measure of central tendency

MEASURE OF CENTRAL TENDENCY

Single value in series of observation which indicate the characteristics of observation.

All data/values clustered around it and used to compare between one series to another.

Measures:

         a. Mean  (Arithmetic/ Geometric/Harmonic)

                 b. Median

                 c. Mode

MEDIAN

Median is what divides the scores in the distribution into two equal parts.

Fifty percent(50%) lies below the median value and 50% lies above the median value.

It is also known as the middle score or 50th percentile.

MEDIAN OF UNGROUPED DATA

•Arrange the scores(from lowest to highest or highest to lowest)

•Determine the middle score in a distribution if n is an odd number and get the average of the two middle most scores if n is an even number.

EXAMPLE

•Find the median score of 7 students in an English class.

                               x(score)

                                  19

                                  17

                                  16

                                  15

                                  10

                                   5

                                   2

 MEDIAN OF GROUPED DATA

Formula:

        Median= l + ( (n/2-m)/f)*c

Where l=Lower limit of the median class

             f=Frequency of the median class

             c=Width of the median class

             N=The total frequency

             m=cumulative frequency of the class

                   preceeding the median class

                PROPERTIES OF MEDIAN

•It may not be an actual observation in the data set.

•It can be applied in ordinal level.

•It is not affected by extreme values because median is a positional measure.

  APPLICATION OF MEDIAN

•It is used to measure the distribution of the earning.

•Use to find the players height e.g.  Football players.

•To find the middle age from the classroom students.

•Also used to find the poverty line.

MODE

The mode or the modal score is a score or scores that occurs in the distribution.

It is classified as unimodal,  bimodal, trimodal or multimodal.

CLASSIFICATION OF MODE

•Unimodal is a distribution of scores that consists of only one mode.

•Bimodal is a distribution of scores that consist of two modes.

•Trimodal is a distribution of scores that consist of three modes or multimodal is a distribution of scores that consist of more than two modes.

MODE FOR GROUPED DATA

Formula:

      Mode= l+     (  f-f1  /2f-f1-f2) *c

Where l=lower limit of the modal class

             f=frequency of modal class

             c=class width of the modal class

             f =frequency of the class just preceeding the

                  modal class

             f2=frequency of the class succedding the

                   modal class

PROPERTIES OF MODE

•It can be used when the data are qualitative as well as quantitative.

•It may not be unique.

•It is affected by extreme values.

It may not exist

APPLICATION OF MODE

•It is used to influx of the public transport.

•The number of games succeeded by any team of players.

•The frequency of the need of infants.

•Used to find the number of the mode is also seen in calculation of the wages,  in the patients going to the hospitals, mode of travels, etc.,

CONCLUSION

• A measure of central tendency is a measure tell us where the middle of a bunch of data lies.

•Median is the number of present in the middle when the number in a set of data are arranged in ascending or descending order.

•Mode is the value that occurs most frequently in a set of data.

ALGEBRA IN MATHEMATICS

                           ALGEBRA

Different meanings of "algebra"

The word "algebra" has several related meanings in mathematics, as a single word or with qualifiers.

• As a single word without an article, "algebra" names a broad part of mathematics.
• As a single word with an article or in plural, "an algebra" or "algebras" denotes a specific mathematical structure, whose precise definition depends on the author. Usually the structure has an addition, multiplication, and a scalar multiplication. When some authors use the term "algebra", they make a subset of the following additional assumptions: associative, commutative, unital, and/or finite-dimensional. In universal algebra, the word "algebra" refers to a generalization of the above concept, which allows for n-ary operations.
• With a qualifier, there is the same distinction:
• Without an article, it means a part of algebra, such as linear algebra, elementary algebra (the symbol-manipulation rules taught in elementary courses of mathematics as part of primary and secondary education), or abstract algebra(the study of the algebraic structures for themselves).
• With an article, it means an instance of some abstract structure, like a Lie algebra, an associative algebra, or avertex operator algebra.
• Sometimes both meanings exist for the same qualifier, as in the sentence: Commutative algebra is the study of commutative rings, which are commutative algebras over the integers.

Algebra as a branch of mathematics

Algebra began with computations similar to those of arithmetic, with letters standing for numbers.^[8] This allowed proofs of properties that are true no matter which numbers are involved. For example, in the quadratic equation

${\displaystyle ax^{2}+bx+c=0,}$

${\displaystyle a,b,c}$ can be any numbers whatsoever (except that ${\displaystyle a}$ cannot be ${\displaystyle 0}$), and the quadratic formula can be used to quickly and easily find the values of the unknown quantity ${\displaystyle x}$ which satisfy the equation. That is to say, to find all the solutions of the equation.

Historically, and in current teaching, the study of algebra starts with the solving of equations such as the quadratic equationabove. Then more general questions, such as "does an equation have a solution?", "how many solutions does an equation have?", "what can be said about the nature of the solutions?" are considered. These questions lead to ideas of form, structure and symmetry.^[10] This development permitted algebra to be extended to consider non-numerical objects, such asvectors, matrices, and polynomials. The structural properties of these non-numerical objects were then abstracted to definealgebraic structures such as groups, rings, and fields.

Before the 16th century, mathematics was divided into only two subfields, arithmetic and geometry. Even though some methods, which had been developed much earlier, may be considered nowadays as algebra, the emergence of algebra and, soon thereafter, of infinitesimal calculus as subfields of mathematics only dates from the 16th or 17th century. From the second half of 19th century on, many new fields of mathematics appeared, most of which made use of both arithmetic and geometry, and almost all of which used algebra.

Today, algebra has grown until it includes many branches of mathematics, as can be seen in the Mathematics Subject Classification^[11] where none of the first level areas (two digit entries) is called algebra. Today algebra includes section 08-General algebraic systems, 12-Field theory and polynomials, 13-Commutative algebra, 15-Linear and multilinear algebra;matrix theory, 16-Associative rings and algebras, 17-Nonassociative rings and algebras, 18-Category theory; homological algebra, 19-K-theory and 20-Group theory. Algebra is also used extensively in 11-Number theory and 14-Algebraic geometry.