Standard Deviation and its Formula

The standard deviation is one of the important concepts in mathematics and can be very useful and also have business value. This can be used to compare two sets of dat. Data collection is one of the important tasks in any research. To carry out any research one has to collect dat. Once the data is collected they have to be analysed. There are many software tools that can be used to analyse dat. The statistical concepts are very useful in the analysis. One of the important concepts is the deviation and interpreting standard deviation correctly is very important. The interpretation part has to be done correctly and accurately, otherwise huge blunders might be possible in the research. The standard deviation sign is very useful and has to be used carefully and properly. An example of standard deviation can help explain the concept better. The examples always help in explaining the concepts better.

The standard deviation statistics can be very useful. There are many formulae that have to be studied in mathematics. Formulae can help one to solve the problems easily. If the formula is not known it has to be derived. The process of derivation can be cumbersome and also takes time. If the formula is known time can be saved. Time is a very important factor while solving problems. Most of the problems can be solved but the time to solve the problems is important. If more time is taken for solving a problem then the whole purpose of solving the problem is not achieved. The standard deviation calculation formula can be very helpful in finding the deviation. If the formula is not available it can be a tough task. To find it one must know the term variance. If one finds the square root of this term then the deviation can be easily found out. So, this is the basic formula to find the deviation. This is quite simple to use. There are many tough formulae in mathematics. This is one of the simplest and can be easily used.

The concept of mean is also very important to study. Only if the mean is found, this topic can be covered. This basically denotes the distance from the central value. First the central value has to be found out. The concept of median and mode has also to be studied. These two concepts are also very important and can also be used in various statistical analysis processes. There are many formulae in statistics. These have to be studied in order to solve the problems. The task of solving the problems can become simpler if these formulae are learnt thoroughly. The answers obtained must be verified also. If they are not verified then there can be problems in later stages. So, they have to be verified in the first place. This forms one of the important steps in arriving at the final answer. So, one has to be very careful while arriving at the final answer.

System linear equations

An equation of a straight line is called a linear equation. When we work with two or more linear equations it is called system of linear equation, the point of intersection of these lines is the solution. The possible number of solutions of a given system of equations depend on how the lines are, if they intersect at only one point there would be one solution, if the lines are parallel then there would be no solutions and if the line is the same then there is possibility of infinite number of solutions which is a rare case.

We can find the solutions of the given linear equations using one of the following methods, graphing method, substitution method, and elimination method. Even a graphic calculator can be used to find the solution.

An equation of the form a1x1+ a2x2 + a3x3+…….+ anxn=b where x1, x2, x3,……xn are the variables and a1, a2,a3,….,an and b are constants which are either real or complex numbers is called a linear equation. Here ai is the coefficient of the variable xi and b is a constant term in the equation.

Coming to a system of linear equations, they are linear equations which have the same variables. For instance, a linear system of m equations in n variables y1, y2, y3,……yn can be given as, a11y1 + a12y2+a13y3+………+a1nyn= b1;
a21y1 + a22y2+a23y3+………+a2nyn= b2 and so on, am1y1 + am2y2+am3y3+………+amnyn= bm. For any system of linear equations there are three possibilities of solutions, a unique solution, no solutions or infinitely many solutions. If the linear system has at least one solution it is said to be consistent and if it has no solution then it is said to be inconsistent. Example of linear systems of equations in two variables is, y=3x+2; y=5x-10

System of Linear Equation solver, while solving linear-equations with more than two variables graphing method of solving cannot be used so in such cases we use algebraic method of substitution or elimination.  In substitution method first one of the equations is written in the form something like ‘y=….’ Where y is one of the variables, next step would be to replace this ‘y’ value in the other equation and then solve the equations.

This method is repeated if necessary. In Elimination method a stepwise elimination of variables is done till there is only one variable left, the value of this variable is substituted in one of the linear equations to get the value of another variable. The method is repeated to get the final solution.

Proper Subset

As well known, a subset of A, say, set B, will have some or all the elements contained in the set A. But if the latter option is removed, then the subset is a proper set. Therefore the difference between a subset - proper subset is that the latter has at least one element less than the number of elements of the main set.
We can also define proper subset as strict subset, because of the analogy to the inequalities. The symbols ‘<’ and ‘>’ mean strict inequalities whereas the symbols ‘=’ and ‘=’ mean only just inequalities as they allow a situation of equality as well.  
On the other hand if B is the subset of A, then A is called super subset of B and if B is the strict subset of A, then A is called as proper superset of B. The subset proper set differences can be identified when the relations are expressed symbolically.
B ?  A (B is subset of A)  B ?  A (B is strict subset of A)B ?  A (B is subset of A)   B ?  A (B is strict subset of A)
Let us look into some proper subsets examples. The set {1, 2, 3} is a strict subset of the set {1, 2, 3, 4} because the former does not have the number 4 which is present in the latter. This is a simple example just to understand the basic difference. Let us see where we can see the difference more effectively. We know the domain of a logarithmic function y = ln (x) is (0,8),in interval form.
Suppose we introduce a set D as{0,8}. One can immediately recognize that x is a strict subset of D because, ‘x’ cannot assume the value of 0.
Similarly, we know the range of a sine function y = sin (x) is [-1, 1], in interval form. Suppose we introduce a set R as {-1, 1}. One can immediately recognize that x is a not a strict subset of R because, ‘y’ can assume all the value of between -1 and 1, both included.
Another practical example can be given from number system. We know all natural numbers are integers. But the converse is not true. That is, the set of integers contain natural numbers and in additionalso contain 0 and negative of whole numbers.
Therefore, set of natural numbers is a strict subset of set of integers. Same way, set of irrational numbers

Prominent Variables

Variables in math are very prominent. It is a part and parcel in math, especially in the topic of algebra. Let us try to figure out what is variable and in the process let us find the variables definition. It takes place of an unknown quantity.

Suppose we say that I am 10 years older than my brother, then at any point of time my age is (x + 10). Since at any point of time my brother’s age is also not known, I assumed that as x. Therefore it can be described as letter that represents the unknown quantity and any value can be assigned to that depending on the situation and need.

It is a general practice to use the small case letters of latter halves of English letters to denote the unknown quantities. Of course, none of the terms ‘small case’, ‘latter half’ or ‘English letters’ is a strict requirement. Though it is a general convention, there are exceptions in many cases. The unknown measures of angles of triangles are denoted in capital letters, to be consistent with the symbols of the respective vertices. Again in the very same case of geometry, the first halves of English alphabets are used for unknown quantities.

For example, the prominent statement of Pythagorean Theorem is,              a2 + b2 = c2. Also, there are many Greek letters are used for the unknowns, mostly in trigonometry. For example, the measure of angles are mostly expressed as ? (in case of units in radians) and as a (in case of units in degrees).

In general two or more of these are used in expressions or functions. As a simple example let us take the case of a linear function y = f(x), where f(x) = mx + b. This is a case of input and output function where variable(s) x denotes the input quantity and the ‘y denotes the corresponding output quantity.
Now by change of variables and solving for the same on the left we can find the behavior of the inverse of the function f(x). Let us now see in what way the variables and expressions are related. Expressions are mostly parts of functions. The degrees (the powers) of these decide the type of functions.

The function behaviors can be predicted from the degree and the sign of the leading coefficient of the function. For example, a quadratic function which of degree 2 and with a single unknown will always have rising characteristics on both ends for a positive leading coefficient.

The Method to Find the Triple Product

In mathematics there are basic operations like addition, subtraction, multiplication and division. Multiplication can also be known as the product of numbers. The numbers can be natural numbers, whole numbers, integers, and real numbers and so on. So, the product of two numbers is nothing but the multiplication of two numbers.

There are different types of quantities in mathematics. There are some which have magnitude and there are some which can have both a magnitude and direction as well. Different names have been allotted to these numbers. The former ones are known as scalars and the latter ones are known as the vectors.

The difference between the two is only about the direction they possess. The scalars do not have a direction attached to them. So, they can be handled more easily than the vectors or carrier. Since they have a direction attached to them a sign is used to represent the direction. This sign shows the difference in direction of two quantities of similar nature.

There is a concept in mathematics called the triple product. From the name itself it is understood three products or three quantities are involved in the operation. Now this can be carried out between scalar quantities or the carriers. Depending on the type of the quantities present in the operation the name of the product changes.

The triple vector product is nothing but an operation in which two cross-products are used. There are three of these are involved. The cross products between the first one and then the second and third are taken together. The vector triple product proof can be given mathematically. Even it can be represented geometrically. The triple vector product proof is easy to understand and requires the basic understanding of addition and subtraction.

A vector triple product example will explain the concept and make it clearer. The examples always make the concept easy to understand. Even difficult examples can be made easy with the help of examples. In geometry there exists a figure known as the parallelepiped.

When the scalar triple product is found out it helps in finding the volume of this geometrical figure. There is geometrical meaning attached to this type of product. It must be carefully understood. The scalar triple product can have different values. The values have different meanings. If it is found to be zero, then the volume of parallelepiped is found to be zero

Simultaneous equations

Simultaneous Equations: If two unknown values had to be solved at the same time then these type of equations are called Simultaneous Equations. Simultaneous forms of equations are two equations with two unknowns. They are called simultaneous because they must both be solved at the same time.

There are various methods to solve Simultaneous-equations.
Simultaneous Equation solver:
How to Solve Simultaneous Equations: These equations can be solved by  by elimination method,  Solving by substation method  and by squaring method.  These equations can also be solved by graphing.

Solving Simultaneous equations.

In the  elimination   method for solving  equations, two equations are simplified by adding them or subtracting them. This eliminates one of the variables so that the other variable can be found.
To add two equations, add the left hand expressions and right hand expressions separately. Similarly, to subtract two equations, subtract the left hand expressions from each other, and subtract the right hand expressions from each other.

Take for example two equations.  x +y=10 and x-y = 2.
See  if by adding or subtracting one variable can be  eliminated. When we add these two equations  +y and –y gets cancelled.
we get 2x =12.

Hence  x = 6  Now plug in the value of x in any one equation say x + y=10
We get 6+y=10, so y = 10-6 which equals 4.  So the solution is  x =6 and  y=4

To solve by substitution , we take one of the equations and solve for x in terms of y  or solve for y in terms of x  Then this value is substituted in the unknown equation and solved for the next variable.
For example  x = 8-y and  2x – y  =  7.
Substitute 8-y  for x in the  2x – y = 7  and solve for y.
2(8-y) - y=7 . now  distribute it.
16 – 2y -y = 7  which equals 16 – 3y  = 7.
We have -3y = -9  Hence  y = 3 .

To find  x, put  value of  y  in the equation 2x – y =7.  We get  2x -3 = 7   so 2x = 10 and x = 5

To solve by graphing bring the two equations to the   y = mx + b form  where m is the slope and b is the y intercept.  Using this first mark the  y  intercept and the slope points.  Plot the points and draw  a straight line connecting  the dots .  Note the point where the two  lines intersects  one another.  From this point of intersection of the lines we can be found the values of  x and y.

Calculating Percentage Formula Using a Formula

There can always be a loss or profit in business. This concept is well explained in mathematics. A profit is obtained only when a product is sold at a greater price than its original cost. A loss is obtained in a business when the product is sold at a lower price than the cost of the product. This is the basic difference between the terms profit and loss. So, the selling price and the cost price of a product are very important in deciding the profit or loss obtained by selling a product. Various techniques are used to earn profit or avoid loss. Both these processes are same in nature. One can earn profit only by avoiding loss.

Now when a loss is obtained in a particular transaction, percent loss can be calculated from it. This helps in better understanding of the whole situation. The percentage loss can be calculated only after the loss in a transaction is found out. So, first the loss is found out. This is found by subtracting the price at which it is sold to the price it is produced. The price at which it is sold is called the selling price and the price at which it is produced is called the cost price. On using this formula in the case of profit a positive value is obtained and in the case of a loss a negative value is obtained. The negative value indicates that the value obtained is loss.

The percent loss formula can be derived once the pct is found out. To calculate percent loss formula, it is necessary to calculate the loss first in a transaction. Once the loss is calculated it is easy to calculate percentage loss formula with its help. The pct can be calculated only when the price at which it is produced is known. The cost at which a product is produced can be taken as the cost price. First the loss is calculated and then it is divided by the cost price.

For the calculation of the pct the answer obtained is always multiplied by 100. Only then the answer obtained is converted into pct. Similar is the case with the calculation of the gain pct. There can be either a gain or a loss in a transaction. Maximum emphasis is given to obtain profit. Profit is nothing but the gain. Both can be used interchangeably.