MCM 602 Quantitative Techniques for Managers

Descriptive Summary Measures - II 95 = 16866 + 632 = 20835 New SD = New  x2 – (New x)2 10 = 20835 – (45)2 10 = 585 = 7.65 Problem 6 The following observations have been obtained while taking a sample. Calculate the sample standard deviation and sample variance for the data given observations. 350, 361, 370, 373, 376, 379, 385, 391, 395. Solution : Arranging the observations in ascending order. Observations Mean (x– x ) (x– x )2 x2 (x) (x) 279 122500 350 377 –27 256 130321 49 136900 361 377 –16 16 139129 370 377 –7 141376 373 377 –4 1 143641 376 377 –1 4 148225 379 377 +2 64 149769 385 377 +8 100 155236 387 377 +10 289 156025 324 394 377 +17 395 377 +18 (x – x )2 = 1832 (x)2 = 1423122 CU IDOL SELF LEARNING MATERIAL (SLM)

96 Quantitative Techniques for Managers Calculating (x – x)2 or s2 = n–1 1832 = 10 – 1 1832 = = 203.55 (sample variance) 9 s2 =  x2 – nx 2 n–1 n–1 = 1423122 – 10  142129 99 = 158124.66 – 157921.11 = 203.55 s = s2 = 203.55 = 14.267 (standard deviation of the sample) 4.4 Summary The unit is summarised by some of its important points as below: Important Terms Used  Coefficient of variation: The ratio of the standard deviation to the arithmetic mean.  Dispersion: The spread of the data about the central value is called dispersion.  Fractile: In a distribution, the location of a value at or above a given fraction is called Fractile.  Inter fractile range: A measure of the spread of data between two fractiles in a distribution.  Inter quartile range: The difference between the values of third and first quartiles. CU IDOL SELF LEARNING MATERIAL (SLM)

Descriptive Summary Measures - II 97  Measure of dispersion: The variation of observations or data in a given data set.  Range: The difference between the highest and the lowest observation or the value of the observation.  Standard Deviation: Positive square root of the variance or the square root of the mean of squared deviations of the given observations from the arithmetic mean.  Standard score: Denoting an observation in terms of the standard deviation above or below the mean value.  Symmetrical: A characterstic of a data frequency distribution in which each half is the mirror image of the other.  Variance: A measure of mean squared variation or distance of the observations from their arithmetic mean. Relationships Used x   = for population N x  x = for ungrouped data n  x =  fi xi for grouped data  fi  Range = Xmax – Xmin  Interquartile range = Q3 – Q1 HGF JKI h N –F First quartile (Q1) = l + f 4 FHG IKJ h 3N – F Third Quartile (Q3) = l + f 4  Quartile deviation (QD) = Q3 – Q1 2 CU IDOL SELF LEARNING MATERIAL (SLM)

98 Quantitative Techniques for Managers  Variance (2) = ( xi – x )2 for sample n –1  Variance (s2) = ( xi – )2 for population N  Standard deviation (s) = (xi – )2 for population N  Standard deviation (s) = (xi – x)2 for sample n –1 =  xi2 – 2 nx (n – 1) (n – 1)   Coefficient of variation = for population   Coefficient of variation = x for sample x  Karl Person’s coefficient of Kurtosis b2 = 4 = 4 and g2 = b2 – 3 = 4 – 32 4 4  2 2 4.5 Key Words/Abbreviations  Range: The difference between the highest and the lowest observation or the value of the observation.  Variance: A measure of mean squared variation or distance of the observations from their arithmetic mean.  Standard deviation: Positive square root of the variance or the square root of the mean of squared deviations of the given observations from the arithmetic mean.  Coefficient variation: The ratio of the standard deviation to the arithmetic mean. CU IDOL SELF LEARNING MATERIAL (SLM)

Descriptive Summary Measures - II 99 4.6 Learning Activity 1. Two workers on the same job show the following results over a long period of time. Worker A Worker B Mean time of completing the job (minutes) : 30 25 Standard deviation (minutes) : 64 (i) Which worker appears to be more consistent in the time he requires to complete the job? (ii) Which worker appears to be faster in completing the job? Explain. 2. Goals scored by two teams A & B in a Football season were as under. Calculate the co- efficient of variation. Which team may be considered more consistent ? Goals scored in a match :0 1 2 34 Number of matches – Team A : 27 9 8 54 Team B : 17 9 6 53 4.7 Unit End Questions (MCQ and Descriptive) A. Descriptive Types Questions 1. Find the average deviation about the median for the following distribution. Demand : 5 15 20 25 30 35 40 45 50 Frequen cy : 3 2 6 8 10 5 8 75 2. From the data collected during industrial survey, as given in the table below, calculate the variance for the distribution. Sales levels : 10 15 20 25 30 Frequency : 10 5 15 4 6 3. From the fluctuations given in two series x and y, find out which of the series show greater fluctuation. Series x : 500 520 524 565 619 635 650 550 Series y : 2,100 2,150 2,340 2,350 2,400 2,300 2,100 2,500 CU IDOL SELF LEARNING MATERIAL (SLM)

100 Quantitative Techniques for Managers 4. Calculate the mean deviation about Arithmetic Mean from the following : Values (x) : 10 11 12 13 Frequency (f ) : 3 12 18 12 5. Compute the quartile deviation and mean deviation from the following data : Height in inches : 58 59 60 61 62 63 64 65 66 No. of students : 15 20 32 35 33 22 22 10 8 6. Compute the mean deviation from the median and from mean for the following distribution of the scores of 50 college students. Score : 140-150 150-160 160-170 170-180 180-190 190-200 Frequency : 5 6 10 13 9 7 7. Calculate median and mean deviations for the following frequency distribution Age (years) : 1-5 6-10 11-15 16-20 21-25 26-30 31-35 36-40 40-45 No. of persons : 7 10 16 32 24 18 10 5 1 8. Compute the coefficient of mean deviation from the following data. Height in inches : 50-53 53-56 56-59 59-62 62-65 65-68 No. of students : 2 7 24 27 13 3 9. Calculate the mean deviation of the following series from the mean. Monthly wages of Workers (in `) 200-250 250-300 300-350 350-400 400-450 450-500 No. of workers : 7 13 15 24 36 50 Wages : 500-550 550-600 600-650 650-700 700-750 above 750 Workers : 25 10 8 6 42 10. Calculate the standard deviation of the following observations on a certain variable. 240.12, 240.13, 240.15, 240.12, 240.17, 240.15, 240.17, 240.16, 240.22, 240.21 CU IDOL SELF LEARNING MATERIAL (SLM)

Descriptive Summary Measures - II 101 11. Find the standard deviation of the following distribution. Age : 20-25 25-30 30-35 35-40 40-45 45-50 No. of persons : 170 110 80 45 40 35 (Take assumed average = 32.5) 12. The mean of 5 observations is 4.4 and the variance is 3.24. If three of the five observations are 1, 2 and 6, find the value of the other two. 13. Find the mean and standard deviation of the first n natural numbers. 14. Calculate the standard deviation from the following data Marks in Cost Accounting : 0-10 10-20 20-30 30-40 40-50 50-60 60-70 No. of students : 5 7 14 12 9 6 2 15. Find out the mean and standard deviation of the following data. Age under : 10 20 30 40 50 60 70 80 No. of person dying : 15 30 53 75 100 110 115 125 16. The standard deviation calculated form a set of 32 observation is 5. If the sum of the observation is 80, what is the sum of the square of these observations. 17. The following table gives the length of life of 400 radio tubes : Length of No. of radio Length of No. of radio life (hrs.) tubes life (hrs.) tubes 1,000-1,199 12 2,000-2,199 55 1,200-1,399 30 2,200-2,399 36 1,400-1,599 65 2,400-2,599 25 1,600-1,799 78 2,600-2,799 9 1,800-1,999 90 Calculate (i) the average length of life of a radio set, (ii) the standard deviation of the life, (iii) the percentage number of tubes whose length of life falls within X ± 2s. CU IDOL SELF LEARNING MATERIAL (SLM)

102 Quantitative Techniques for Managers 18. The study of the age of 100 film stars grouped in the intervals of 10-12, 12-14… etc. revealed the mean age and standard deviation to be 32.02 and 13.18 respectively. While checking, it was discovered that the age 57 was misread as 27. Calculate the correct mean age and standard deviation. 19. The mean and standard deviation of a sample of 100 observations were calculated as 40 and 5.1 respectively by a student, who took by mistake 50 instead of 40 for one observation. Calculate the correct mean and standard deviation. 20. The arithmetic mean of runs scored by three batsmen Vijay, Subhash, and Kumar in the same series of 10 innings are 50, 48 and 12 respectively. The standard deviations of their runs are respectively 15, 12, and 2. Who is the most consistent of the three ? If one of the three is to be selected, who will be selected ? 21. You are supplied the following data about heights of boys and girls in a college. Boys Girls Number 3,372 4,538 Average height 68 ” 61” Variance 296 456 You are required to find (i) Combined average of heights of boys and girls (ii) Coefficient of variation for each group. 22. Find the standard deviation and the coefficient of variation for the following frequency distribution and clearly state the fundamental difference between these two measures of variation : Class Interval : 1-3 3-5 5-7 7-9 9-11 11-13 13-15 Frequency : 3 9 25 35 17 10 1 CU IDOL SELF LEARNING MATERIAL (SLM)

Descriptive Summary Measures - II 103 23. Given the following results relating to two groups containing 20 and 30 observations, calculate the coefficient of variation of all the 50 observations by combining both the groups. Group I II x 45 55 x2 118 132 24. An analysis of monthly wages paid to the workers in two firms A & B belonging to the same industry gives the following results. Firm A Firm B Number of workers 500 600 Average monthly wage ` 186 ` 175 Variance of distribution of wages 81 100 (i) Which firm A or B has a larger wage bill? (ii) In which firm A or B, is there greater variability in individual wage (iii) Calculate (a) the average monthly wage (b) the variance of the distribution of wages of all workers in the firms A and B taken together. 25. What advantages can be taken of for the quartile deviation for data analysis? 26. How do you make use of the coefficient of variation for identification of two groups of data? 27. Describe the importance of standard deviation in business data. B. Multiple Choice/Objective Type Questions 1. The measure of dispersion which uses only two observations is called _____________. (a) mean (b) median (c) range (d) coefficient of variation CU IDOL SELF LEARNING MATERIAL (SLM)

104 Quantitative Techniques for Managers 2. Which of the following is an absolute measure of dispersion? (a) coefficient of variation (b) standard deviation (c) coefficient of dispersion (d) coeficient of skewness 3. The range of the values –5, –8, –10, 0, 6, 10 is ________________. (a) 0 (b) 10 (c) –10 (d) 20 4. The average of squared deviations from mean is called _________________. (a) variance (b) mean diviation (c) standard deviation (d) coefficient of variation 5. Which of the following is a unit free quantity? (a) Range (b) Standard deviation (c) coefficient of variation (d) Arithmetic mean. Answers 1. (c), 2. (b), 3. (d), 4. (a), 5. (c) 4.8 References References of this unit have been given at the end of the book. CU IDOL SELF LEARNING MATERIAL (SLM)

Descriptive Summary Measures - III 105 UNIT 5 DESCRIPTIVE SUMMARYMEASURES - III Structure: 5.0 Learning Objectives 5.1 Introduction 5.2 Skewness 5.3 Kurtosis 5.4 Summary 5.5 Key Words/Abbreviations 5.6 LearningActivity 5.7 Unit End Questions (MCQ and Descriptive) 5.8 References 5.0 Learning Objectives After studying this unit, you will be able to:  Explain the deviation of data through skewness and kurtosis  Discuss about the moments.  Make capable of calculating the mean 5.1 Introduction Introduction Measures of dispersion describe the spread of individual values in a data set around a central value. Such descriptive analysis of a frequency distribution remains incomplete until we measure the degree to which these individual values in the data set deviate from symmetry on both sides of the central value and the direction in which these are distributed. This analysis is CU IDOL SELF LEARNING MATERIAL (SLM)

106 Quantitative Techniques for Managers important due to the fact that sets may have the same mean and standard deviation but the frequency curves may differ in their shape. The shape of any uni-model frequency distribution may vary in two aspects:(a) Degree of asymmetry (Skewness) (b) Flatness of mode (Kurtosis) 5.2 Skewness Skewness means “Lack of Symmetry”. Thus, the study of the shape of the curve drawn with the help of frequency distribution is helpful in understanding how the observations are varying about the mean values. If the distribution curve is symmetrical about the mean value, it is a symmetric bell- shaped curve for a symmetrical distribuiton. Fig. 5.1: Symmetrical Distributions Negatively Skewed Positively Skewed When curve is not symmetrical, the values of Mean, Mode and Median fall at different points. The curve may shift its bulk of the bell-shape either to the right or left of the Mean Value. These are called skewness to the left (positive) or right (negative) of the mean, such as Fig. 5.2: Skewed Distribution Measure of Skewness (SK) = Mean – Median = M – Md 1. SK = Mean – Mode = M – Mo or, SK CU IDOL SELF LEARNING MATERIAL (SLM)

Descriptive Summary Measures - III 107 2. SK = (Q3 – Md) – (Md – Q1) = Q3 + Q1 – 2Md These are the absolute values of Skewness and not generally used : 3. Karl Pearson’s Coefficient of Skewness It is given by SK = Mean – Mode SD = M – Mo )  But Mo = 3Md – 2M Hence this can be substituted in the above relationship to get SK = 3( M – Md )  4. Bowley’s coefficient of skewness Prof. A.L. Bowley’s coefficient of skewness is given as follows SK = (Q3 – Md) – (Md – Q1) (Q3 – Md) + (Md – Q1) = Q3  Q1 – 2 Md Q3 – Q1 Subject to the following limits : GFH IJK|a – b|  |a + b|  a – b  1 for a, b two real positive numbers a b Hence | SK (Bowley)|  1 or –1  SK (Bowley)  1 5.3 Kurtosis Having studied the three measures the Central tendency, dispersion and skewness, we now describe Kurtosis to define a distribution in case all these measures are not in a position to define distribution completely. CU IDOL SELF LEARNING MATERIAL (SLM)

108 Quantitative Techniques for Managers Let us consider figure 5.3 in which three curves have been drawn and all three are symmetrical about the mean with the same range. In this case, we need one more measure to define this distribution completely, which is termed as Kurtosis. Prof. Karl Pearson called it is as Convexity of the curve or Kurtosis. While skewness defines the right or left tail of the distribution curve, Kurtosis enables us to identify the shape and nature of the middle portion of the curve. It, therefore, indicates the measure of flatness of the frequency. In the Fig. 5.3, curve B is neither flat, nor sharply peaked. It is called the normal curve. Curve with normal type of hump is called MesoKurtic, whereas curve, which has higher peak than the normal curve is called LeptoKurtic and curves with less flatness is called Platy Kurtic. A = Lapto Kurtic B = Meso Kurtic C = Platy Kurtic Fig. 5.3: Curves with same mean As a measure of Kurtosis, Karl Pearson described coefficient b2 (Beta two) or its derivative 2(gama two) as under 2 = 4 = 4 4  2 2 and 2 = 2 – 3 = 4 – 34 4 For a normal curve, 2 = 3 and 2 = 0 For lepto curve A, b2 > 3 and for platy curve C, b2 < 3. Where m2 and m4 are 2nd and 4th moment about a point menu. CU IDOL SELF LEARNING MATERIAL (SLM)

Descriptive Summary Measures - III 109 5.4 Summary The unit is summarised by some of its important points as below:  Skewness: The lack of symmetry of the distribution, indicating the extent to which a distribution of data is concentrated at one end or the other.  Kurtosis: A measure enabling to identify the shape and nature of the middle value of the distribution curve i.e. its peakedness. 5.5 Key Words/Abbreviations  Mesokurtic: Curve with normal type of hump is called MesoKurtic  Leptokurtic: Curve, which has higher peak than the normal curve is called LeptoKurtic  Platy Kurtic: Curves with less flatness is called Platy Kurtic 5.6 Learning Activity 1. Calculate coefficient of variation from the following data Income (`) less than : 700 800 900 1,000 1,100 1,200 No. of families : 12 30 50 75 110 120 2. From the data given below, state which team (A or B) is more consistent. No. of goals scored in a match : 0 1 2 3 4 Number of matches - Team A : 27 9 8 5 1 Team B : 1 5 8 9 27 5.7 Unit End Questions (MCQ and Descriptive) A. Descriptive Types Questions 1. What is skewness? Write down it measure and define it. 2. Write the Karl Pearson's Kurtosis coefficient. CU IDOL SELF LEARNING MATERIAL (SLM)

110 Quantitative Techniques for Managers 3. Lives of two models of refrigerators in a recent survey are : Life (No. of years) : 0-2 2-4 4-6 6-8 8-10 10-12 Number of refrigerators Model A : 5 16 13 7 54 Model B : 2 7 12 19 9 1 What is the average life of each model of these refrigerators ? Which model has greater uniformity ? 4. A purchasing agent received samples of envelops from two suppliers. He had the sample tested for tearing weight with the following results. Find out which company’s envelops are more uniform. Tearing weight (in (lbs) : 50-59.9 60-69.9 70-79.9 80-89.9 Samples from Company A : 3 42 12 3 Company B : 10 16 26 8 5. For a group of 50 male workers, the mean and standard deviation of their weekly wages are ` 63 and ` 9 respectively. For a group of 40 female workers, these were ` 54 and ` 6 respectively. Find the standard deviation of the combined group of 90 workers. 6. Evaluate an appropriate measure of dispersion for the following data Income (in Rs.) 50 50-70 70-90 90-110 110-130 130-150 above 150 less than 100 140 Number of 54 300 230 125 51 persons : [Calicut University, B.Com., 1975] 7. You are supplied by the following data about heights of boys and girls in a college Number Boys Girls 3,372 4,538 Average height 68” 61” Variance 296 456 You are required to find (i) Combined average of heights of boys and girls (ii) Coefficient of variation for each group [Osmania University, B.Com. III, Oct., 1983] CU IDOL SELF LEARNING MATERIAL (SLM)

Descriptive Summary Measures - III 111 8. Calculate coefficient of variation from the following data Income (Rs.) less than : 700 800 900 1,000 1,100 1,200 No. of families : 12 30 50 75 110 120 [Himachal University, B.Com., April 1982] B. Multiple Choice/Objective Type Questions 1. If 1 = 9 ; 2 = 11 then coefficient of skewness is __________. (a) 0.589 (b) 0.689 (c) 0.489 (d) 0.889 2. If first quartile and third quartile are as 32 and 35 respectively with median of 20 then distribution is skewed to __________. (a) lower tail (b) upper tail (c) close end tail (d) open end tail 3. If median is 12, mean is 15 and standard deviation of data is 3 then karl Pearson’s coefficient of skewness is __________. (a) 17 (b) 27 (c) 15 (d) 3 4. According to beta, platykurtic distribution is one in which the __________. (a) beta three is greater than three (b) beta two is greater than three (c) beta two is greater than two (d) beta three is less than three 5. Considering mean, mode and skewness of data, value of skewness will be positive if __________. (a) mean < median (b) mean > mode (c) mean < mode (c) mean > median Answers 1. (b), 2. (a), 3. (d), 4. (d), 5. (c) 5.8 References References of this unit have been given at the end of the book. CU IDOL SELF LEARNING MATERIAL (SLM)

112 Quantitative Techniques for Managers UNIT 6 CORRELATION AND REGRESSION ANALYSIS Structure: 6.0 Learning Objectives 6.1 Introduction 6.2 Correlation Concept 6.3 Definition of Correlation 6.4 Types of Correlation 6.5 CorrelationAnalysis 6.6 Properties of Correlation Coefficient 6.7 Lines of Regression 6.8 Coefficient of Regression 6.9 The Method of Least Square 6.10 Solved Problems 6.11 Summary 6.12 Key Words/Abbreviations 6.13 LearningActivity 6.14 Unit End Questions (MCQ and Descriptive) 6.15 References CU IDOL SELF LEARNING MATERIAL (SLM)

Correlation and Regression Analysis 113 6.0 Learning Objectives After studying this unit, you will be able to:  Define the concept of bivariate data relationship.  Explain the correlation concept.  Discuss the definition and types of correlation.  Analyse the correlation analysis through various methods.  Differentiate the lag and lead in correlation.  Describe the lives of Repression.  Discuss the coefficient of Repression.  Explain the method of least squares.  Define the method Regression Analysis.  Use of these concepts through solved problems.  Analyse the capability assessment through self-assessment problems. 6.1 Introduction Business managers need the information about future trends of various parameters like demand, expenditure, cash flows, salary of workers etc. to take decisions about their future operations. In day today life also, we would like to find out whether any price increase would change our behaviour towards buying certain house hold items. For such decisions, we either use intuitive sense or a definite calculative relationship for effective and arithmetic prediction. The term ‘Regression’ was first used by Sir Francis Galton in connection with the studies on estimation of the stature of the sons of tall parents to its effect on the mean population. Now this expression is used in statistics for estimation or prediction of an unknown value of one variable from the known value of the other variable. This powerful tool is now used extensively in natural, social or physical sciences. In business, it is used to study the relationship between two or more variables that are related. Prediction or estimation of future is an important area for the benefit of planning and looking deeper into business. Regression Analysis is one of the scientific techniques for making such a CU IDOL SELF LEARNING MATERIAL (SLM)

114 Quantitative Techniques for Managers prediction. In life. we come across many inter-related events such as sales and price of the product, rainfall and yield of the crops, salary and expenditure of individuals, Rates and demand of the transport means etc. When the estimation is related to the study of only two variables at one time, it is called Simple Regression whereas relationships of more than two variables is called Multiple Regression. In Regression analysis, there are two types of variables, namely, dependent and independent variables. The variable which influences the value of the other variable is called Independent variable, whereas the variable, whose value is influenced by the independent variable is termed as Dependent Variable. If the regression curve obtained by plotting the values of two variables is a straight line, it is called a linear Regression. If no such clear shape of the curve can be established or is different from that of a straight line, it is called curved or Non-linear Regression. In this book, we are restricting the discussion to linear regression most commonly used by Business Community for their future predictions or forecasting. 6.2 Correlation Concept Having discussed the cases of univariates in terms of average, dispersion and skewness etc, we now wish to discuss the cases of observation sets of two or more variables. For example, the heights of individuals are generally related to weights and weights are related to the age of the individuals. When we study such cases, then we think in terms of correlation. Such distribution, in which each unit of the series assumes two values is called a bi-variate distribution and if we have more than two variables on each unit of a distribution, it is termed as multi-variate distribution. In case of bi-variate distribution, we may be looking for the answers to the following questions; 1. Is there an association between the two variables? If so, to what extent? 2. Is there any definite relationship between the two variables? In case of an increase or decrease of one variable, any definite effect on the other variable needs be studied, if any such effect exists. Its extent also needs to be established. This concept of relationship between two variables is denoted by Correlation. Correlation is a statistical tool which studies this relationship of two variables. Correlation Analysis involves various methods and techniques used for studying and measuring the extent of the realtionship of two CU IDOL SELF LEARNING MATERIAL (SLM)

Correlation and Regression Analysis 115 variables. The cause and effect relationship i.e., the extent to which variation in one variable effects the other variable is answered by “Regression technique”, which will be discussed in the next chapter. To illustrate the concept, we list out certain examples for correlation. 1. In a business scenario, the sales revenue and related advertising expenditure can be cited as two interdependent variables, the effect of one may not be the result of the other always in the same proportion. 2. In an examination, the series of marks in two subjects may not have any definite relationship, but these can be compared with reference to different schools from a similar class. 3. The age details of wives and husbands in a sample of selected married couples can be a good measure of relationship of two variables. 4. The comparision of Price and its effect on demand of a particular commodity may establish relationship for future decision making. 5. The heights and weights of a selected community personnel may be good measure for designing clothes, vehicles or type of food criterion. 6.3 Definition of Correlation In a bi-variate distribution analysis, we would be interested in knowing whether there exists a relationship between the two variables under discussion and study. Correlation has been defined as follows: “Correlation is an analysis of the covariation between two or more variables” – A.M. Tutle “When the relationship is of a quantitative nature, the appropriate statistical tool for discovering and measuring the relationship and expressing it in a brief formula is known as correlation” – Craxton and Cowden “Correlation analysis contributes to the understanding of economic behaviour, aids in locating the critically important variables on which others depend, may reveal to the economist the connections by which disturbances spread and suggest to him the paths through which stabilising forces may become effective.” – W.A. Neiswanger CU IDOL SELF LEARNING MATERIAL (SLM)

116 Quantitative Techniques for Managers “The effect of correlation is to reduce the range of uncertainty of out prediction” – Tippett Thus simply putting it – “Correlation analysis is the statistical tool we can use to describe the degree to which one variable is linearly related to another”. 6.4 Types of Correlation Positive and Negative Correlation While studying the realtionships of any two related variables, if we find the deviations of the values of variables in the same direction i.e., if one variable increases, the corresponding values of the second variable also increase, then it is called a Positive Correlation. This is true even if decreasing values of one variable correspond to the decreasing corresponding values of the other variable. The amount of rain fall and yield of the crop may be one such example. Some other examples of positive correlation are heights and weights of human beings, price and expenditure on luxury items etc. On the other hand, if the variables deviation is in opposite directions such as decrease in one causes increases in the values of the other or vice-versa, it is called the Negative Correlation. Volume and pressure of a perfect gas or money and purchases in the market on a particular day may represent a negative correlation. In addition, price and demand of a commodity, temperatures and sales of woolen clothes can be cited as examples of negative correlation – The two deviations can be represented as under : Fig. 6.1:Positive and Negative Correlation CU IDOL SELF LEARNING MATERIAL (SLM)

Correlation and Regression Analysis 117 Linear and Non-Linear Correlation When we establish the rate of change of one variable with respect to the other and find it to be constant, it is called a Linear Correlation. When this change does not remain the same, the correlation will be termed as Non-Linear Correlation. Fig. 6.2:Linear andNon-Linear Correlation If we obtain a data for two variables x and y as follows x 0 3 6 9 12 y 3 15 27 39 51 General relationship that can be established is y = 4x + 3, which is a linear trend. When this relationship does not establish a linear curve, i.e. a straight line passing through all the points, then it will be called a non-linear correlation. In such cases, the rate of change of one variable is not the same in all the cases. For example of simplicity, we would like to concentrate only on linear trends because non- linear relationship analysis is complicated. Even linear relationship is not always easy specially in social and economic sciences, where formulation of mathematical models may not always be possible. 6.5 Correlation Analysis Correlation Analysis is a statistical technique used to indicate the degree of relationship existing between one variable and the other. It is also used along with Regression Analysis (described in the next chapter) to measure how well the regression line explains the variations of the dependent variable with the independent variable. CU IDOL SELF LEARNING MATERIAL (SLM)

118 Quantitative Techniques for Managers For correlation analysis, we use the following methods 1. Scatter Diagram Method 2. Karl Pearson’s Coefficient of Correlation 3. Bi-viariate Correlation Method 4. Spearman’s Rank Method 5. Concurrent Deviations Method 6.5.1 Scatter Diagram Method It is the simplest method of representation of the bi-variate distribution. If points of relationships of two variables are very close, a good measure of correlation can be established as in case of Fig. 6.3. But when the points are scattered in a haphazard fashion, it is not easy to indicate the correlation. Straight line correlation established with the help of scatter diagram is a useful method for a large number of business decisions. When the distribution is plotted on a graph in the ungrouped form, we call it as a scatter diagram. The scatter diagram can indicate two types of information. (a) (b) (c) (d) (e) Fig. 6.3:Scatter Diagram The scatter diagrams given above in Fig. 6.4 indicate different relationships. Fig. 6.4 (a) indicates a positive linear relation while Fig. 6.4 (b) shows a negative linear relation. Similarly, Fig. 6.4 (c) and CU IDOL SELF LEARNING MATERIAL (SLM)

Correlation and Regression Analysis 119 (d) are non-linear relationships, (c) being positive while (d) as negative. When.we study Fig. 6.4 (e), we find that no definite relationship of x and y can be established. From the above discussion, we can easily refer that using scatter diagram is comparatively easier method as it is quite comprehensible and it helps us to form general opinion about the nature of relationship between the two variables only by visual inspection of the relationship graph. When number of observations are very large and don’t necessarily create a trend or a pattern easily, it may not be a useful method for establishing correlation. This method also does not indicate exact measure of relationship. 6.5.2 Karl Pearson’s Coefficient of Correlation The linear relationship of correlation between two variables has been suggested by Karl Pearson (1867-1936) a British Biometrician and Statistician and it is the most widely used method of correlation being practiced. It is denoted by r or rxy or r(x, y) denoting the measure of correlation between two variables x and y. It is the ratio of the co-variance between x and y written as cov(x, y) to the product of the standard deviations of x and y. It can be written as : cov(x, y) r = xy When (x1, y1), (x2 y2)…… (xn, yn) are the n-pairs of observations of the variables x and y in a bivariate distribution. cov (x, y) 1 y) Then = (x – x ) (y – n x = 1 (x – x)2 n and y = 1 (y – y)2 n CU IDOL SELF LEARNING MATERIAL (SLM)

120 Quantitative Techniques for Managers Thus, we can write the Karl Pearson’s coefficient of correlation as : 1 (x – x)(y – y) r= n 1 (x – x)2 1 (y – y)2 nn (x – x )( y – y) = (x – x)2 ( y – y)2 If we denote dx = x – x and dy = y – y Then  dx dy r=  dx2  dy2 It can also be written as 1 n  xy  ( x  y) n2 r= 1 1 n2 n  x2  ( x)2 n2 n  y2  ( y)2 n  xy –  x  y b g= n  x2 –  x 2 n  y2 – ( y)2 6.5.3 Spearman’s Rank Correlation Method In addition to the Karl Pearson’s Coefficient of Correlation, we use another approach for ascertaining the degree of correlation between the two variables under consideration. It is based on the ranks of the values of each variable. (The values are to be graded or ranked as per their ascending or descending sequence). The coefficient of correlation determined on this basis is called the Spearman’s Rank Correlation Coefficient and is denoted by the relationship indicated below : CU IDOL SELF LEARNING MATERIAL (SLM)

Correlation and Regression Analysis 121 n 6 di2  =1– i 1 n(n2 – 1) where  = Spearman’s Rank Correlation Coefficient and n = The difference of ranks of a value of the variables The value of the Spearman’s Rank Correlation Coefficient i.e. r(Pronounced as rho) also varies between –1 to 1. i.e. –1  1 The ranks or gradings are alloted to the values of x and y, the two variables, according to the sequence in a particular series i.e., x or y. If least value of x series is alloted rank 1, the next higher, value as 2 and so on. It is similarly done for variable y also. It can be done even in the reverse order. These ranks for each pair of variables are then subtracted to get the difference in ranks i.e., di for a particular pair i of the variables. There may arise a situation when the value of the variable is repeated in the series. In such cases, the ranks are shared at equal level by the similar values. For example, if there are two values having magnitude of say 15 each, and the normal sequential ranks come to 4, the other value 15 could be ranked 5. Actual ranks are allotted as the average value of the ranks i.e. 45 = 4.5 each 2 for calculation of Rank Correlation Coefficient. The modified relationship is adopted as follows : 6MLNM d 2  mGFH m2 – 1JKI OPPQ 12  =1– n(n2 – 1) LM GF IJ OPA correction factor m m2 – 1 has been added to compensate for the repeated ranks in this NM H KQP12 expression. The value m = number of times an observation is repeated. Thus, if we have only the value, say 15 repeated twice, the ranks are given as 4.5 (above example) and then m = 2 to be incorporated in the correction factor. CU IDOL SELF LEARNING MATERIAL (SLM)

122 Quantitative Techniques for Managers For a possibility of having more than one observation repeated, the correction factor gets modified as under FHG KIJm1 m12 – 1 GFH IJK+ m2m22 – 1etc. 12 12 where m1, m2 = Observation 1 and 2 repeated m1 and m2 times. In order to show that such a relationship is valid, let us establish the proof of this relationship. Since the method is based on ranks of x and y, we get x  x 1  2  3n == nn n 1 =2 n 1 similarly y = 2 b g2x1 2 = n  x2 – x F I1 G J= [12 + 22 + ……n2] – n1 3 H Kn 2 L O1 MN P= n(n  1)(2n  1) – (n  1)2 Qn 6 4 (n  1) = [2(2n + 1) – 3(n + 1)] 12 (n  1) = [4n + 2 – 3n – 3] 12 n2 –1 = 12 CU IDOL SELF LEARNING MATERIAL (SLM)

Correlation and Regression Analysis 123 Same value can be obtained for s2y n2 –1  2x = 2y = 12 Now d =x–y b g b g b g= x  x – y  y linear x  y  2 = (x  x)  (y  y 2 b g= x  x 2  (y  y)2  2(x  x)( y  y) b g 2  (y  2  2(x  x) (y  y)  xx y) d 2 (x – x)2 ( y – y)2 (x – x)( y – y) and = + – 2 nn n n  d 2 2 (x – x)( y – y) n = 2x + 2y – n Since Spearman’s Rank correlation coefficient is given by = (x – x )( y – y) n x  y  . xy = (x – x)( y – y) n d 2  n = 2x + 2y – 2 xy. = 2x + 2x – 2r 2x (Since x = y) = 22x [1 – r]  (1 – r) = d 2 2n 2 x d 2 = 2n(n2 – 1) / 12 CU IDOL SELF LEARNING MATERIAL (SLM)

124 Quantitative Techniques for Managers 6d 2 = n(n2 – 1) Thus 6d 2  = 1 – n(n2 – 1) Spearman’s limits for r (coefficient of Rank correlation) lies between –1 and +1, which can be establish by taking value of d2 = n(n + 1) + 4x2 – 4(x + 1) x n(n2 – 1) = 3 Since d2 is non-negative and so is n, r max will be 1 – (Some non-negative) = 1 Similarly 6n(n2 – 1) / 3  = 1 – n(n2 – 1) = 1 – 2 = –1 Hence –1  1 6.6 Properties of Correlation Coefficient Property I : The value of correlation coefficient varies between –1 and 1 i.e. –1  r < 1 also –1  1 Property II : Correlation coefficient is independent of the change of origin and scale. Thus the concept of assumed mean or the scale of observations can be used to advantage. x– A If u = h CU IDOL SELF LEARNING MATERIAL (SLM)

Correlation and Regression Analysis 125 and y–B Then v= h rxy = (x – x)( y – y) can be transformed as (x – x)2 ( y – y)2 (u – u)(v – v) ruv = (u – u )2 (v – v )2 = rxy  rxy = ruv with change of orign and scale Property III : The independent variables are uncorrelated, but the converse is not true. Thus 1 and cov (x, y) = n xy – ( x . y) rxy = 0 for independent variables 6.7 Lines of Regression The line of Regression is the graphical or relationship representation of the best estimate of one variable for any given value of the other variable. The nomenclature of the line depends on the independent and dependent variables. If x and y are two variables of which relationship is to be indicated, a line which gives best estimate of y for any value of x, it is called regression line of y on x. If the dependent variable changes to x, then best estimate of x by any value of y is called regression line of x on y. The values of the variables are to be connected with the help of best fit curve and this is based on the principle of least square. The principle of least square is that we minimise the sum of the squares of the deviations or the errors of estimates. Thus the deviations between the given observed values of the variable and their corresponding estimated values are given by the line of Best fit. Line of Regression of y on x Let us assume the observed values of two variables under consideration for regression analysis as (x1, y1); (x2, y2); (x3, y3)……(xn, yn). If we also assume a linear relationship between the two variables, the equation of the line of Regression can be written as y = a + bx. CU IDOL SELF LEARNING MATERIAL (SLM)

126 Quantitative Techniques for Managers (x (x 1’ i’ y) Cy ii ’ ) 1 Fig. 6.4: Regression Line For any given point (xi, yi), let Bi be a point indicating the x-position of xi, on line and BiAi as the y position of Bi. Then CiBi = CiAi – BiAi = yi – (a + bxi) This is the error or deviation of point Ci(xi, yi) for this ith point. We can, similarly, obtain the deviation of all the observed values of points from the line of best fit y = a + bx. These deviations will be positive for all points above the line and negative for the points below this line. Now we apply the principle of least squares to obtain the values of a and b. For finding the values of a and b, the sum of the squares of the deviations or errors should be minimum. Hence E n = (Ci Bi )2 i 1 n = yi  (a  bxi ) 2 i 1 To get the values of a and b, we use the concept of maxima and minima and we write partial derivatives of E with respect to a and b. E E a = 0 and b = 0 2 E This is only a requirement for maxima or minima of E. For E to be minimum, a2 > 0 and 2E b2 > 0. (Satisfied by normal equations for the values of a and b) CU IDOL SELF LEARNING MATERIAL (SLM)

Correlation and Regression Analysis 127  or 2(y–a–bx) (y–a–bx) = 0 a  and 2(y–a–bx) (y–a–bx) = 0 b or 2(y–a–bx) (–1) = 0 and 2(y–a–bx) (–x) = 0 From where, we get the relationship as, (y–a–bx) = 0 x (y–a–bx) = 0 or y–na–bx = 0 and xy – ax – bx2 = 0 or  y = na + bx …(i) and xy = ax + bx2 …(ii) These equations are called the normal equations for estimation of a and b. The values of other expressions can be obtained from the given equations and then can be substituted in the above equations to obtain the values of a and b. We can also write x2y – xxy a = nx2 – (x)2 nxy – xy and b = nx2 – (x)2 We can extend the same relationship to other measures such as x , y ,  x ,  y and the correlation coefficient rxy. Equation (i) can be coverted as 1 x Sy = a + b nn CU IDOL SELF LEARNING MATERIAL (SLM)

128 Quantitative Techniques for Managers  y =a+bx …(iii) This means that the regression line of y on x passes through its mean point ( x , y ) i.e. the point ( x , y ) lies on the regression line of y on x. Since x = nx a f1 Since x2 = x2 – x2  n and, x2 = n[x2 + ( x )2] …(iv) or …(v) Also, 1 Hence cov (x, y) = xy – x y  n xy = n[cov (x, y) + x y ] x y =ax +bx2 cov(x, y) = bs2x cov(x, y) b=  2 x Now y – y = b(x– x ) = cov(x, y) (x– x )  2 x cov(x, y) But rxy =  x y  cov(x, y) = r.xy ...(vi) ...(vii)  y –y = r x y (x– x ) = r y (x– x ) x …(viii)  2 …(ix) x These are important relationships for data analysis. Similar relationship can be obtained for the regression line of x and y such as  x = nA + By and xy = ny + By2 CU IDOL SELF LEARNING MATERIAL (SLM)

Correlation and Regression Analysis 129 A= dy2ib gx – bygbxyg …(x) ny2 – (y)2 nxy – (x)(y) and B = ny2 – (y)2 …(xi) Also B = cov(x, y) = rx …(xii) y  2 y and x– x = rx (y– y ) …(xiv) y This also establishes that in case of perfect correlation (r = ±1), both the lines of regression will coincide. Angle between the Regression Lines Since the regression lines can be represented as (y – y ) = r y (x –x) x and (x – x ) = r x (y – y ) y Then y – y =  y (x – x ) r x Now we can write the slope of these lines as m1 = r y and m2 = y x r x If q is the angle between these lines, GFH KIJtan q x y r2 –1 = mi – m2 = r 1  m1m2  2   2 x y CU IDOL SELF LEARNING MATERIAL (SLM)

130 Quantitative Techniques for Managers  MLMN FGH JIKOPQPq = tan–1xy r2 –1 …(xv) r  2   2y x 6.8 Coefficient of Regression If line of Regression is represented as y = a + bx, then the coefficient ‘b’, which is the slope of the line of Regression of y on x is called the coefficient of regression, From equation drawn in Para 14.2, we have cov(x, y) r y x byx =  2 = x This represents the coefficient of Regression for line y on x. Similarly coefficient of regression for line x on y can be written as bxy = r x y Thus we can establish that r2 = byx. bxy. ...(xvi) and r = ± byx.bxy In this relationship, if the regression coefficients are positive, we adopt positive sign for the correlation coefficient and vice-versa. If one of the regresssion coefficients is greater than 1, the other must be less than unity. Since, r2 = byx . bxy > 1 It is impossible as 0  r2  1  bxy 1 <1 …(xvii) byx We can also establish that the arithmetic mean of the regression coefficients is greater than the correlation coefficient. Since AM > GM CU IDOL SELF LEARNING MATERIAL (SLM)

Correlation and Regression Analysis 131 ab or 2 > ab Substituting a = byx and b = bxy, we have 1 2 (byx + bxy) > byx .bxy 1 …(xviii) or, 2 (byx + bxy) > r Regression coefficients are also independent of change of the origin, not of scale Let us have x–a y–b u = and v = hk Since correlation coefficients are independent of the origin and scale, we know rxy = ruv Here x = hu and y = kv Since bxy = rxy. y = ruv. k v = k ruv. v x h u h u k = h buv h and bxy = k bvu Thus we can say that regression coefficients are independent of origin but not of scale Hence nuv – (u)(v) …(xix) byx = bvu = nu2 – (u)2 nuv – (u)(v) and bxy = buv = nv2 – (v)2 …(xx) CU IDOL SELF LEARNING MATERIAL (SLM)

132 Quantitative Techniques for Managers 6.9 The Method of Least Square We have already worked out the method of determining the equation of a straight line. This was fitting the line through various points on the scatter diagram. Now we achieve the same thing through mathematical means to establish the line of ‘best fit’. The concept was used in deciding on the lines of regression under Para 17.2. It is called the line of best fit, if it minimises the error between the estimated points on the line and the actual given point used to draw the line. Let us study the scatter diagram given below. Fig. 6.5 If we used the value of y (estimate) the value deviation from points A, B, C and D used to draw the line are taken as AA' = 3, BB' = –2, CC' = 2, DD' = –3, where A', B', C' and D' are error points with reference to the line drawn. Total error = 3 – 2 + 2 – 3 = 0 If we use the concept of sum of the squares Sum of squares = 32 + (–2)2 + (2)2 + (–3)2 = 9 + 4 + 4 + 9 = 26 Now, in order to establish the straight line for minimising the sum of squares of these errors, we call the method as “Method of least squares”. Statisticians have derived the following equations for the best fit concept CU IDOL SELF LEARNING MATERIAL (SLM)

Correlation and Regression Analysis 133 xy – nx y d ib = x2 – n x 2 Where b = slope of the best fit line x = values of the independent variables y = values of the dependent variable x = Mean of the values of the independent variable and y = Mean of the values of the dependent variable n = number of observations or data points Another relationship obtained is a = y –bx a = y intercept of the line b = slope of the line y = Mean of the values of dependent variables x = Mean of the values of independent variable Standard error of estimate (Linear Regression) In order to establish the reliability of the estimating lines, we use the concept of the standard error of estimate. This measures the variability or the spread of the observed values from the given regression line. Thus, standard error of estimate is given by  (y – y)2 Se = n–2 Where y = values of the dependent variable of y.  = estimated value of the variable corresponding to each value y CU IDOL SELF LEARNING MATERIAL (SLM)

134 Quantitative Techniques for Managers y = number of data points. It can also be written as follows Se = y2 – ay2 – bxy n–2 c h c h1 1 = y 1  r2 2 or x 1  r2 2 6.10 Solved Problems Problem 1 While taking the data of a certain group of employees in a company, we can establish the relationship between the length of service and corresponding level of their montly salary. The sample chosen is completely random. One such sample is illustrated below : Employee Code No. Length of Service (yrs.) Monthly Salary (`) (x) (y) 92001 1 3,000 92002 2 4,000 92003 3 4,500 92004 4 5,500 92005 5 6,000 92006 5 6,500 92007 2 6,000 92008 4 7,000 92009 3 8,000 92010 5 5,000 92011 4 6,000 92012 1 4,000 92013 2 7,000 92014 6 9,000 CU IDOL SELF LEARNING MATERIAL (SLM)

Correlation and Regression Analysis 135 92015 10 9,500 92016 9 9,500 92017 5 5,500 92018 6 7,500 92019 9 5,500 92020 8 9,000 For all the 20 employees, we have both the value of length of service and their monthly salary. Taking length of service as x and the monthly salary as y, draw the frequency distribution table for both the variables separately. Solution: Length ofService of 20 Employees Length of Service (x) Frequency ( f ) 1<4 7 4<7 9 7 < 10 3 10 < 13 1 MONTHLYSALARIESOF20EMPLOYEES Monthly Salary (y) Frequency ( f ) 3,000 < 4,000 1 4,000 < 5,000 3 5,000 < 6,000 4 6,000 < 7,000 4 7,000 < 8,000 3 8,000 < 9,000 1 9,000< 10,000 4 CU IDOL SELF LEARNING MATERIAL (SLM)

136 Quantitative Techniques for Managers BIVARIATEFREQUENCYDISTRIBUTIONOF20EMPLOYEES y 3,000 4,000 5,000 6,000 7,000 8,000 9,000 f(x) x <4,000 <5,000 <6,000 <7,000 <8,000 13 <9,000 <10,000 7 1<4 3 1 1 9 4<7 13 1 3 2 1 3 7<10 1 1 10<13 44 3 2 20 f(y) 1 14 Problem 2 Calculate the coefficient of correlation between x and y series from the following data. Series xy No. of pairs of observations 15 15 Arithmetic mean 25 18 Standard deviation 3.01 3.03 Sum of squares of deviations from mean 136 138 Summation of product deviations of x and y series from thier respective arithmetic mean = 122 Solution: The given information can be summarised as follows n = 15; x = 25; y = 18 x = 3.01; y = 3.03 S(x – x )2 = 136 S(y – y )2 = 138 and S(x – x )(y – y ) = 122 CU IDOL SELF LEARNING MATERIAL (SLM)

Correlation and Regression Analysis 137 Karl Pearson’s correlation coefficient between x and y series is given by the relationship (x – x)(y – y) r= n x y 122 = 15  3.01  3.03 = 0.8917 The other data are found to be irrelevant. Problem 3 Calculate the coeffcient of correlation from the following results n = 10; Sx = 140; Sy = 150 (x – x )2 = 180 (y – y )2 = 215 (x – 10) (y – 15) = 60 Solution: Substituting u = (x–10) and v = (y–15) We get  u = (x–10) = x – 10n = 140 – 100 = 40 Similarly  v = (y–15) = y – 15n = 150 – 150 = 0 u2 = (x–10)2 = 180 v2 = (y–15)2 = 215 and uv = (x–10)(y–15) = 60 Karl Pearson’s correlation coefficient is calculated as under rxy = ruv = nuv – uv b gnu2 – (u)2 nv2 – v 2 10  60 – 40  0 = 10  180 – (40)2 10  215 – 0 CU IDOL SELF LEARNING MATERIAL (SLM)

138 Quantitative Techniques for Managers 600 = 200  2150 = 0.91 Problem 4 A computer, while calculating correlation between two variables x and y from 25 pairs of observations, obtained the following results. n = 25; x = 125; x2 = 650; y = 100 ; y2 = 460 and xy = 508 It was, however, discovered at the time of checking that two pairs of observations were incorrectly copied. They were taken as (6,14) and (8,6) while the correct values were (8, 12) and (6, 8). Prove that the correct value of the correlation should be 2/3. Solution: As per the data given above, we obtain Correct value of x = 125 – 6 – 8 + 8 + 6 = 125 Correct value of y = 100 – 14 – 6 + 12 + 8 = 100 Correct value of x2 = 650 – 62 – 82 + 82 + 62 = 650 Correct value of y2 = 460 – 142 – 62 + 122 + 82 = 436 and correct vlaue of xy = 508 – 6 × 14 – 8 × 6 + 8 × 12 + 6 × 8 = 520 nxy – (xy) d i b grxy = nx2 – x2 ny2 – y 2 25  520 – 125  100 = 250  650 – 1252 25  436 – 1002 2 = 3 CU IDOL SELF LEARNING MATERIAL (SLM)

Correlation and Regression Analysis 139 Problem 5 Find Karl Pearson’s coefficient of correlation between sales and expenses of the following ten firms. Firms : 12 3 4 5 6 7 8 9 10 Sales in thousand units : 50 50 55 60 60 65 65 60 60 50 Expenses in thousand ` : 11 13 14 16 16 15 15 14 13 13 Solution: We use sales as x and expenses as y variable : x – 60 Let us substitute u = 5 and v = y – 14 Then, we can build the table as follows : Firms x y u v u2 v2 uv 6 1 50 11 –2 –3 4 9 2 0 2 50 13 –2 –1 4 1 0 0 3 55 14 –1 0 1 0 1 1 4 60 16 0 2 0 4 0 0 5 60 16 0 2 0 4 2 6 65 15 1 1 1 1 uv = 12 7 65 15 1 1 1 1 8 60 14 0 0 0 0 9 60 13 0 –1 0 1 10 50 13 –2 –1 4 1 x = 580 y = 140 u = –5 v = 0 u2 = 15 v2 = 22 These values can be used for calculating coefficient of correlation as follows : nuv – uv rxy = ruv = nu2 – (u)2 nv2 – (v)2 CU IDOL SELF LEARNING MATERIAL (SLM)

140 Quantitative Techniques for Managers 10  12 – (–5)  0 = 10  15 – (–5)2 10  22 – 02 120 = (150 – 25)(220) 120 = = 0.73 125  220 Problem 6 (1) Compute the correlation coefficient between the corresponding values of x and y in the following table. x 2 4 5 6 8 11 y 18 12 10 8 7 5 (2) Multiply each x value in the table by 2 and add 6. Multiply each value of y in the table by 3 and subtract 15. Find the correlation coefficient between the two new sets of values. Explain why you do or donot obtain the same result as in (1). Solution: (1) For computation of correlation coefficient, we obtain the values from the following table. x y x– x y– y (x – x )2 (y – y )2 (x – x )(y – y ) =x–6 = y – 10 2 18 –4 8 16 64 –32 4 12 –2 –4 24 4 5 10 –1 01 0 0 68 0 –2 0 4 0 87 2 –3 4 9 –6 11 5 5 –5 25 25 –5 x = 36 y = 60 x – x = 0 y – y = 0 (x – x )2 (y – y )2 (x – x ) = 50 (y – y = 67 = 106 CU IDOL SELF LEARNING MATERIAL (SLM)

Correlation and Regression Analysis 141 Because x 36  x = = =6 66 y y 60 = 6 = 6 = 10 (x – x)( y – y) b g b grxy =  x  x 2  y – 2 –67 = = – 0.92 50  106 Thus variables x and y are having Negative high correlation (2) New variables can be defined as u = 2x + 6 and v = 3y – 15 The revised table with the new defined variables can be drawn as. x y u = 2x+6 v=3y–15 u2 v2 uv 390 2 18 10 39 100 1521 294 240 4 12 14 21 196 441 162 132 5 10 16 15 256 225 0 6 8 18 9 324 81 uv = 1218 8 7 22 6 484 36 11 5 28 0 784 0 u = 108 v = 108 u2 = 2144 v2 = 2034 New coefficient of correlation is given by nuv – (uv) b gruv = nu2 – u 2 nv2 – (v)2 6  1218 – 108  90 –2412 = = = –0.92 6  2144 – 1082 6  2304 – 902 6868800 Since coefficient of correlation is independent of origin, the two values are found to be the same. CU IDOL SELF LEARNING MATERIAL (SLM)

142 Quantitative Techniques for Managers Problem 7 For a given sets of 10 observations, find the Rank correlation coefficient. x : 5 4 3 8 10 6 6 7 8 5 y : 6 2 1 5 6 3 9 8 10 7 Solution: The data can be presented in the tabular form as follows x y Rank of Rank of d = (y–x) d2 xy –2 4 0 0 5 6 7.5 5.5 0 0 4.5 20.25 42 9 9 4.5 20.25 2.5 6.25 3 1 10 10 –3.5 12.25 –1 1 8 5 2.5 7 –1.5 2.25 –3.5 12.25 10 6 1 5.5 6 3 5.5 8 6 9 5.5 2 78 4 3 8 10 2.5 1 5 7 7.5 4 d2 = 78.5  Rank correlation coefficient  = 1 – 6d 2 n(n2 – 1) = 1 – 6  78.5 10  99 = 0.525 CU IDOL SELF LEARNING MATERIAL (SLM)

Correlation and Regression Analysis 143 Problem 8 The coefficient of rank correlation of the marks obtained by 10 students in two particular subjects are found to be 0.5. It was later discovered that the difference in ranks in two subjects obtained by one student was wrongly taken as 3 instead of 7, what should be the correct value of the coefficient of Rank correlation? Solution: Given here, n = 10, r = 0.5 By using the correlation Rank formula, 6d 2 0.5 = 1 – n(n2 – 1) 6d 2 = 1 – 10(102 – 1)  d2 = 990 = 82.5 62 Since one difference (d) was taken as 3 instead of 7, the corrected value of d2 = 82.5 – 32 + 72 = 122.5 \\Corrected value of Rank Correction Coefficient  =1– 6  122.5 = 0.2576. 10  99 Problem 9 Ten competitors in a beauty contest are ranked by three judges in the following order 1st Judge 1 6 5 10 3 2 4 9 7 8 2nd Judge 3 5 8 4 7 10 2 1 6 9 3rd Judge 6 4 9 8 1 2 3 10 5 7 CU IDOL SELF LEARNING MATERIAL (SLM)

144 Quantitative Techniques for Managers Use the rank correlation coefficient to determine which pair of judges has the nearest approach to common tastes in beauty. Solution: The respective ranks by three judges are already given and these can be identified as R1, R2, R3 as ranks by the first, second and the third judge. The tables are worked out as under R R R d =R –R d =R –R d = R – R d2 d2 d2 1 2 3 12 1 2 13 1 3 23 2 3 12 13 23 1 36 –2 –5 –3 4 25 9 6 54 1 2 1 14 1 5 89 –3 –4 –1 9 16 1 10 4 8 6 2 –4 36 4 16 3 71 –4 2 6 16 4 36 2 10 2 –8 0 8 64 0 64 4 23 2 1 –1 41 1 9 1 10 8 –1 9 64 1 81 7 65 1 2 1 14 1 8 97 –1 1 2 11 4 200 60 214 Using n = 10, 6  200 12 = 1 – 10  99 = –0.2121 6  60 Similarly 13 = 1– = 0.6363 10  99 23 =1– 6  214 = –0.2970 10  99 Since 13 has the maximum value, the first and third judge have shown the nearest approach to common tastes in beauty and since 12 and 23 are negative, it indicates that judges (1, 2) and (2, 3) have opposite tastes for beauty. CU IDOL SELF LEARNING MATERIAL (SLM)