Meaning and Definition: Quantitative techniques may be defined as those techniques which provide the decision makes a systematic and powerful means of analysis, based on quantitative data. It is a scientific method employed for problem solving and decision making by the management. With the help of quantitative techniques, the decision maker is able to explore policies for attaining the predetermined objectives. In short, quantitative techniques are inevitable in decision-making process.
Classification of Quantitative Techniques: There are different types of quantitative techniques. We can classify them into three categories. They are: 1. Mathematical Quantitative Techniques 2. Statistical Quantitative Techniques 3. Programming Quantitative Techniques Mathematical Quantitative Techcniques: A technique in which quantitative data are used along with the principles of mathematics is known as mathematical quantitative techniques. Mathematical quantitative techniques involve: 1. Permutations and Combinations: Permutation means arrangement of objects in a definite order.
The number of arrangements depends upon the total number of objects and the number of objects taken at a time for arrangement. The number of permutations or arrangements is calculated by using the following formula:= n! n r ! Combination means selection or grouping objects without considering their order. The number of combinations is calculated by using the following formula:= n! n r ! 2. Set Theory:Set theory is a modern mathematical device which solves various types of critical problems. Quantitative Techniques for Business 5 School of Distance Education 3. Matrix Algebra:
Matrix is an orderly arrangement of certain given numbers or symbols in rows and columns. It is a mathematical device of finding out the results of different types of algebraic operations on the basis of the relevant matrices. 4. Determinants: It is a powerful device developed over the matrix algebra. This device is used for finding out values of different variables connected with a number of simultaneous equations. 5. Differentiation: It is a mathematical process of finding out changes in the dependent variable with reference to a small change in the independent variable. 6. Integration: Integration is the reverse process of differentiation. . Differential Equation: It is a mathematical equation which involves the differential coefficients of the dependent variables. Statistical Quantitative Techniques: Statistical techniques are those techniques which are used in conducting the statistical enquiry concerning to certain Phenomenon. They include all the statistical methods beginning from the collection of data till interpretation of those collected data. Statistical techniques involve: 1. Collection of data: One of the important statistical methods is collection of data. There are different methods for collecting primary and secondary data. . Measures of Central tendency, dispersion, skewness and Kurtosis Measures of Central tendency is a method used for finding he average of a series while measures of dispersion used for finding out the variability in a series. Measures of Skewness measures asymmetry of a distribution while measures of Kurtosis measures the flatness of peakedness in a distribution. 3. Correlation and Regression Analysis: Correlation is used to study the degree of relationship among two or more variables. On the other hand, regression technique is used to estimate the value of one variable for a given value of another.
Quantitative Techniques for Business 6 School of Distance Education 4. Index Numbers: Index numbers measure the fluctuations in various Phenomena like price, production etc over a period of time, They are described as economic barometres. 5. Time series Analysis: Analysis of time series helps us to know the effect of factors which are responsible for changes: 6. Interpolation and Extrapolation: Interpolation is the statistical technique of estimating under certain assumptions, the missing figures which may fall within the range of given figures. Extrapolation provides estimated figures outside the range of given data. . Statistical Quality Control Statistical quality control is used for ensuring the quality of items manufactured. The variations in quality because of assignable causes and chance causes can be known with the help of this tool. Different control charts are used in controlling the quality of products. 8. Ratio Analysis: Ratio analysis is used for analyzing financial statements of any business or industrial concerns which help to take appropriate decisions. 9. Probability Theory: Theory of probability provides numerical values of the likely hood of the occurrence of events. 10. Testing of Hypothesis
Testing of hypothesis is an important statistical tool to judge the reliability of inferences drawn on the basis of sample studies. Programming Techniques: Programming techniques are also called operations research techniques. Programming techniques are model building techniques used by decision makers in modern times. Programming techniques involve: 1. Linear Programming: Linear programming technique is used in finding a solution for optimizing a given objective under certain constraints. 2. Queuing Theory: Queuing theory deals with mathematical study of queues. It aims at minimizing cost of both servicing and waiting.
Quantitative Techniques for Business 7 School of Distance Education 3. Game Theory: Game theory is used to determine the optimum strategy in a competitive situation. 4. Decision Theory: This is concerned with making sound decisions under conditions of certainty, risk and uncertainty. 5. Inventory Theory: Inventory theory helps for optimizing the inventory levels. It focuses on minimizing cost associated with holding of inventories. 6. Net work programming: It is a technique of planning, scheduling, controlling, monitoring and co-ordinating large and complex projects comprising of a number of activities and events.
It serves as an instrument in resource allocation and adjustment of time and cost up to the optimum level. It includes CPM, PERT etc. 7. Simulation: It is a technique of testing a model which resembles a real life situations 8. Replacement Theory: It is concerned with the problems of replacement of machines, etc due to their deteriorating efficiency or breakdown. It helps to determine the most economic replacement policy. 9. Non Linear Programming: It is a programming technique which involves finding an optimum solution to a problem in which some or all variables are non-linear. 10.
Sequencing: Sequencing tool is used to determine a sequence in which given jobs should be performed by minimizing the total efforts. 11. Quadratic Programming: Quadratic programming technique is designed to solve certain problems, the objective function of which takes the form of a quadratic equation. 12. Branch and Bound Technique It is a recently developed technique. This is designed to solve the combinational problems of decision making where there are large number of feasible solutions. Problems of plant location, problems of determining minimum cost of production etc. are examples of combinational problems.
Quantitative Techniques for Business 8 School of Distance Education Functions of Quantitative Techniques: The following are the important functions of quantitative techniques: 1. To facilitate the decision-making process 2. To provide tools for scientific research 3. To help in choosing an optimal strategy 4. To enable in proper deployment of resources 5. To help in minimizing costs 6. To help in minimizing the total processing time required for performing a set of jobs USES OF QUANTITATE TECHNIQUES Business and Industry Quantitative techniques render valuable services in the field of business and industry.
Today, all decisions in business and industry are made with the help of quantitative techniques. Some important uses of quantitative techniques in the field of business and industry are given below: 1. Quantitative techniques of linear programming is used for optimal allocation of scarce resources in the problem of determining product mix 2. Inventory control techniques are useful in dividing when and how much items are to be purchase so as to maintain a balance between the cost of holding and cost of ordering the inventory 3.
Quantitative techniques of CPM, and PERT helps in determining the earliest and the latest times for the events and activities of a project. This helps the management in proper deployment of resources. 4. Decision tree analysis and simulation technique help the management in taking the best possible course of action under the conditions of risks and uncertainty. 5. Queuing theory is used to minimize the cost of waiting and servicing of the customers in queues. 6. Replacement theory helps the management in determining the most economic replacement policy regarding replacement of an equipment.
Limitations of Quantitative Techniques: Even though the quantitative techniques are inevitable in decision-making process, they are not free from short comings. The following are the important limitations of quantitative techniques: Quantitative Techniques for Business 9 School of Distance Education 1. Quantitative techniques involves mathematical models, equations and other mathematical expressions 2. Quantitative techniques are based on number of assumptions. Therefore, due care must be ensured while using quantitative techniques, otherwise it will lead to wrong conclusions. 3. Quantitative techniques are very expensive. . Quantitative techniques do not take into consideration intangible facts like skill, attitude etc. 5. Quantitative techniques are only tools for analysis and decision-making. They are not decisions itself. Quantitative Techniques for Business 10 School of Distance Education CHAPTER – 2 CORRELEATION ANALYSIS Introduction: In practice, we may come across with lot of situations which need statistical analysis of either one or more variables. The data concerned with one variable only is called univariate data. For Example: Price, income, demand, production, weight, height marks etc are concerned with one variable only.
The analysis of such data is called univariate analysis. The data concerned with two variables are called bivariate data. For example: rainfall and agriculture; income and consumption; price and demand; height and weight etc. The analysis of these two sets of data is called bivariate analysis. The date concerned with three or more variables are called multivariate date. example: agricultural production is influenced by rainfall, quality of soil, fertilizer etc. For The statistical technique which can be used to study the relationship between two or more variables is called correlation analysis.
Definition: Two or more variables are said to be correlated if the change in one variable results in a corresponding change in the other variable. According to Simpson and Kafka, “Correlation analysis deals with the association between two or more variables”. Lun chou defines, “ Correlation analysis attempts to determine the degree of relationship between variables”. Boddington states that “Whenever some definite connection exists between two or more groups or classes of series of data, there is said to be correlation. In nut shell, correlation analysis is an analysis which helps to determine the degree of relationship exists between two or more variables. Correlation Coefficient: Correlation analysis is actually an attempt to find a numerical value to express the extent of relationship exists between two or more variables. The numerical measurement showing the degree of correlation between two or more variables is called correlation coefficient. Correlation coefficient ranges between -1 and +1. SIGNIFICANCE OF CORRELATION ANALYSIS Correlation analysis is of immense use in practical life because of the following reasons: 1.
Correlation analysis helps us to find a single figure to measure the degree of relationship exists between the variables. 2. Correlation analysis helps to understand the economic behavior. Quantitative Techniques for Business 11 School of Distance Education 3. Correlation analysis enables the business executives to estimate cost, price and other variables. 4. Correlation analysis can be used as a basis for the study of regression. Once we know that two variables are closely related, we can estimate the value of one variable if the value of other is known. 5.
Correlation analysis helps to reduce the range of uncertainty associated with decision making. The prediction based on correlation analysis is always near to reality. 6. It helps to know whether the correlation is significant or not. This is possible by comparing the correlation co-efficient with 6PE. It ‘r’ is more than 6 PE, the correlation is significant. Classification of Correlation Correlation can be classified in different ways. The following are the most important classifications 1. Positive and Negative correlation 2. Simple, partial and multiple correlation 3.
Linear and Non-linear correlation Positive and Negative Correlation Positive Correlation When the variables are varying in the same direction, it is called positive correlation. In other words, if an increase in the value of one variable is accompanied by an increase in the value of other variable or if a decrease in the value of one variable is accompanied by a decree se in the value of other variable, it is called positive correlation. Eg: 1) A: 10 B: 80 2) X: 78 Y: 20 Negative Correlation: When the variables are moving in opposite direction, it is called negative correlation.
In other words, if an increase in the value of one variable is accompanied by a decrease in the value of other variable or if a decrease in the value of one variable is accompanied by an increase in the value of other variable, it is called negative correlation. Eg: 1) A: B: 16 5 10 10 15 8 20 6 25 2 12 20 100 60 18 30 150 52 14 40 170 46 10 50 200 38 5 Quantitative Techniques for Business School of Distance Education 2) X: Y: 40 2 32 3 25 5 20 8 10 12 Simple, Partial and Multiple correlation Simple Correlation In a correlation analysis, if only two variables are studied it is called simple correlation. Eg. he study of the relationship between price & demand, of a product or price and supply of a product is a problem of simple correlation. Multiple correlation In a correlation analysis, if three or more variables are studied simultaneously, it is called multiple correlation. For example, when we study the relationship between the yield of rice with both rainfall and fertilizer together, it is a problem of multiple correlation. Partial correlation In a correlation analysis, we recognize more than two variable, but consider one dependent variable and one independent variable and keeping the other Independent variables as constant.
For example yield of rice is influenced b the amount of rainfall and the amount of fertilizer used. But if we study the correlation between yield of rice and the amount of rainfall by keeping the amount of fertilizers used as constant, it is a problem of partial correlation. Linear and Non-linear correlation Linear Correlation In a correlation analysis, if the ratio of change between the two sets of variables is same, then it is called linear correlation. For example when 10% increase in one variable is accompanied by 10% increase in the other variable, it is the problem of linear correlation.
X: 10 15 Y: 50 75 30 150 60 300 Here the ratio of change between X and Y is the same. When we plot the data in graph paper, all the plotted points would fall on a straight line. Non-linear correlation In a correlation analysis if the amount of change in one variable does not bring the same ratio of change in the other variable, it is called non linear correlation. X: Y: 2 8 4 10 6 18 10 22 15 26 Here the change in the value of X does not being the same proportionate change in the value of Y. Quantitative Techniques for Business 13
School of Distance Education This is the problem of non-linear correlation, when we plot the data on a graph paper, the plotted points would not fall on a straight line. Degrees of correlation: Correlation exists in various degrees 1. Perfect positive correlation If an increase in the value of one variable is followed by the same proportion of increase in other related variable or if a decrease in the value of one variable is followed by the same proportion of decrease in other related variable, it is perfect positive correlation. g: if 10% rise in price of a commodity results in 10% rise in its supply, the correlation is perfectly positive. Similarly, if 5% full in price results in 5% fall in supply, the correlation is perfectly positive. 2. Perfect Negative correlation If an increase in the value of one variable is followed by the same proportion of decrease in other related variable or if a decrease in the value of one variable is followed by the same proportion of increase in other related variably it is Perfect Negative Correlation. For example if 10% rise in price results in 10% fall in its demand the correlation is perfectly negative.
Similarly if 5% fall in price results in 5% increase in demand, the correlation is perfectly negative. 3. Limited Degree of Positive correlation: When an increase in the value of one variable is followed by a non-proportional increase in other related variable, or when a decrease in the value of one variable is followed by a nonproportional decrease in other related variable, it is called limited degree of positive correlation. For example, if 10% rise in price of a commodity results in 5% rise in its supply, it is limited degree of positive correlation.
Similarly if 10% fall in price of a commodity results in 5% fall in its supply, it is limited degree of positive correlation. 4. Limited degree of Negative correlation When an increase in the value of one variable is followed by a non-proportional decrease in other related variable, or when a decrease in the value of one variable is followed by a nonproportional increase in other related variable, it is called limited degree of negative correlation. For example, if 10% rise in price results in 5% fall in its demand, it is limited degree of negative correlation.
Similarly, if 5% fall in price results in 10% increase in demand, it is limited degree of negative correlation. 5. Zero Correlation (Zero Degree correlation) If there is no correlation between variables it is called zero correlation. In other words, if the values of one variable cannot be associated with the values of the other variable, it is zero correlation. Quantitative Techniques for Business 14 School of Distance Education Methods of measuring correlation Correlation between 2 variables can be measured by graphic methods and algebraic methods. I Graphic Methods 1) 2) Scatter Diagram Correlation graph
II Algebraic methods (Mathematical methods or statistical methods or Co-efficient of correlation methods): 1) Karl Pearson’s Co-efficient of correlation 2) Spear mans Rank correlation method 3) Concurrent deviation method Scatter Diagram This is the simplest method for ascertaining the correlation between variables. Under this method all the values of the two variable are plotted in a chart in the form of dots. Therefore, it is also known as dot chart. By observing the scatter of the various dots, we can form an idea that whether the variables are related or not.
A scatter diagram indicates the direction of correlation and tells us how closely the two variables under study are related. The greater the scatter of the dots, the lower is the relationship Y X X X X X X X X Y X X X X X X X X 0 X 0 X Perfect Positive Correlation Perfect Negative Correlation Quantitative Techniques for Business 15 School of Distance Education Y X X X X X X X X X X X X X X X X Y X X X X X X X X X X X X X X X 0 X 0 X High Degree of Positive Correlation High Degree of Negative Correlation Y X X X X X X X X X X X X X X X X X X X X X X X X X X X X Y X X X
X X X X X X X X X X X X X X X X X X X X X X X X X X 0 X 0 X Low Degree of Positive Correlation Low Degree of Negative Correlation Y X X X X X X X X X X X X X X X X X X X X X X X X X X 0 X No Correlation (r = 0) Quantitative Techniques for Business 16 School of Distance Education Merits of Scatter Diagram method 1. It is a simple method of studying correlation between variables. 2. It is a non-mathematical method of studying correlation between the variables. It does not require any mathematical calculations. 3. It is very easy to understand. It gives an idea about the correlation between variables even to a layman. . It is not influenced by the size of extreme items. 5. Making a scatter diagram is, usually, the first step in investigating the relationship between two variables. Demerits of Scatter diagram method 1. It gives only a rough idea about the correlation between variables. 2. The numerical measurement of correlation co-efficient cannot be calculated under this method. 3. It is not possible to establish the exact degree of relationship between the variables. Correlation graph Method Under correlation graph method the individual values of the two variables are plotted on a graph paper.
Then dots relating to these variables are joined separately so as to get two curves. By examining the direction and closeness of the two curves, we can infer whether the variables are related or not. If both the curves are moving in the same direction( either upward or downward) correlation is said to be positive. If the curves are moving in the opposite directions, correlation is said to be negative. Merits of Correlation Graph Method 1. This is a simple method of studying relationship between the variable 2. This does not require mathematical calculations. 3.
This method is very easy to understand Demerits of correlation graph method: 1. A numerical value of correlation cannot be calculated. 2. It is only a pictorial presentation of the relationship between variables. 3. It is not possible to establish the exact degree of relationship between the variables. Karl Pearson’s Co-efficient of Correlation Karl Pearson’s Coefficient of Correlation is the most popular method among the algebraic methods for measuring correlation. This method was developed by Prof. Karl Pearson in 1896. It is also called product moment correlation coefficient. Quantitative Techniques for Business 17
School of Distance Education Pearson’s coefficient of correlation is defined as the ratio of the covariance between X and Y to the product of their standard deviations. This is denoted by ‘r’ or rxy r = Covariance of X and Y (SD of X) x (SD of Y) Interpretation of Co-efficient of Correlation Pearson’s Co-efficient of correlation always lies between +1 and -1. The following general rules will help to interpret the Co-efficient of correlation: 1. When r – +1, It means there is perfect positive relationship between variables. 2. When r = -1, it means there is perfect negative relationship between variables. . When r = 0, it means there is no relationship between the variables. 4. When ‘r’ is closer to +1, it means there is high degree of positive correlation between variables. 5. When ‘r’ is closer to – 1, it means there is high degree of negative correlation between variables. 6. When ‘r’ is closer to ‘O’, it means there is less relationship between variables. Properties of Pearson’s Co-efficient of Correlation 1. If there is correlation between variables, the Co-efficient of correlation lies between +1 and -1. 2. If there is no correlation, the coefficient of correlation is denoted by zero (ie r=0) 3.
It measures the degree and direction of change 4. If simply measures the correlation and does not help to predict cansation. 5. It is the geometric mean of two regression co-efficients. i. e r= · Computation of Pearson’s Co-efficient of correlation: Pearson’s correlation co-efficient can be computed in different ways. They are: a b c Arithmetic mean method Assumed mean method Direct method Arithmetic mean method:Under arithmetic mean method, co-efficient of correlation is calculated by taking actual mean. Quantitative Techniques for Business 18 School of Distance Education r= or r= hereas x-x- and y=y- Calculate Pearson’s co-efficient of correlation between age and playing habits of students: Age: No. of students Regular players 20 500 400 21 400 300 22 300 180 23 240 96 24 200 60 25 160 24 Let X = Age and Y = Percentage of regular players Percentage of regular players can be calculated as follows:400 x 100 = 80; 300 x 100 = 75; 180 x 100 = 60; 96 x 100 = 40 , 500 400 300 240 100 = 30; and 100 15 Pearson’s Coefficient of Correlation (r) = ? ? . . Computation of Pearson’s Coefficient of correlation Age x 20 21 22 23 24 25 135 % of Regular Player y 80 75 60 40 30 15 300 = = = 22. 19 (x-22. 5) -2. 5 -1. 5 -0. 5 0. 5 1. 5 2. 5 (y-50) 30 25 10 -10 -20 -35 ) -75. 0 -37. 5 – 5. 0 – 5. 0 -30. 0 -87. 5 -240 6. 25 2. 25 0. 25 0. 25 2. 25 6. 25 17. 50 900 625 100 100 400 1225 3350 Quantitative Techniques for Business School of Distance Education = r = Assumed mean method: = . = 50 v = v , = v . = -0. 9912 Under assumed mean method, correlation coefficient is calculated by taking assumean only. r= Where dx = deviations of X from its assumed mean; dy= deviations of y from its assumed mean Find out coefficient of correlation between size and defect in quality of shoes:
Size No. of shoes Produced : : 15-16 200 150 16-17 270 162 17-18 340 170 18-19 360 180 19-20 400 180 20-21 300 114 No. of defectives: Let x = size (ie mid-values) y = percentage of defectives x values are 15. 5 , 16. 5, 17. 5, 18. 5, 19. 5 and 20. 5 y values are 75, 60, 50, 50, 45 and 38 Take assumed mean: x = 17. 5 and y = 50 Computation of Pearson’s Coefficient of Correlation x 15. 5 16. 5 17. 5 18. 5 19. 5 20. 5 y 75 60 50 50 45 38 dx -2 -1 0 1 2 3 3 18 Quantitative Techniques for Business dy 25 10 0 0 -5 -12 dxdy -50 -10 0 0 -10 -36 106 x2 4 1 0 1 4 9 19 dy2 625 100 0 0 25 144 894 20 School of Distance Education r= v r= = = Direct Method: = . = -0. 9485 Under direct method, coefficient of correlation is calculated without taking actual mean or assumed mean r= From the following data, compute Pearson’s correlation coefficient: Price : 10 12 41 14 48 15 60 19 50 Demand (Qty) 40 Let us take price = x and demand = y Computation of Pearson’s Coefficient of Correlation Price (x) 10 12 14 15 19 = 70 Demand (y) 40 41 48 60 50 = 239 xy xy 400 492 672 900 950 3414 x x2 100 144 196 225 361 1026 y 2 1600 1681 2304 3600 2500 11685 21 Quantitative Techniques for Business School of Distance Education r= r= , v v , r= = . = +0. 621 Probable Error and Coefficient of Correlation Probable error (PE) of the Co-efficient of correlation is a statistical device which measures the reliability and dependability of the value of co-efficient of correlation. Probable Error [ = standard error = 0. 6745 x standard error Standard Error (SE) = v 1 v PE = 0. 6745 2 If the value of coefficient of correlation ( r) is less than the PE, then there is no evidence of correlation.
If the value of ‘r’ is more than 6 times of PE, the correlation is certain and significant. By adding and submitting PE from coefficient of correlation, we can find out the upper and lower limits within which the population coefficient of correlation may be expected to lie. Uses of PE: 1) PE is used to determine the limits within which the population coefficient of correlation may be expected to lie. 2) It can be used to test whether the value of correlation coefficient of a sample is significant with that of the population If r = 0. 6 and N = 64, find out the PE and SE of the correlation coefficient.
Also determine the limits of population correlation coefficient. Sol: r = 0. 6 N=64 Quantitative Techniques for Business 22 School of Distance Education PE = 0. 6745 SE SE = v = . v . = . = 0. 08 P. E = 0. 6745 0. 08 = 0. 05396 Limits of population Correlation coefficient = r PE = 0. 6 0. 05396 = 0. 54604 to 0. 6540 Qn. 2 r and PE have values 0. 9 and 0. 04 for two series. Find n. Sol: PE = 0. 04 0. 6745 0. 04 . . v . v . v . = = 0. 0593 = 0. 0593 v 0. 0593 v = 0. 19 v v . . 3. 2 N = 3. 2 = 10. 266 N = 10 Quantitative Techniques for Business 23 School of Distance Education
Coefficient of Determination One very convenient and useful way of interpreting the value of coefficient of correlation is the use of the square of coefficient of correlation. The square of coefficient of correlation is called coefficient of determination. Coefficient of determination = r2 Coefficient of determination is the ratio of the explained variance to the total variance. For example, suppose the value of r = 0. 9, then r2 = 0. 81=81% This means that 81% of the variation in the dependent variable has been explained by (determined by) the independent variable.
Here 19% of the variation in the dependent variable has not been explained by the independent variable. Therefore, this 19% is called coefficient of non-determination. Coefficient of non-determination (K2) = 1 – r2 K2 = 1- coefficient of determination Qn: Sol:Calculate coefficient of determination and non-determination if coefficient of correlation is 0. 8 r =0. 8 Coefficient of determination = = 0. 82 = 0. 64 = 64% Co efficient of non-determination =1– = 1- 0. 64 = 0. 36 = 36% Merits of Pearson’s Coefficient of Correlation:1. This is the most widely used algebraic method to measure coefficient of correlation. . It gives a numerical value to express the relationship between variables 3. It gives both direction and degree of relationship between variables 4. It can be used for further algebraic treatment such as coefficient of determination coefficient of non-determination etc. 5. It gives a single figure to explain the accurate degree of correlation between two variables Demerits of Pearson’s Coefficient of correlation 1. It is very difficult to compute the value of coefficient of correlation. 2. It is very difficult to understand Quantitative Techniques for Business 24 School of Distance Education . It requires complicated mathematical calculations 4. It takes more time 5. It is unduly affected by extreme items 6. It assumes a linear relationship between the variables. But in real life situation, it may not be so. Spearman’s Rank Correlation Method Pearson’s coefficient of correlation method is applicable when variables are measured in quantitative form. But there were many cases where measurement is not possible because of the qualitative nature of the variable. For example, we cannot measure the beauty, morality, intelligence, honesty etc in quantitative terms.
However it is possible to rank these qualitative characteristics in some order. The correlation coefficient obtained from ranks of the variables instead of their quantitative measurement is called rank correlation. This was developed by Charles Edward Spearman in 1904. Spearman’s coefficient correlation (R) = 1Where D = difference of ranks between the two variables N = number of pairs Qn: Find the rank correlation coefficient between poverty and overcrowding from the information given below: Town: Poverty: Over crowing: A 17 36 B 13 46 C 15 35 D 16 24 E 6 12 F 11 18 G 14 27 H 9 22
I 7 2 J 12 8 Sol: Here ranks are not given. Hence we have to assign ranks R = 1N = 10 Quantitative Techniques for Business 25 School of Distance Education Computation of rank correlation Co-efficient Town A B C D E F G H I J Poverty Over crowding 17 13 15 16 6 11 14 9 7 12 36 46 35 24 12 18 27 22 2 8 R1 1 5 3 2 10 7 4 8 9 6 R2 2 1 3 5 8 7 4 6 10 9 D 1 4 0 3 2 0 0 2 1 3 D2 1 16 0 9 4 0 0 4 1 9 44 R=1= 1= 1 – 0. 2667 + 0. 7333 = Qn:- Following were the ranks given by three judges in a beauty context. Determine which pair of judges has the nearest approach to Common tastes in beauty.
Judge I: Judge I: Judge I: R 1 3 6 = 1= 6 5 4 5 8 9 10 4 8 3 7 1 2 10 2 4 2 3 9 1 10 7 6 5 8 9 7 N= 10 Quantitative Techniques for Business 26 School of Distance Education Judge I (R1) 1 6 5 10 3 2 4 9 7 8 Computation of Spearman’s Rank Correlation Coefficient Judge II Judge III R1-R2 R2-R3 R1-R3 (D1) (R2) (R3) (D2) (D3) 3 5 8 4 7 10 2 1 6 9 6 4 9 8 1 2 3 10 5 7 2 1 3 6 4 8 2 8 1 1 3 1 1 4 6 8 1 9 1 2 5 2 4 2 2 0 1 1 2 1 4 1 9 36 16 64 4 64 1 1 200 R=19 1 1 16 36 64 1 81 1 4 214 25 4 16 4 4 0 1 1 4 1 60 Rank correlation coefficient between I & II = = 1= 1- 1. 121 = – 0. 2121 Rank correlation Coefficient between II & III judges = 1 = 1= Rank correlation coefficient between I& II judges =1 = 1- 0. 364 = +0. 636 Quantitative Techniques for Business 27 – 0. 297 = 1- School of Distance Education The rank correlation coefficient in case of I & III judges is greater than the other two pairs. Therefore, judges I & III have highest similarity of thought and have the nearest approach to common taste in beauty. Qn: The Co-efficient of rank correlation of the marks obtained by 10 students in statistics & English was 0. 2.
It was later discovered that the difference in ranks of one of the students was wrongly takes as 7 instead of 9 Find the correct result. R = 0. 2 R = 16 2 = 0. 2 . . = = 103 10 6? Correct ? = 90 0. 8= 792 = = 132 – 7 + 9 = 164 Correct R 1= = 1- = 1 = 1 – 0. 9939 = 0. 0061 Qn: The coefficient of rank correlation between marks in English and maths obtained by a group students is 0. 8. If the sum of the squares of the difference in ranks is given to be 33, find the number of students in the group. R=1ie, 11-08 = Sol: = 0. 8 = 0. 8 0. 2 ( -N) = 198 28 Quantitative Techniques for Business
School of Distance Education N3 – N = N = 10 . 990 Computation of Rank Correlation Coefficient when Ranks are Equal There may be chances of obtaining same rank for two or more items. In such a situation, it is required to give average rank for all. Such items. For example, if two observations got 4th rank, each of those observations should be given the rank 4. 5 (i. e 4. 5) Suppose 4 observations got 6th rank, here we have to assign the rank, 7. 5 (ie. to each of the 4 observations. When there is equal ranks, we have to apply the following formula to compute rank correlation coefficient: …………………..
R= 1- Where D – Difference of rank in the two series N – Total number of pairs m – Number of times each rank repeats Qn:Obtain rank correlation co-efficient for the data:- X: Y: 68 62 64 58 75 68 50 45 64 81 80 60 75 68 40 48 55 50 64 70 Here, ranks are not given we have to assign ranks Further, this is the case of equal ranks. R= 1- 1 6 ? 2 12 3 3 ………….. R= 1- ………….. Quantitative Techniques for Business 29 School of Distance Education Computation of rank correlation coefficient x 68 64 75 50 54 80 75 40 55 64 y 62 58 68 45 81 60 68 48 50 70 R1 4 6 2. 5 9 6 1 2. 10 8 6 R2 5 7 3. 5 10 1 6 3. 5 9 8 2 ? D(R1-R2) 1 1 1 1 5 5 1 1 0 4 D2 1 1 1 1 25 25 1 1 0 16 72 R=1- = 1=1=1= 1= 0. 5455 = 1 – 0. 4545 Merits of Rank Correlation method 1. Rank correlation coefficient is only an approximate measure as the actual values are not used for calculations Quantitative Techniques for Business 30 School of Distance Education 2. It is very simple to understand the method. 3. It can be applied to any type of data, ie quantitative and qualitative 4. It is the only way of studying correlation between qualitative data such as honesty, beauty etc. . As the sum of rank differences of the two qualitative data is always equal to zero, this method facilitates a cross check on the calculation. Demerits of Rank Correlation method 1. 2. 3. 4. Rank correlation coefficient is only an approximate measure as the actual values are not used for calculations. It is not convenient when number of pairs (ie. N) is large Further algebraic treatment is not possible. Combined correlation coefficient of different series cannot be obtained as in the case of mean and standard deviation.
In case of mean and standard deviation, it is possible to compute combine arithematic mean and combined standard deviation. Concurrent Deviation Method: Concurrent deviation method is a very simple method of measuring correlation. Under this method, we consider only the directions of deviations. The magnitudes of the values are completely ignored. Therefore, this method is useful when we are interested in studying correlation between two variables in a casual manner and not interested in degree (or precision).
Under this method, the nature of correlation is known from the direction of deviation in the values of variables. If deviations of 2 variables are concurrent, then they move in the same direction, otherwise in the opposite direction. The formula for computing the coefficient of concurrent deviation is: r= Where N = No. of pairs of symbol C = No. of concurrent deviations (ie, No. of + signs in ‘dx dy’ column) Steps: 1. Every value of ‘X’ series is compared with its proceeding value. Increase is shown by ‘+’ symbol and decrease is shown by ‘-‘ 2. The above step is repeated for ‘Y’ series and we get ‘dy’ 3.
Multiply ‘dx’ by ‘dy’ and the product is shown in the next column. The column heading is ‘dxdy’. Quantitative Techniques for Business 31 School of Distance Education 4. Take the total number of ‘+’ signs in ‘dxdy’ column. ‘+’ signs in ‘dxdy’ column denotes the concurrent deviations, and it is indicated by ‘C’. 5. Apply the formula: r= If 2c , 2 , . Qn:- Calculate coefficient if correlation by concurrent deviation method:Year Supply Price Sol: : : : 2003 2004 2005 160 292 164 280 172 260 2006 2007 182 234 166 266 2008 170 254 2009 178 230 2010 192 190 2011 186 200
Computation of coefficient of concurrent Deviation Supply (x) 160 164 172 182 166 170 178 192 186 Price (y) 292 280 260 234 266 254 230 190 200 dx + + + + + + + dy + + dxdy C=0 r= = = Quantitative Techniques for Business == = -1 32 School of Distance Education Merits of concurrent deviation method: 1. It is very easy to calculate coefficient of correlation 2. It is very simple understand the method 3. When the number of items is very large, this method may be used to form quick idea about the degree of relationship 4. This method is more suitable, when we want to know the type f correlation (ie, whether positive or negative). Demerits of concurrent deviation method: 1. This method ignores the magnitude of changes. Ie. Equal weight is give for small and big changes. 2. The result obtained by this method is only a rough indicator of the presence or absence of correlation 3. Further algebraic treatment is not possible 4. Combined coefficient of concurrent deviation of different series cannot be found as in the case of arithmetic mean and standard deviation. Quantitative Techniques for Business 33 School of Distance Education CHAPTER – 3 REGRESSION ANALYSIS
Introduction:Correlation analysis analyses whether two variables are correlated or not. After having established the fact that two variables are closely related, we may be interested in estimating the value of one variable, given the value of another. Hence, regression analysis means to analyse the average relationship between two variables and thereby provides a mechanism for estimation or predication or forecasting. The term ‘Regression” was firstly used by Sir Francis Galton in 1877. The dictionary meaning of the term ‘regression” is “stepping back” to the average.
Definition: “Regression is the measure of the average relationship between two or more variables in terms of the original units of the date”. “Regression analysis is an attempt to establish the nature of the relationship between variables-that is to study the functional relationship between the variables and thereby provides a mechanism for prediction or forecasting”. It is clear from the above definitions that Regression Analysis is a statistical device with the help of which we are able to estimate the unknown values of one variable from known values of another variable.
The variable which is used to predict the another variable is called independent variable (explanatory variable) and, the variable we are trying to predict is called dependent variable (explained variable). The dependent variable is denoted by X and the independent variable is denoted by Y. The analysis used in regression is called simple linear regression analysis. It is called simple because three is only one predictor (independent variable). It is called linear because, it is assumed that there is linear relationship between independent variable and dependent variable.
Types of Regression:There are two types of regression. They are linear regression and multiple regression. Linear Regression: It is a type of regression which uses one independent variable to explain and/or predict the dependent variable. Multiple Regression: It is a type of regression which uses two or more independent variable to explain and/or predict the dependent variable. Quantitative Techniques for Business 34 School of Distance Education Regression Lines: Regression line is a graphic technique to show the functional relationship between the two variables X and Y.
It is a line which shows the average relationship between two variables X and Y. If there is perfect positive correlation between 2 variables, then the two regression lines are winding each other and to give one line. There would be two regression lines when there is no perfect correlation between two variables. The nearer the two regression lines to each other, the higher is the degree of correlation and the farther the regression lines from each other, the lesser is the degree of correlation. Properties of Regression lines:1. The two regression lines cut each other at the point of average of X and average of Y ( i. X and Y ) 2. When r = 1, the two regression lines coincide each other and give one line. 3. When r = 0, the two regression lines are mutually perpendicular. Regression Equations (Estimating Equations) Regression equations are algebraic expressions of the regression lines. Since there are two regression lines, therefore two regression equations. They are :1. Regression Equation of X on Y:- This is used to describe the variations in the values of X for given changes in Y. 2. Regression Equation of Y on X :- This is used to describe the variations in the value of Y for given changes in X.
Least Square Method of computing Regression Equation: The method of least square is an objective method of determining the best relationship between the two variables constituting a bivariate data. To find out best relationship means to determine the values of the constants involved in the functional relationship between the two variables. This can be done by the principle of least squares: The principle of least squares says that the sum of the squares of the deviations between will be the observed values and estimated values should be the least. In other words, ? the minimum.
With a little algebra and differential calculators we can develop some equations (2 equations in case of a linear relationship) called normal equations. By solving these normal equations, we can find out the best values of the constants. Regression Equation of Y on X:Y = a + bx The normal equations to compute ‘a’ and ‘b’ are: ? ? Quantitative Techniques for Business ? ? ? 35 School of Distance Education Regression Equation of X on Y:X = a + by The normal equations to compute ‘a’ and ‘b’ are:? ? ? y ? ? Qn:- Find regression equations x and y and y on x from the following:X: Y: Sol: 25 18 30 24 35 30 40 36 45 42 50 48 55 54
Regression equation x on y is: x = a + by Normal equations are: ? ? ? ? ? 2 Computation of Regression Equations x 25 30 35 40 45 50 55 ? 280 ? y 18 24 30 36 42 48 54 252 ? 2 x2 625 900 1225 1600 2025 2500 3025 11900 ? 2 y2 324 576 900 1296 1764 2304 2916 10080 ? xy 450 720 1050 1440 1890 2400 2970 10920 280 = 7a+ 252 b ————( 1) ———–(2) 10920 = 252a+10080 b Eq. 1 36 (2) (3) Quantitative Techniques for Business 10080 = 252a + 9072b ————-(3) 10920 = 252a + 10080b ————- (2) 840 = 0 + 1008 b 36 School of Distance Education 1008 b b Substitute b = 840 = = 0. 3 = 0. 83 in equation ( 1 ) 280 280 = 7a + (252 0. 83) = 7 a + 209. 16 7a+ 209. 116 = 280 7a a Substitute a = 280-209. 160 = . = 10. 12 = 10. 12 and b =0. 83 in regression equation: . . Regression equation Y on X is: y = a + bx Normal Equations are: ? ? ? ? ? ——- (1) 252 = 7a + 280 b 10920 = 280 a+ 11900 b ——- (2) (1) 40 10080 = 280 a + 11200 b ——- (3) 10920 = 280 a+ 11900 b ——- (2) 840 = 0 + 700 b 700 b = 840 b= = 1. 2 (2) – (3) Substitute b = 1. 2 in equation (1 ) 252 = 7a + (280×1. 2) 252 = 7a + 336 7a + 336 = 252 Quantitative Techniques for Business 37
School of Distance Education 7a = 252 – 336 = -84 a= = -12 Substitute a = -12 and b = 1. 2 in regression equation y = -12+1. 2x Qn:- From the following bivariate data, you are required to: (a) Fit the regression line Y on X and predict Y if x = 20 (b) Fir the regression line X on Y and predict X if y = 10 X: Y: 4 14 12 4 8 2 6 2 4 4 4 6 16 4 8 12 Computation of regression equations x 4 12 8 6 4 4 16 8 ? 62 ? y 14 4 2 2 4 6 4 12 48 ? x2 16 144 64 36 16 16 256 64 612 ? y2 196 16 4 4 16 36 16 144 432 ? xy 56 48 16 12 16 24 64 96 332 Regression equation y on x y = a + bx Normal equations are: ? ? ? ? Quantitative Techniques for Business 38 School of Distance Education 48 = 8a + 62 b ………(1) 332 = 62a + 612 b ……………(2) eq. 1 eq. 2 eq. 3 62 8 eq. 4 2,976 2,976 320 496 496 0 3844 … . . 3 4896 … . . 4 1052 -1052 b = 320 b= Substitute b = -0. 304 in eq (1) 48 = 8a + (62 x -0. 304) 48 = 8a + -18. 86 48 + 18. 86 = 8a a = 66. 86 a = . = 8. 36 Substitute a = 8. 36 and b = -0. 304 in regression equation y on x : y = 8. 36 + -0. 3042 x y = 8. 36 – 0. 3042 x If x = 20, then, y=8. 36 – (0. 3042×20) = 8. 36 – 6. 084 = 2. 76 (b) Regression equation X on Y: X=a + by Normal equations: ? ? Quantitative Techniques for Business ? ? ? 39 School of Distance Education 62 = 8a + 48 b ………. (1) 332 = 48 a + 432 b ………. (2) eq 1 eq 2 6 3 372 40 48 0 288 … . . 3 144 332 = 48 a + 432 b ……(2) 144 b = -40 b= Substitute b = -0. 2778 in equation (1) 62 = 8a + (48 0. 2778 62 = 8a + -13. 3344 62+13. 3344 = 8 a 8a = 75. 3344 a= . = – 0. 2778 = 9. 4168 Substitute a = 9. 4168 and b = -0. 2778 in regression equation: x = 9. 4168 + -0. 2778 y x = 9. 4168 + -0. 2778 y If y=10, then x=9. 4168 – (0. 2778×10) x= 9. 4168 – 2. 778 x = 6. 388 Regression Coefficient method of computing Regression Equations: Regression equations can also be computed by the use of regression coefficients. Regression coefficient X on Y is denoted as bxy and that of Y on X is denoted as byx. Regression Equation x on y: x – x= bxy (y – ) i. e x – x = . 40 Quantitative Techniques for Business School of Distance Education Regression Equation y on x: y – y= byx (x – ) i. e y – y = . Properties of Regression Coefficient: 1. There are two regression coefficients. They are bxy and byx 2. Both the regression coefficients must have the same signs.
If one is +ve, the other will also be a +ve value. 3. The geometric mean of regression coefficients will be the coefficient of correlation. r= . . 4. If x and the same. are the same, then the regression coefficient and correlation coefficient will be Computation of Regression Co-efficients Regression co-efficients can be calculated in 3 different ways: 1. Actual mean method 2. Assumed mean method 3. Direct method Actual mean method:Regression coefficient x on y (bxy ) = Regression coefficient y on x (b yx ) = Where x = x– y = y? ? ? ? Assumed mean method: Regression coefficient x on y (bxy) .
Regression coefficient y on x (byx) Where dx = deviation from assumed mean of X dy = deviation from assumed mean of Y Quantitative Techniques for Business . 41 School of Distance Education Direct method:Regression Coefficient x on y (bxy ) . Regression Coefficient y on x (byx) Qn: . Following information is obtained from the records of a business organization:91 15 53 8 45 7 76 12 89 17 95 25 80 20 65 13 Sales ( in ‘000): Advertisement Expense ( in ‘000) You are required to:1. Compute regression coefficients under 3 methods 2. Obtain the two regression equations and 3. Estimate the advertisement expenditure for a sale of Rs. ,20,000 Let x = sales y = Advertisement expenditure Computation of regression Coefficients under actual mean method x 91 53 45 76 89 95 80 65 ? 594 ? y 15 8 7 12 17 25 20 13 117 x16. 75 -21. 65 -29. 25 1. 75 14. 75 20. 75 5. 75 -9. 25 y0. 375 -6. 625 -7. 625 -2,625 -2. 375 10. 375 5. 375 -1. 625 ? xy 6. 28 140. 78 223. 03 -4. 59 35. 03 215. 28 30. 91 15. 03 661. 75 ? 2 x2 280. 56 451. 56 855. 56 3. 06 217. 56 430. 56 33. 06 85. 56 2357. 48 ? y2 0. 14 43. 89 58. 14 6. 89 5. 64 107. 64 28. 89 2. 64 253. 87 42 Quantitative Techniques for Business School of Distance Education X= Y= = = = 74. 25 = 14. 25 ? ? = . . Regression coefficient x on y = 2. 61 Regression coefficient Y on X ( = . . ? ? = 0. 28 Computation of Regression Coefficient under assured mean method x 91 53 45 76 89 95 80 65 y 15 8 7 12 17 25 20 13 ? x-70 (dx) 21 -17 -25 6 19 25 10 -5 34 ? y-15 (dy) 0 -7 -8 -3 2 10 5 -2 3 ? dxdy 0 +119 +200 -18 +38 +250 +50 +10 649 ? 441 289 625 36 361 625 100 25 2502 ? 0 49 64 9 4 100 25 4 255 Regression Coefficient x on y (bxy) . Quantitative Techniques for Business 43 School of Distance Education = 8 = = 649 34 3 8 255 3 5192 2040 102 9 = = 2. 61 Regression coefficient y on x (byx) = 8 .