The Stem And Leaf Plots

Stem and Leaf Plot Definition:- The stem and leaf plot is used for the presentation of the quantitative data in the graphical formats. It is similar to the histogram by which the shape of the distribution of the data can be found. It is the useful tool which can be used in exploratory data analysis. This plot was popular during the type writer time. In modern computer the machine language is zero and one.

So this technique of layout of the data is obsolete in modern computers. The stem and leaf display is also called as the stem plot. The stem and leaf displays retain the data to at least two significant digits. This display contains two columns which are separated by a vertical line. The left column contains the stem and the right column contains the leaf of the data.

For example in the nub thirty two the stem is left most digits which are three and the leaf is the rightmost digit which is two is the leaf. In numb ten one is the stem and zero is the leaf of the number. In number twenty nine the leftmost digit is two and rightmost digit is nine. The number two is called the stem and the number nine is called the leaf.

To construct the stem-leaf displays the data numbs are arranged in the ascending orders.  The data value may be rounded to a particular place value that can be used as the leaf. The remaining digits to the left of the rounding digit will be used as the stem. Stem and leaf plots can be used to find the range, median, mean and media. Other statistics parameters can also be calculated with this available data.

Irregular Polygon Definition:- Polygon is the plane figure which is bounded by the finite chain of straight line segments and closed in a loop to form a closed chain or circuit. These straight line segments are known as edges or sides. The junction of the two sides is called the vertex or corner. Polygon means a shape which has many sides and angles.

A regular polygon means the shapes which have many sides of equal lengths and many angles which are equal in measurements. Irregular polygons are those in which length of each side is not equal and the measurement of angles is also not equal.  The polygon in which one or more interior angles are greater than one hundred and eighty degrees is called as concave polygon.

The polygon which has only three sides cannot be concave. The convex polygon has opposite properties to the concave. It means one or more angles of the polygon are less than one hundred and eighty degrees.

A line which is drawn through the concave polygon can intersect it more than two places. It is also possible that some of the diagonals lie outside of the polygons.  In convex polygon all diagonal lie inside the polygon.  The area of the concave polygon can be found by assuming it as other irregular polygon.

Spherical Geometry

Spherical Geometry :- The geometrical symmetry is associated with the two dimensional surface of the sphere. It is not Euclidean. The main application of the spherical geometry is in navigation and astronomy. The plane geometry is associated with the point and lines. On the sphere, points are defined as usual sense. The straight lines are not defined as the usual sense. In spherical geometry angles can be defined between great circles, resulting a spherical trigonometry which is differ from the ordinary trigonometry in many respect; the sum of the interior angles of a triangle exceeds one hundred and eighty degrees.

Let r is the radius of the sphere. The volume of the sphere is 4/3 pi r3 cubic meter.  To find the volume of the sphere we can divide it into number of infinitesimally small circular disk of the thickness dx. The calculation of the volume of the sphere can be done as below. The surface area of the disk is equal to pi r².

Now the volume of the sphere can be found by finding the integral of the area within the limits of minus r to plus r. The formula can be derived more quickly by finding the surface area and then by integrating it within the limit of zero to r. The spherical geometry is not the elliptical geometry but shares with the geometry the property that a line has no parallel through a given point. The real projected plane is closely associated with the spherical geometry.

Midpoint Formula Geometry : - The point which is exactly at the centre of the two points is known as the middle point. This point divides a given line segment between the two equal halves. The middle point is called the midpoint in the geometry. The midpoint formula has been taken from the section formula. We have to find a point which divides a line in a particular ratio in the section formula.

In the section formula a point which divides the line in the m and n ratio is to find out. If the value of the two ratios which are represented by m and n is equal or m= n then we get the mid points at the given line. 
Let AB is a line and P is a point anywhere in between A and B. Let the coordinates of point A are (x₁, y₁) and coordinate of point B are (x₂, y₂). P is the point which lies in between points A and B has the coordinates (x, y). The coordinates on the mid-point of the line segment joining the points (x₁, y₁) and B(x₂, y₂) are given by  {(x₁+x₂)/2  ,(y₁+y₂)/2}. these are obtained by replacing l and m in the above formula. If a point P divides the given line segment  joining the points (x₁, y₁) and B(x₂, y₂) , then the coordinate of point P are given by  {(x₁+kx₂)/(k+1)  ,(y₁+ky₂)/(k+1)} These are obtained by dividing the numerator and denominator in the above expression the  replacing l/m the by k.

Obtuse Angle

                                                                                                          
In this article we shall learn about What is an Obtuse Angle? And also about Obtuse Scalene Triangle. Before studying about obtuse angle let us learn the definition of ‘angle’. In terms of geometry, an angle is defined as a measure between the circular arc and its radius and it’s shape is formed by two rays called as the sides (arms) of the angle and it shares an end point which is common to the two rays i.e., vertex of the angle. The units to measure an angle are degrees in sexagesimal system, radians in circular system and grades in centismal system.

We have various types of angles such like acute angles, right angles, obtuse angles, supplementary angles, straight angle, reflex angles etc. There are few properties which are listed below:

•    The measure of it can be positive or negative.
•    The measure of it can exceed 360°.
•    When we have two angles such as 30° and 390° where 390° = 360° + 30° where it means that the two terminal sides 30° and 390° belongs to same plane. Hence these two angles are called Coterminal angles.
•    It is expressed in terms of degrees and radians.

So far, we have learnt about definition of angle and its properties, now let us define obtuse angle?
It is defined as an angle whose measure will be greater than 90 degrees but less than 180 degrees. If the measure of this is not between 90 degrees to 180 degrees is not considered as an obtuse angle. Even if the measure is exactly 90 degrees is called as a right angle but not an obtuse. In other words, it is defined as the swipe between the quarter and the half rotation of a circle whose measure varies from 90 degrees to 180 degrees.

Now let us see about Obtuse Scalene Triangle. Firstly, Scalene Triangle is defined as a triangle, whose all sides are unequal and all angles are unequal. Where as an obtuse scalene triangle definition is similar to a scalene triangle where one of its angle is greater than 90°. An obtuse scalene triangle has one obtuse angle and two acute angles, where the two acute angles may be equal or unequal. If the two acute angles are equal then it is known as an obtuse isosceles triangle. There are few facts about scalene triangle such as

•    All interior angles are different.
•    The side which is opposite to the smallest angle will be the shortest side.
•    Similarly, the side which is opposite to the largest angle will be the longest side.

In order to find the area of an obtuse triangle the best formula to use is “heron’s formula”.

According to heron’s formula, the area A of a triangle whose sides are a, b, c is as follows:

A = √s(s-a)(s-b)(s-c), where a, b, c are sides of triangle and ‘s’ is the semi perimeter of the triangle. i.e., S = (a + b + c) / 2.

Polynomials

An expression with a single term is called as monomial, with two terms as binomial and with three terms as trinomial. If the number of terms is more, then such expressions are given a general name as polynomials, the word ‘poly’ means ‘many’. So it can also be written as poly-nomial to emphasize the meaning. Therefore, a poly-nomial is an expression that contains a number of terms consisting variables with constant coefficients. The terms of expression are usually arranged in descending order of the variable powers. Since a constant can also be expressed with the variable power as 0, a constant term can also be a part of a poly-nomial.
But as per convention in algebra, the definition of a polynomial includes certain restrictions. A polynomial can be built up with variables using all operations except division. For example, x³ – (2x + 3)/(x) + 7 cannot be called as a polynomial. However, this restriction applies only division by a variable and not for division by any constant, because such divisions can be considered as equivalent to fractional coefficients. The other restriction is that the exponents of the variables can only be non-negative integers.

A polynomial is generally an expression but acts as part both in equations and functions and such equations and functions are named with prefix ‘poly’ in general. In fact we convert a polynomial function to an equation while attempting to find its zeroes. Therefore, it is imperative to know how to factor a polynomial so that the zero product property can be used to determine the solutions. Using the zero product way is the easiest way to find the solutions of the variables.

Polynomial equations with a single variable with degrees 1 and 2 can easily be solved and the latter type is more popularly known as ‘quadratic’ equation. Mostly quadratic equations are possible to solve by factoring but even otherwise it can be done by using the quadratic formula. But equations with higher degrees are not all that easy to solve. But thanks to the great work by the mathematicians, there are ways to do that. Let us see some of the helpful concepts enunciated by the mathematicians.

As per fundamental theorem of algebra, the number of roots (the number of solutions of variables when equated to 0) of a polynomial is same as the degree of the same. This concept guides us to do the complete solution. We must also be aware that in some cases the solutions or some of the solutions may be imaginary. To get an idea on this, Descartes’s rule of sign changes helps. As per this rules we can figure out the number of real solutions, both positive and negative. Subsequently we can figure out the imaginary solutions with the help of fundamental theorem of algebra.

The rational zero theorem helps us to know what are the possible zeroes of a function. By a few trials, we can figure out a few zeroes and can reduce the polynomial to the level of a quadratic. Thereafter, the remaining zeroes can easily be figured out.

In addition, these days there are many websites advertising as ‘Factor Completely Calculator’ to help us in finding the solutions of a polynomial.

Correlations

Zero Order Correlation :- Zero order correlation means there is no correlation between the two quantities. They vary independently.  If classes in one variable are associated by the classes in the other, then the variables are called correlated. The correlations is said to be perfect if the ratio of two variable deviations is constant. Numerical measure of correlation is called co- efficient of correlations. A group of n individuals may be arranged in the order of merit with respect to some characteristics.

The same group would give different order for different characteristics. Consider order corresponding to two characteristics A and B. The correlations between these n pairs of ranks is called rank correlation in characteristics A and B for the group of individuals.  When the variation of the value of one variable becomes the cause of the variation of the value of other variable then it is called the correlation between the two variables. When the variation between two quantities is directly proportional then it is called positive correlations. It means when one variable increases other also increases or when one variable decreases other variable also decreases.

When the variation between two quantities is indirectly proportional then it is called negative correlations. It means when one variable increases other decreases or when one variable decreases other variable increases. For example the production of any quantity is indirectly proportional to the cost of that quantity. When the temperature increases the charge of the electricity bill also increases. This is the example of positive correlations.

On the other hand during the summer season the length of night hours decreases. It means the day hours are increasing and night hours are decreasing. During the cold season the day hours are decreasing whereas the night hours are increasing. There is a negative correlation between the day and the night hours. The reception of the radio signals is indirectly proportional to the distance from the transmitter. The reception near the transmitter is better than the reception at a distance. As the distance between the transmitter and the receiver increases the signal strength in the receiver decreases. It means the signal strength is indirectly proportional to the distance from the transmitter.

This is an example of the negative correlation. Zero order correlation means there is no correlation and all the quantities have their own graph.

Correlation Equation: - Correlation equations are written to find the order of correlations. We can find the perfect positive correlation, negative correlation or zero order correlations.  The numerical measure of correlation is called co- efficient of correlation and is defined as
r = (∑XY)/(n σx σy ) =(∑XY)/√(∑X² ∑Y²), where X and Y are the deviation from the mean positions.

Deviations are used to find the lower and upper bound of that quantity. Let a resistance value is written as 200 ± 5%.

The lower value of the resistance is equal to 190 and upper value is 210 ohm.
σ²x=(∑x²)/n  ,   σ²y=(∑y²)/n

Where, X = deviation from mean, ¯x=x-¯x

Y = deviation from mean, ¯y=y-¯y
σ_(x =  Standard deviation of x series )
σ_(y =  Standard deviation of y series )
n = number of values in two variables

We can also use the direct Method by substituting the value of σ_(x ) and  σ_(y ) in the above mentioned formula.

r = (∑XY)/(n σx σy ) = (∑XY)/√(∑X² ∑Y²)          or (n∑xy- ∑x∑y)/√(n∑x²-(∑x )²4 ×{n∑y²-∑x²}

Where r is the coefficient of correlation which can be used to find the rank of two quantities.