoms on the left side. Let's check the hydrogen. Two hydrogen atoms and 2 more hydrogen atoms. So there will be four hydrogen atoms on the left side. And on the other side there are also four hydrogen atoms. All elements are equal. We put the equal sign.
Next example.
Here the example is interesting because parentheses appear. They say that if a factor is behind the brackets, then each element in the brackets is multiplied by it. You need to start with nitrogen, since there is less of it than oxygen and hydrogen. On the left there is one nitrogen, and on the right, taking into account the brackets, there are two.
There are two hydrogen atoms on the right, but four are needed. We get out of this by simply multiplying water by two, resulting in four hydrogens. Great, hydrogen equalized. There is oxygen left. Before the reaction there are 8 atoms, after - also 8.
Great, all the elements are equal, we can set “equal”.
Last example.
Next up is barium. It is equalized, you don’t need to touch it. Before the reaction there are two chlorines, after it there is only one. What needs to be done? Place 2 in front of the chlorine after the reaction.
Now, due to the coefficient that was just set, after the reaction we got two sodiums, and before the reaction we also got two. Great, everything else is equalized.
You can also equalize reactions using the electronic balance method. This method has a number of rules by which it can be implemented. The next step is to arrange the oxidation states of all elements in each substance in order to understand where oxidation occurred and where reduction occurred.
