Business mathematics and statistics by G.Reka

statistics

1. BUSINESS MATHEMATICS AND STATISTICS G.REKA ASSISTANT PROFESSOR DEPARTMENT OF COMMERCE JAMAL MOHAMED COLLEGE TRICHY - 20 INDIA

2. UNIT-I INTRODUCTION  The word ‘statistics’ is derived from the Latin word ‘status' or Italian word‘ statista ' or the German word ‘ statistik ’ or French word ‘ statisque ’ each of which means a political state. Statistics is not a new discipline but is as old as the human activity itself. DEFINITION  A.L. Bowley defines "Statistics are numerical statements of facts in any department of enquiry placed in relation to each other“

3. Characteristics or Features of Statistics Statistics are Aggregate of Facts Only data related to facts are termed as statistics. This data should be plural one. Single or unrelated figures are not statistics. Since single figure could not be compared, it cannot be termed as statistics Statistics are Affected to a Marked Extent of Multplicity of causes Only those facts , which are consequential to multiple causation are statistics. Statistics must be Numerically Expressed Statistics is the study of only those facts which are being capable of being stated in numbers or quantity .

4. Statistics must be Enumerated or Estimated According to Reasonable Standard of Accuracy Enumeration means collecting data by actual counting or measurement complete accuracy may not be required on all occasion hence , the data may have to be estimated. This estimation should be done with the reasonable standard of accuracy. Statistics should be Collected in a Systematic Manner The data collected would be a systematic one. Proper arrangement in a systematic manner would minimize the time, as well as provide accurate information.

5. SCOPE OF STATISTICS OR APPLICATION OF STATISTICS Statistics and Business Statistics is most commonly used in business. The statistical data regarding the demand and supply of products can be collected and analysed to take a decision regarding the new business. Thus it helps to take companies decision regarding whether a company can start a new business. The existing company can also make a comparative study about their performance with the performance of other companies through statistical analysis. Statistics and Economics Some of the uses of statistics in economics are as follows: * Measures of gross national product and input -output analysis have greatly advanced overall economic knowledge and opened up entire new field of study *Financial statistics are basic in the field of money and banking, short term credit, consumer studies finance and public finance. * statistical studies of business cycle , long term growth and seasonal fluctuations serve to expand our knowledge of economic instability and to modify older theories.

6. Statistics and Physical Sciences The physical sciences are making increasing use of statistics, especially astronomy, chemistry, geology, meteorology and physics. Statistics and Natural Sciences Statistical techniques have proved to be extremely useful in the study of all natural sciences like biology. medicine, zoology, botany. Statistics and Research Statistics is indispensable in research work. Most of the research findings in various disciplines of knowledge have great importance along with subject of statistics. Statistics and Computer Statistical tools like SPSS package, multiple discriminate analysis, multiple regression analysis will help all the fields of the management with the help of computer.

7. Statistics and Management Most of the managerial decisions are taken with the help of statistics. The data regarding the performance of a company will facilitate to take decision regarding future. Statistical techniques like correlation analysis, regression analysis and time series technique can be used in this regard. Statistical techniques can also be used for the payment of wages to the employees of the company. Statistics in Banking and Finance Statistics are mostly used in banking and finance. In banks, statistical data regarding loan, the customer deposit etc. Are represented in statistical data. Financial institution like industrial development bank of India , state financial corporation of India also use statistics in projecting the future and to solve various statistical problems.

8. DIAGRAMMATIC REPRESENTATION INTRODUCTION Classification refers to grouping of data into homogeneous class and categories. Tabulation is the process of presenting the classified data in tables. Classification and tabulation are applied in order to make the collected data understandable. Many figures may be uninteresting and even confusing So, a better way of representing data is by diagram and graph. DIAGRAM A diagram n is a visual form for presentation of statistical data. Diagrams refer to the various types of devices such as bars, circles, maps, pictorials, cartograms. These devices can take many attractive forms.

9. TYPES OF DIAGRAMS The following are the important types of diagrams: (I)One-dimensional diagrams (II) Two-dimensional diagrams (III) Three-dimensional diagrams (IV) Pictograms and Cartograms (I) One-dimensional Diagrams One-dimensional diagram can also be called as bar diagram. In bar diagrams, only the length is considered. The following are the important types of bar diagram ( i ) Simple Bar diagrams (ii) Sub-divided Bar diagrams (iii) Multiple Bar diagrams (iv) Percentage Bar diagrams (v) Deviation Bar diagrams

10. ( i )Simple Bar Diagrams Simple bar diagram represents only one variable. It gives much importance to one characteristic of the data. The figures like production, sales in factories number of students in a college year after can be represented by such bars. The width of the bar is not given any importance. (ii) Sub-divided Bar Diagrams Sub-divided bars are used to present such data which are to be shown in the parts or which are totals of various sub-division. Each part may explain different characters of the data. For example, the number of students of a college may be divided course-wise.

11. (III)Multiple Bar Diagrams The techniques of simple bar diagrams can be extended to represent two or more sets of inter-related data in one diagram. It supplies information about one phenomenon (iv)Percentage Bar Diagrams In percentage bar diagram, the length of all the bars are equal. Various parts of each bar are converted into percentage. (v)Deviation Bar Diagrams Deviation bar diagrams represent only the difference of (deviations of) figures which is shown in the shape of bars. Bars representing positive differences are shown on one side and those representing negative difference on the other side

12. (II)Two-dimensional Diagrams In one-dimensional diagram, only one dimension using heights (length) is considered. But in two-dimensional diagram, both lengths as well as width are taken into account. It is also called as area diagram or surface diagram. The important types of two-dimension diagram are; ( i ) Rectangles (ii) Squares (iii) Circles (iv) Pie Diagram ( i ) Rectangles In a rectangle diagram, both the dimensions (length and width) of the bars are taken into account. A rectangle is a two-dimension diagram because it is based on the area principle (length and breadth)

13. (ii) Squares Under this method, each bar diagram is represented in the form of squares. First we convert the square root of each value of the variable. Then the value should be represented in the form of bar diagram. iii) Circles Circle diagrams are more attractive and appealing than square diagrams. The area of the circle is directly proportional to its radius. Each value is taken as area of a circle. The radius is found for each circle, by dividing with π (227) and then taking the circle, based upon the radius, circle diagram can be drawn. (iv) Pie Diagram Pie diagram means sub-divided circle diagram. It is a representative of various data on the basis of different segments or sections. It gives a clear idea about the percentage of the component part to the total. The percentage/value of any component part is calculated by applying the following formula, because the angle at the centre of the circle is 360°. It is also known as angular diagram.

14. (III)Three-dimensional Diagrams Three-dimensional diagrams are those in which three dimension breadth and height are taken into account. They are constructed in the form ofcubes , spheres, cylinders and blocks. (IV)Pictograms and Cartograms ( i ) Pictograms Pictograms is the technique of presenting statistical data through appropriate pictures. Pictures are more attractive and appealing to the eye. The number of pictures drawn or the size of the pictures being proportional to the values of the different magnitude to be presented. ii) cartograms in cartograms, statistical facts are presented through maps accompanied by various types of diagrammatic representatives. It is the presentation of data in geographical basis. It is also called as statistical maps.

15. GRAPHICAL REPRESENTATION Frequency distribution related to discrete and continuous series can be well drawn in a graph. A graph is a visual form of presentation. Graphical presentation of statistical data gives a pictorial effect. Graphs are very useful for studying time series. Graphs are drawn on a special type of paper known as graph sheet. The special feature of the graphs is that they are more obvious, accurate and precise diagram. Classification of Graphs It is classified into two major heads: ( i ) Graphs of frequency distribution (a) Histogram (b) Frequency Polygon (c) Frequency Curve (d) Ogives or cumulative frequency curve (ii) Graphs of time series (a) Nature Scale Method ( i ) Line Graph or Line Chart for one variable (ii) Line Graph or Line chart for two or more variables (b) Ratio Scale Method

16. ( i ) Graphs of Frequency Distribution (a) Histogram It is one of the major popular and commonly used devices for drawing continuous frequency distribution. It is a set of vertical bars. Frequencies representing the variables should be drawn in a graph in the form of vertical bars. It is also called as graphs of time series (b) Frequency Polygon It is another device of distribution. It gives a curve instead of bars. It is an improved method of histogram. It is drawn for both discrete series and continuous series. In case of discrete frequency distribution, frequency polygon is obtained by plotting the frequencies on the Y axis against the corresponding values of the variable on the X axis and joining the points so obtained by straight lines. c) Frequency Curve A frequency curve is a smooth, free hand curve drawn through the vertices of a frequency polygon. The object of this curve is to eliminate the erratic ups and downs. For drawing the frequency curve, first of draw a frequency polygon by joining mid-points of each class interval. Then the frequency polygon should be smoothed.

17. (d) Ogives or Cumulative Frequency Curve Ogives , pronounced as olive, is the chart. a graphical presentation of the cumulative frequency of continuous series. It is drawn by connecting plots of the cumulative frequency and the class intervals Ogives can be constructed in two methods: ( i ) less than ogives (ii) more than ogives . ( i ) Less than Ogives This consists in plotting the 'less than' cumulative frequencies against the upper class boundaries of the respective classes. The points so obtained are joined by a smooth, free hand curve to give less than ogive . This curve is an increasing curve, sloping upwards from left to right. (ii) More than Ogives Similarly, more than ogives are plotted against the lower class boundaries of the respective classes. The points so obtained are joined by a smooth free hand curve to give 'more than olives'. It is a decreasing curve and slope downwards from left to right.

18. (ii)Graphs of Time Series Time series is concerned with the representation of data for different periods of time. Time (year, month and day) may be takes in x axis and variables (population, demand, production) may be taken in y axis. There are two methods for constructing graphs for Time series. They are Nature Scale Method Ratio Scale Method

19. CLASSIFICATION AND TABULATION INTRODUCTION During every statistical investigation, the collected data, also known as raw data or ungrouped data, are always in an unorganised form and need to be organised and presented in meaningful form in order to facilitate further statistical analysis. The first step in the analysis and interpretation of data is classification and tabulation. Classification means arranging the data into different groups on the basis of their similarities. The next step is tabulation which is concerned with the systematic arrangement and presentation of classified data. DEFINITION Classification is the process of arranging the collected data into classes and sub-classes according to their common characteristics. It can be defined as follows: Classification is the process of arranging things (either actually or notionally) in the groups according to their resemblances and affinities and given expression to the unity of attributes that may subsist amongst a diversity of individuals. -Prof. Cornor Classification is the process of arranging data into sequences and groups according to their common characteristics or separating them into different but related parts. _ Secrist

20. TYPES OF CLASSIFICATION The collected data are classified on the basis of the purpose and objectives of the investigation or enquiry. Generaly , the data can be classified on the basis of the following criteria l. Geographical Classification 2. Chronological Classification 3. Qualitative Classification 4. Quantitative Classification 5. Conditional Classification 1 Geographical Classification It is also known as spatial classification. Here the data are classified on the basis of geographical or vocational differences such as state, cities, districts zones or villages between various items of the data set.   2 Chronological Classification When data are classified on the basis of differences in time such as years, months, weeks, days, hours etc., the classification is known as chronological classification.

21. 3. Qualitative Classification When data are classified according to some qualitative phenomena like honesty. employment, intelligence, literacy, beauty, caste, etc., the classification is termed itative or descriptive or by attributes. Here the data are classified according to the presence or absence of the attributes. (a) Simple Classification When classification is done with respect to one attribute, two classes are formed. One possessing the attribute and the other not possessing the attribute. This type of classification is called simple or dichotomous classification. (b) Manifold Classification Here, classification is done simultaneously with for example, sex and literacy. The population is first classified with respect to ‘sex, into 'males, and females. Each of these classes may further be classified into 'literate' and 'illiterate'. This type of classification manifold classification

22. 4. Quantitative Classification If data are classified on the basis of phenomenon which is capable of quantitative measurement like height, weight, income, expenditure, sales, profits etc., is termed as quantitative classification 5 . Conditional Classification When the data are classified according to certain conditions, other than geo- graphical or chronological, it is called a conditional classification.