E-LESSON-8 , 9 , 10

1 IDOL Institute of Distance and Online Learning ENHANCE YOUR QUALIFICATION, ADVANCE YOUR CAREER. www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

2 BBA/BCM Business Mathematics and Statistics Course Code: BBA/BCOM 102 Semester: First SLM Unit: 8,9,10 E-Lesson: 5 www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

Business Mathematics and 33 Statistics OBJECTIVES INTRODUCTION To make students aware of the basic concepts of In this unit we are going to learn about statistics. statistics and its concept. To develop an understanding of Classification of Under this you will learn and understand the data. types of classification of data. To make students understand the concept of In this unit you will learn the concept of Measures of Central tendency and Dispersion. Measures of Central tendency and Dispersion www.cuidol.in Unit 8,9,10(BBA /BCOM 1021)02) AllINSrTigIThUtTEaOreF DrIeSsTAeNrCvEeAdNwD iOtNhLICNUE L-EIDARONLING

Topics To Be Covered 4  Introduction to Statistics  Classification of data  Types of classification of data  Measures of Central Tendency  Measures of Dispersion www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

5 STATISTICS AND ITS CONCEPTS www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

Meaning of Statistics 6 • Statistics is the study of the collection, analysis, interpretation, presentation, and organization of data. • Statistics can be defined as the collection presentation and interpretation of numerical data.- Croxton and Cowden. www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

 Collection of data : The collection of data is the first step of statistical 7 investigation. It must be collected very carefully. So, the data must be covered, if not the conclusion will not be reliable.  Organization : The data may be obtained either from primary source or the secondary source. If the data is to be obtained from the primary source, then it needs organization. The data are organized by editing, classifying and tabulating them.  Presentation : After the collection and organization of data, they are presented in systematic form such as table, diagram and graphical form.  Analysis : After the collection, organization and presentation of data, the next step is to analyze the data. To analyze the data we use average, correction, regression, time series etc. The statistical tools of analysis depend upon the nature of data.  Interpretation : The last step of a statistical method is the interpretation of the result obtained from the analysis. Interpretation means to draw the valid conclusion. www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

Characteristics of 8 statistics  Aggregate of facts  Numerically expressed  Predetermined purpose  Affected by multiplicity of causes  Collected in systematic manner www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

Scope and Importance 9 of Statistics  In Planning  In Business  In Economics  In Administration  In Research www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

Application of Statistics in 10 Managerial Decision Making 1. Marketing: Statistical analysis are frequently used in providing information for making decision in the field of marketing it is necessary first to find out what can be sold and the to evolve suitable strategy, so that the goods which to the ultimate consumer. A skill full analysis of data on production purchasing power, man power, habits of compotators, habits of consumer, transportation cost should be consider to take any attempt to establish a new market. 2. Production: In the field of production statistical data and method play a very important role. The decision about what to produce? How to produce? When to produce? For whom to produce is based largely on statistical analysis. www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

11 3. Finance: The financial organization discharging their finance function effectively depend very heavily on statistical analysis of peat and tigers. 4. Banking: Banking institute have found if increasingly to establish research department within their organization for the purpose of gathering and analysis information, not only regarding their own business but also regarding general economic situation and every segment of business in which they may have interest. 5. Investment: Statistics greatly assists investors in making clear and valued judgment in his investment decision in selecting securities which are safe and have the best prospects of yielding a good income. www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

12 6. Purchase: the purchase department in discharging their function makes use of statistical data to frame suitable purchase policies such as what to buy? What quantity to buy? What time to buy? Where to buy? Whom to buy? 7. Accounting: statistical data are also employer in accounting particularly in auditing function, the technique of sampling and destination is frequently used. www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

Limitations 13  Involves mathematical models, equations and other mathematical expressions  Based on number of assumptions  Very expensive  Do not take into consideration intangible facts like skill, attitude etc  Only tools for analysis and decision-making.  They are not decisions itself. www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

CLASSIFICATION 14 OF DATA www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

DEFINITION 15 “Classification is the process of arranging things in groups or classes according to their resemblances and affinities and give expressions of the unity attributes that may subsist amongst a diversity individuals”. -Conner www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

Objectives of Classification 16 of Data  To condense the mass of data  To prepare the data for tabulation  To study the relationships  To facilitate comparison www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

Types Of Classifications 17  Geographical classification  Chronological classification  Qualitative classification  Quantitative classification  Alphabetical classification www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

GEOGRAPHICAL (OR 18 SPATIAL) CLASSIFICATION When the data classified according to geographical location or region (like states, cities, regions, zones , areas etc) It is called a geographical classification. For example, the production of food grains in INDIA may be presented state- wise in following manner: www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

State-wise estimates of production of food grains 19 S.NO. Name of states Total food grains 1 (thousands tones) ANDHRA PARDESH 2 1093.90 3 BIHAR 4 KARNATAKA 12899.89 5 1834.78 PUNJAB 21788.20 UTTER PRADESH 41828.30 www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

CHRONOLOICAL 20 CLASSIFICATION  When data are observed over a period of time the type of classification is known as chronological classification (on the basis of its time of occurrence ).  Various the serious such as National income figures , annual output of wheat monthly expenditure of a house hold , daily consumptions of milk, etc. are some examples of chronological classification.  For example, we may present the figures of population (or production , sales,etc.) as follows: www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

Population of India 1941 to 1991 21 S.No. Year Population in crores 1 1941 31.87 2 1951 36.11 3 1961 43.91 4 1971 54.82 5 1981 68.33 www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

Qualitative Classification 22 We may first divide the population in to males and females on the basis of the attribute ‘gender’, each of this class may be further subdivide into ‘literate’ and ‘illiterate’ on the basis of attribute ‘literacy’ further classification can be made on the basis of same other attribute, say, employment. www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

Quantitative 23 Classification Quantitative classification refers to the classification of data according to some characteristics that can be measured, such as height, weight ,income, sales profit, production, etc. For example, the student of a college may be classified according to weight as follows: www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

QUANTITATIVE 24 CLASSIFICATION Weight (kg) No. of Students 40-50 60 50-60 50 60-70 28 70-80 20 80-90 12 Total 170 www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

Alphabetical 25 Classification When the data are arranged according to alphabetical order, it is called alphabetical classification. For example state-wise density of population in India is depicted in an alphabetical order below: www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

Showing the Density of Population in an Alphabetical Order 26 Names of States Density of Population (Per Sq. Km) Andhra Pradesh 157 Assam 150 Bihar 324 Gujarat 136 Haryana 225 62 Himachal Pradesh 548 Kerala wwwww.wcu.ciudiodol.lc.ionm Unit B8,B9A,101(0B2B/AB/BCCMOM101202) All rightAarllerriegshetrsveredsweirtvheCdUw-IDitOhLCU-IDOL

Measures of Central 27 Tendency  Measures of central tendency are statistical measures which describe the position of a distribution  They are also called statistics of location, and are the complement of statistics of dispersion, which provide information concerning the variance or distribution of observations.  In the univariate context, the mean, median and mode are the most commonly used measures of central tendency. www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

Central Tendency Or 28 Average  An average is a single value which represents the whole set of figures and all other individual items concentrate around it.  It is neither the lowest value in the series nor the highest it lies somewhere between these two extremes.  The average represents all the measurements made on a group, and gives a concise description of the group as a whole.  When two are more groups are measured, the central tendency provides the basis of comparison between them. www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

Definition Of Central 29 Tendency  Simpson and Kafka defined it as “ A measure of central tendency is a typical value around which other figures congregate”  Waugh has expressed “An average stand for the whole group of which it forms a part yet represents the whole”. www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

Functions Of Average 30  Brief Description  Helpful in Comparison  Helpful in formulation of policies  Basis of Statistical Analysis  Representation of the Universe www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

CHARACTERISTICS OF A 31 GOOD AVERAGE  Easy to understand  Simplified  Uniquely defined  Represent the whole group or data  Not affected by extreme values  Capable of further algebraic treatment www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

TYPES OF AVERAGES 32 www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

ARITHMETIC MEAN 33  Most popular & widely used measure.  It is defined as the value which is obtained by adding all the items of a series and dividing this total by the number of items.  It is also referred as mean and is denoted as .  Formula: www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

Properties of Arithmetic 34 Mean Property 1 : If all the observations assumed by a variable are constants, say \"k\", then arithmetic mean is also \"k\". Property 2 : The algebraic sum of deviations of a set of observations from their arithmetic mean is zero. That is, for unclassified data, ∑(x - x̄ ) = 0. And for a grouped frequency distribution, ∑f(x - x̄ ) = 0. www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

Properties of Arithmetic 35 Mean Property 3 : Arithmetic mean is affected due to a change of origin and/or scale which implies that if the original variable \"x\" is changed to another variable \"y\" effecting a change of origin, say \"a\" and scale, say \"b\", of \"x\". That is y = a + bx. Then we have, Arithmetic mean of \"y\" = a + bx̄ Property 4 : If there are two groups containing n₁ and n₂ observations x̄ 1 and x̄ 2 are the respective arithmetic means, then the combined arithmetic mean is given by: x̄ = (n1x̄ 1 + n2x̄ 2) / (n1 + n2) www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

Merits and Demerits of 36 Arithmetic Mean MERITS OF MEAN DEMERITS OF MEAN  It is rigidly defined.  It cannot be calculated if any  It is easy to understand & observations are missing. easy to calculate.  It cannot be calculated for the  It is based upon all values data with open end classes. of the given data.  It is affected by extreme values.  It is capable of further  It cannot be located graphically. mathematical treatment.  It is not much affected by  It may be number which is not present in the data. sampling fluctuations. www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

Median- Introduction 37  Median is a central value of the distribution, or the value which divides the distribution in equal parts, each part containing equal number of items. Thus it is the central value of the variable, when the values are arranged in order of magnitude.  Connor has defined as “ The median is that value of the variable which divides the group into two equal parts, one part comprising of all values greater, and the other, all values less than median” www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

Median- Meaning 38  The point or the value which divides the data in to two equal parts., or when the data is arranged in numerical order.  The data must be ranked (sorted in ascending order) first. The median is the number in the middle.  Depending on the data size we define median as: It is the middle value when data size N is odd. It is the mean of the middle two values, when data size N is even.  Ungrouped Frequency Distribution Find the cumulative frequencies for the data. The value of the variable corresponding to which a cumulative frequency is greater than (N+1)/2 for the first time. (Where N is the total number of observations.) www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

Median 39 Grouped Frequency Distribution First obtain the cumulative frequencies for the data. Then mark the class corresponding to which a cumulative frequency is greater than (N)/2 for the first time. (N is the total number of observations.) Then that class is median class. Then median is evaluated by interpolation formula. Median, m = L + [ (N/2 – F) / f ]C L means lower boundary of the median class N means sum of frequencies F means cumulative frequency before the median class. Meaning that the class before the median class what is the frequency f means frequency of the median class C means the size of the median class www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

Quartiles 40  The data can be divided in to four equal parts by three points. These three points are known as quartiles. The quartiles are denoted by Qi , i = 1,2,3  Qi is the value corresponding to (iN/4)th observation after arranging the data in the increasing order.  Formulas: www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

 For grouped data : First obtain the cumulative frequencies for the data. Then 41 mark the class corresponding to which a cumulative frequency is greater than (iN)/4 for the first time. (Where N is total number of observations.). Then that class is Qi class. Then Qi is evaluated by interpolation formula. www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

Merits and Demerits of 42 Median MERITS OF MEDIAN DEMERITS OF MEDIAN  It is rigidly defined.  It is not based upon all values of the given data.  It is easy to understand & easy to calculate.  For larger data size the arrangement of data in the  It is not affected by extreme values. increasing order is difficult process.  Even if extreme values are not known median can be calculated.  It is not capable of further mathematical treatment.  It can be located just by inspection in many cases.  It is insensitive to some changes in the data values.  It can be located graphically. www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

MODE 43  The mode is the most frequent data value.  Mode is the value of the variable which is predominant in the given data series.  In case of discrete frequency distribution, mode is the value corresponding to maximum frequency.  Sometimes there may be no single mode if no one value appears more than any other.  There may also be two modes (bimodal), three modes (trimodal), or more than three modes (multi-modal). www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

MODE 44  For grouped frequency distributions, the modal class is the class with the largest frequency. After identifying modal class mode is evaluated by using interpolated formula. This formula is applicable when classes are of equal width. www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

Merits and Demerits of 45 Mode MERITS OF MODE DEMERITS OF MODE  It is easy to understand & easy to  It is not rigidly defined. calculate.  It is not based upon all values of  It is not affected by extreme values the given data. or sampling fluctuations.  It is not capable of further  Even if extreme values are not mathematical treatment. known mode can be calculated.  It can be located just by inspection in many cases.  It is always present within the data.  It can be located graphically. www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

Geometric Mean 46  A geometric mean is a mean or average which shows the central tendency of a set of numbers by using the product of their values. For a set of n observations, a geometric mean is the nth root of their product. The geometric mean G.M., for a set of numbers x1, x2, … , xn is given as G.M. = (x1. x2 … xn)1⁄n or, G. M. = (π i = n xi) 1⁄n = n√( x1, x2, … , xn). 1  The geometric mean of two numbers, say x, and y is the square root of their product x×y. For three numbers, it will be the cube root of their products i.e., (x y z) 1⁄3. www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

Relation Between 47 Geometric Mean and Logarithms In order to make calculation easy and less time consuming the concept of logarithms is used in the calculation of geometric means. Since, G.M. = (x1. x2 … xn) 1⁄n Taking log on both sides, we have log G.M. = 1⁄n (log ((x1. x2 … xn)) or, log G.M. = 1⁄n (log x1 + log x2 + … + log xn) or, log G.M. = (1⁄n) ∑ i= n log xi 1 or, G.M. = Antilog(1⁄n (∑ i= n log xi)). 1 www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

Merits and Demerits of 48 Geometric Mean MERITS OF GEOMETRIC MEAN DEMERITS OF GEOMETRIC MEAN  It is based upon all values of the  It is not easy to understand & not given data. easy to calculate  It is capable of further mathematical  It is not well defined. treatment.  If anyone data value is zero then GM  It is not much affected by sampling is zero. fluctuations.  It cannot be calculated if any observations are missing.  It cannot be calculated for the data with open end classes.  It is affected by extreme values. www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

MEASURES 49 OF DISPERSION www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL

RANGE (R) 50  It is the simplest measures of dispersion  It is defined as the difference between the largest and smallest values in the series. R=L–S Where: R = Range, L = Largest Value, S = Smallest Value Coefficient of Range = ������ −������ ������+������ www.cuidol.in Unit 8,9,10(BBA /BCOM 102) All right are reserved with CU-IDOL