MBA SEM 1 Decision science 1

Graphical method for Location of median Median can be located with the help of the cumulative frequency curve or ‘ogive’. The procedure for locating median in a grouped data is as follows: Example: Draw ogive curves for the following frequency distribution and determine the median. 100 CU IDOL SELF LEARNING MATERIAL (SLM)

Solution: 101 CU IDOL SELF LEARNING MATERIAL (SLM)

The median value from the graph is 42 Merits It is easy to compute. It can be calculated by mere inspection and by the graphical method It is not affected by extreme values. It can be easily located even if the class intervals in the series are unequal Limitations It is not amenable to further algebraic treatment It is a positional average and is based on the middle item It does not take into account the actual values of the items in the series 3.3.5 Mode According to Croxton and Cowden, ‘The mode of a distribution is the value at the point around which the items tend to be most heavily concentrated’. In a busy road, where we take a survey on the vehicle - traffic on the road at a place at a particular period of time, we observe the number of two wheelers is more than cars, buses and other vehicles. Because of the higher frequency, we say that the modal value of this survey is ‘two wheelers’ Mode is defined as the value which occurs most frequently in a data set. The mode obtained may be two or more in frequency distribution. 102 CU IDOL SELF LEARNING MATERIAL (SLM)

Computation of mode: (a) For Ungrouped or Raw Data: The mode is defined as the value which occurs frequently in a data set EXAMPLE: The following are the marks scored by 20 students in the class. Find the mode 90, 70, 50, 30, 40, 86, 65, 73, 68, 90, 90, 10, 73, 25, 35, 88, 67, 80, 74, 46 Solution: Since the marks 90 occurs the maximum number of times, three times compared with the other numbers, mode is 90. EXAMPLE: A doctor who checked 9 patients’ sugar level is given below. Find the mode value of the sugar levels. 80, 112, 110, 115, 124, 130, 100, 90, 150, 180 Solution: Since each value occurs only once, there is no mode. EXAMPLE: Compute mode value for the following observations. 2, 7, 10, 12, 10, 19, 2, 11, 3, 12 Solution: Here, the observations 10 and 12 occurs twice in the data set, the modes are 10 and 12. For discrete frequency distribution, mode is the value of the variable corresponding to the maximum frequency. Solution: Here, 7 is the maximum frequency, hence the value of x corresponding to 7 is 8. Therefore 8 is the mode. (b) Mode for Continuous data: The mode or modal value of the distribution is that value of the variate for which the frequency is maximum. It is the value around which the items or observations tend to be most heavily concentrated. The mode is computed by the formula 103 CU IDOL SELF LEARNING MATERIAL (SLM)

EXAMPLE: The following data relates to the daily income of families in an urban area. Find the modal income of the families. Solution: 104 CU IDOL SELF LEARNING MATERIAL (SLM)

The modal income of the families is 375. Determination of Modal class: For a frequency distribution modal class corresponds to the class with maximum frequency. But in any one of the following cases that is not easily possible. (i) If the maximum frequency is repeated. (ii) If the maximum frequency occurs in the beginning or at the end of the distribution (iii) If there are irregularities in the distribution, the modal class is determined by the method of grouping. Steps for preparing Analysis table: We prepare a grouping table with 6 columns (i) In column I, we write down the given frequencies. (ii) Column II is obtained by combining the frequencies two by two. (iii) Leave the Ist frequency and combine the remaining frequencies two by two and write in column III (iv) Column IV is obtained by combining the frequencies three by three. (v) Leave the Ist frequency and combine the remaining frequencies three by three and write in column V (vi) Leave the Ist and 2nd frequencies and combine the remaining frequencies three by three and write in column VI Mark the highest frequency in each column. Then form an analysis table to find the modal class. After finding the modal class use the formula to calculate the modal value. EXAMPLE: Calculate mode for the following frequency distribution: Solution: 105 CU IDOL SELF LEARNING MATERIAL (SLM)

Analysis Table: 106 CU IDOL SELF LEARNING MATERIAL (SLM)

(d) Graphical Location of Mode The following are the steps to locate mode by graph (i) Draw a histogram of the given distribution. (ii) Join the rectangle corner of the highest rectangle (modal class rectangle) by a straight line to the top right corner of the preceding rectangle. Similarly, the top left corner of the highest rectangle is joined to the top left corner of the rectangle on the right. (iii) From the point of intersection of these two diagonal lines, draw a perpendicular line to the x –axis which meets at M. (iv) The value of x coordinate of M is the mode. EXAMPLE: Locate the modal value graphically for the following frequency distribution Solution: Merits of Mode: 107 It is comparatively easy to understand. It can be found graphically. It is easy to locate in some cases by inspection. It is not affected by extreme values. It is the simplest descriptive measure of average. Demerits of Mode: It is not suitable for further mathematical treatment. It is an unstable measure as it is affected more by sampling fluctuations. Mode for the series with unequal class intervals cannot be calculated. In a bimodal distribution, there are two modal classes and it is difficult to determine the values of the mode. CU IDOL SELF LEARNING MATERIAL (SLM)

3.4 EMPIRICAL RELATIONSHIP AMONG MEAN, MEDIAN AND MODE A frequency distribution in which the values of arithmetic mean, median and mode coincide is known of symmetrical distribution, when the values of mean, median and mode are not equal the distribution is known as asymmetrical or skewed. In moderately skewed asymmetrical distributions a very important relationship exists among arithmetic mean, median and mode. Karl Pearson has expressed this relationship as follows Mode = 3 Median – 2 Arithmetic Mean EXAMPLE: In a moderately asymmetrical frequency distribution, the values of median and arithmetic mean are 72 and 78 respectively; estimate the value of the mode. Solution: The value of the mode is estimated by applying the following formula EXAMPLE: In a moderately asymmetrical frequency distribution, the values of mean and mode are 52.3 and 60.3 respectively, Find the median value. Solution: The value of the median is estimated by applying the formula: 108 CU IDOL SELF LEARNING MATERIAL (SLM)

3.5 RANGE Range is defined as difference between the largest and smallest observations in the data set. Range(R) = Largest value in the data set (L) –Smallest value in the data set(S) R= L – S Grouped Data: For grouped frequency distribution of values in the data set, the range is the difference between the upper class limit of the last class interval and the lower class limit of first class interval. Coefficient of Range: The relative measure of range is called the coefficient of range Co efficient of Range = (L- S) / (L+ S) 3.6 SUMMARY 109 • A central tendency is a single figure that represents the whole mass of data CU IDOL SELF LEARNING MATERIAL (SLM)

• Arithmetic mean or mean is the number which is obtained by adding the values of all the items of a series and dividing the total by the number of items. • When all items of a series are given equal importance than it is called simple arithmetical mean and when different items of a series are given different weights according with their relative importance is known as weighted arithmetic mean. • Median is the middle value of the series when arranged in ascending order • When a series is divided into more than two parts, the dividing values are called partition values. • Mode is the value which occurs most frequently in the series, that is modal value has the highest frequency in the series 3.7 KEYWORDS • Weighted Arithmetic mean: • There are situations in which values of individual observations in the data set are not of equal importance. Then weighted arithmetic mean will be used. • Geometric mean: • • Harmonic Mean: Harmonic Mean is defined as the reciprocal of the arithmetic mean of reciprocals of the observations • Median: Median is the value of the variable which divides the whole set of data into two equal parts. • Mode: The mode of a distribution is the value at the point around which the items tend to be most heavily concentrated’ 3.8 LEARNING ACTIVITY Check property of arithmetic mean for the following example: X: 4 6 8 10 12 In the above example if mean is increased by 2, then what happens to the individual observations? ___________________________________________________________________________ ____________________________________________________________________ 3.9 UNIT END QUESTIONS A. Descriptive Questions 110 CU IDOL SELF LEARNING MATERIAL (SLM)

Short Questions 1. What is meant by measure of central tendency? 2. What are the desirable characteristics of a good measure of central tendency? 3. What are the merits and demerits of the arithmetic mean? 4. Express weighted arithmetic mean in brief. 5. Define Median. Discuss its advantages and disadvantages? Long Questions 1. The mean of 100 items are found to be 30. If at the time of calculation two items are wrongly taken as 32 and 12 instead of 23 and 11. Find the correct mean. 2. A cyclist covers his first three kms at an average speed of 8 kmph. Another two kms at 3 kmph and the last two kms at 2 kmph. Find the average speed for the entire journey. 3. The mean marks of 100 students were found to 40. Later it was discovered that a score of 53 was misread as 83. Find the corrected mean corresponding to the corrected score. 4. In a moderately asymmetrical distribution the values of mode and mean are 32.1 and 35.4 respectively. Find the median value. 5. Find the mean and median: Wages (`) 60 – 70 50 – 60 40 – 50 30 – 40 20 – 30 No. of labourers 5 10 20 5 3 B. Multiple choice questions 1. Which of the following is a measure of central value? a. Median b. Deciles c. Quartiles d. Percentiles 2. Geometric Mean is better than other means when a. the data are positive as well as negative b. the data are in ratios or percentages c. the data are binary d. the data are on interval scale 3. The median of the variate values 11, 7, 6, 9, 12, 15, 19 is a. 9 b. 12 c. 15 d. 11 4. The middle values of an ordered series is called 111 CU IDOL SELF LEARNING MATERIAL (SLM)

a. 50th percentile b. 2nd quartile c. 5th decile d. All of these 5. Mode is that value in a frequency distribution which possesses a. minimum frequency b. maximum frequency c. frequency one d. None of these Answer 1) a 2) b 3) d 4) d 5) b 3.10 REFERENCES Textbooks / Reference Books • T1: Levine, D., Sazbat, K. and Stephan, D. 2013. Business Statistics, 7thEdition, Pearson Education, India, ISBN: 9780132807265. • T2; Gupta, C. and Gupta, V. 2004. An Introduction to Statistical Methods, 23rdEdition, Vikas Publications, India, ISBN: 9788125916543. • R1: Croucher, J. 2011. Statistics: Making Business Decisions, 13thEdition, Tata McGraw Hill, ISBN: 9780074710419. • R2 Gupta, S. 2011. Statistical Methods, 4thEdition, Sultan Chand & Sons, ISBN: 8180548627. 112 CU IDOL SELF LEARNING MATERIAL (SLM)

UNIT 4 SITUATIONAL/DESCRIPTIVE STATISTICS Structure 113 4.0 Learning Objectives 4.1 Introduction 4.2 Characteristics of A Good Measure of Dispersion 4.3 Types of Measures of Dispersions 4.3.1 Range 4.3.2 Inter Quartile Range and Quartile Deviation 4.3.3. Mean Deviation 4.3.4 Standard Deviation 4.4 Combined Mean and Combined Standard Deviation 4.5 Relative Measures 4.5.1 Coefficient of Variation 4.6 Moments 4.6.1 Raw moments: 4.6.2 Central Moments: 4.6.3 Relation between raw moments and central moments 4.7 Skewness and Kurtosis 4.7.1 Skewness 4.8 Box Plot 4.9 Summary 4.10 Keywords 4.11 Learning Activity 4.12 Unit End Questions 4.13 References 4.0 LEARNING OBJECTIVES After studying this unit, students will be able to, • Provides the importance of the concept of variability (dispersion) • Measures the spread or dispersion and Identifiers the causes of dispersion CU IDOL SELF LEARNING MATERIAL (SLM)

• Describes the spread - range and standard deviations • Describes the role of Skewness and Kurtosis • Explains about moments • Illustrates the procedure to draw Box plot. 4.1 INTRODUCTION The measures of central tendency describes the central part of values in the data set appears to concentrate around a central value called average. But these measures do not reveal how these values are dispersed (spread or scattered) on each side of the central value. Therefore while describing data set it is equally important to know how for the item in the data are close around or scattered away from the measures of central tendency. 4.2 CHARACTERISTICS OF A GOOD MEASURE OF DISPERSION An ideal measure of dispersion is to satisfy the following characteristics. (i) It should be well defined without any ambiguity. (ii) It should be based on all observations in the data set. (iii) It should be easy to understand and compute. (iv) It should be capable of further mathematical treatment. (v) It should not be affected by fluctuations of sampling. (vi) It should not be affected by extreme observations. 4.3 TYPES OF MEASURES OF DISPERSION Range, Quartile deviation, Mean deviations, Standard deviation and their Relative measures The measures of dispersion are classified in two categories, namely (i) Absolute measures (ii) Relative measures. Absolute Measures It involves the units of measurements of the observations. For example, (i) the dispersion of salary of employees is expressed in rupees, and (ii) the variation of time required for workers is expressed in hours. Such measures are not suitable for comparing the variability of the two data sets which are expressed in different units of measurements. 4.3.1 Range Raw Data: Range is defined as difference between the largest and smallest observations in the data set. Range(R) = Largest value in the data set (L) –Smallest value in the data 114 CU IDOL SELF LEARNING MATERIAL (SLM)

set(S) R= L – S Grouped Data: For grouped frequency distribution of values in the data set, the range is the difference between the upper-class limit of the last class interval and the lower-class limit of first-class interval. Coefficient of Range The relative measure of range is called the coefficient of range Co efficient of Range = (L- S) / (L+ S) Example The following data relates to the heights of 10 students (in cms) in a school. Calculate the range and coefficient of range. 158, 164, 168, 170, 142, 160, 154, 174, 159, 146 Solution: Example Calculate the range and the co-efficient of range for the marks obtained by 100 students in a school. Solution: Merits: 115 CU IDOL SELF LEARNING MATERIAL (SLM)

• Range is the simplest measure of dispersion. • It is well defined, and easy to compute. • It is widely used in quality control, weather forecasting, stock market variations etc. Limitations: • The calculations of range are based on only two values – largest value and smallest value. • It is largely influenced by two extreme values. • It cannot be computed in the case of open-ended frequency distributions. • It is not suitable for further mathematical treatment. 4.3.2 Inter Quartile Range and Quartile Deviation The quartiles Q1, Q2 and Q3 have been introduced and studied in Chapter 5. Inter quartile range is defined as: Inter quartile Range (IQR) = Q3 – Q1 Quartile Deviation is defined as, half of the distance between Q1 and Q3. Quartile Deviation Q.D = Q Q 3 1 -2 It is also called as semi-inter quartile range. Coefficient of Quartile Deviation: The relative measure corresponding to QD is coefficient of QD and is defined as: Coefficient of Quartile Deviation Merits: • It is not affected by the extreme (highest and lowest) values in the data set. • It is an appropriate measure of variation for a data set summarized in open ended class intervals. • It is a positional measure of variation; therefore it is useful in the cases of erratic or highly skewed distributions. Limitations: The QD is based on the middle 50 per cent observed values only and is not based on all the observations in the data set, therefore it cannot be considered as a good measure of variation. It is not suitable for mathematical treatment. It is affected by sampling fluctuations. 116 CU IDOL SELF LEARNING MATERIAL (SLM)

The QD is a positional measure and has no relationship with any average in the data set. 4.3.3. Mean Deviation The Mean Deviation (MD) is defined as the arithmetic mean of the absolute deviations of the individual values from a measure of central tendency of the data set. It is also known as the average deviation. The measure of central tendency is either mean or median. If the measure of central tendency is mean (or median), then we get the mean deviation about the mean (or median). Example The following are the weights of 10 children admitted in a hospital on a particular day. Find the mean deviation about mean, median and their coefficients of mean deviation. 7, 4, 10, 9, 15, 12, 7, 9, 9, 18 Solution: 117 CU IDOL SELF LEARNING MATERIAL (SLM)

4.3.4 Standard Deviation Consider the following data sets. It is obvious that the range for the three sets of data is 8. But a careful look at these sets clearly shows the numbers are different and there is a necessity for a new measure to address the real variations among the numbers in the three data sets. This variation is measured by standard deviation. The idea of standard deviation was given by Karl Pearson in 1893. Ungrouped data 1. Actual mean method: 2. Assumed mean method: Grouped Data (Discrete) Grouped Data (continuous) 118 CU IDOL SELF LEARNING MATERIAL (SLM)

Example The following data gives the number of books taken in a school library in 7 days find the standard deviation of the books taken 7, 9, 12, 15, 5, 4, 11 Solution: Actual mean method 119 CU IDOL SELF LEARNING MATERIAL (SLM)

Merits: The value of standard deviation is based on every observation in a set of data. It is less affected by fluctuations of sampling. It is the only measure of variation capable of algebraic treatment. Limitations: Compared to other measures of dispersion, calculations of standard deviation are difficult. While calculating standard deviation, more weight is given to extreme values and less to those near mean. It cannot be calculated in open intervals. If two or more data set were given in different units, variation among those data set cannot be compared. Example Raw Data: Weights of children admitted in a hospital is given below calculate the standard deviation of weights of children. 13, 15, 12, 19, 10.5, 11.3, 13, 15, 12, 9 Solution: 120 CU IDOL SELF LEARNING MATERIAL (SLM)

Example Find the standard deviation of the first ‘n’ natural numbers. Solution: The first n natural numbers are 1, 2, 3…, n. The sum and the sum of squares of these n numbers are Example: 121 CU IDOL SELF LEARNING MATERIAL (SLM)

The wholesale price of a commodity for seven consecutive days in a month is as follows: Calculate the variance and standard deviation. Solution: The computations for variance and standard deviation is cumbersome when x values are large. So, another method is used, which will reduce the calculation time. Here we take the deviations from an assumed mean or arbitrary value A such that d = x – A In this question, if we take deviation from an assumed A.M. =255. The calculations then for standard deviation will be as shown in below Table; Example The mean and standard deviation from 18 observations is 14 and 12 respectively. If an additional observation 8 is to be included, find the corrected mean and standard deviation. Solution: 122 CU IDOL SELF LEARNING MATERIAL (SLM)

The sum of the 18 observations is = n #x = 18 × 14 = 252. The sum of the squares of these 18 observations Example A study of 100 engineering companies gives the following information Calculate the standard deviation of the profit earned. Solution: A=35 C=10 123 CU IDOL SELF LEARNING MATERIAL (SLM)

4.4 COMBINED MEAN AND COMBINED STANDARD DEVIATION Combined arithmetic mean can be computed if we know the mean and numb of items in each group of the data 124 CU IDOL SELF LEARNING MATERIAL (SLM)

Example From the analysis of monthly wages paid to employees in two service organizations X and Y, the following results were obtained Which organization pays a larger amount as monthly wages? (ii) Find the combined standard deviation? Solution: (i) For finding out which organization X or Y pays larger amount of monthly wages, we have to compare the total wages: Total wage bill paid monthly by X and Y is Organization Y pays a larger amount as monthly wages as compared to organization X. (ii) For calculating the combined variance, we will first calculate the combined mean 125 CU IDOL SELF LEARNING MATERIAL (SLM)

4.5 RELATIVE MEASURES It is a pure number independent of the units of measurements. This measure is useful especially when the data sets are measured in different units of measurement. For example, suppose a nutritionist would like to compare the obesity of school children in India and England. He collects data from some of the schools in these two countries. The weight is normally measured in kilograms in India and in pounds in England. It will be meaningless, if we compare the obesity of students using absolute measures. So it is sensible to compare them in relative measures. 4.5.1 Coefficient of Variation The standard deviation is an absolute measure of dispersion. It is expressed in terms of units in which the original figures are collected and stated. The standard deviation of heights of students cannot be compared with the standard deviation of weights of students, as both are expressed in different units, i.e., heights in centimetre and weights in kilograms. Therefore the standard deviation must be converted into a relative measure of dispersion for the purpose of comparison. The relative measure is known as the coefficient of variation. If we want to compare the variability of two or more series, we can use C.V. The series or groups of data for which the C.V is greater indicate that the group is more variable, less stable, less uniform, less consistent or less homogeneous. If the C.V is less, 126 CU IDOL SELF LEARNING MATERIAL (SLM)

it indicates that the group is less variable, more stable, more uniform, more consistent or more homogeneous. Merits: The C.V is independent of the unit in which the measurement has been taken, but standard deviation depends on units of measurement. Hence one should use the coefficient of variation instead of the standard deviation. Limitations: If the value of mean approaches 0, the coefficient of variation approaches infinity. So, the minute changes in the mean will make major changes. Example If the coefficient of variation is 50 per cent and a standard deviation is 4, find the mean. Solution: Example The scores of two batsmen, A and B, in ten innings during a certain season, are as under: Find which of the batsmen is more consistent in scoring. Solution: 127 CU IDOL SELF LEARNING MATERIAL (SLM)

Example The weekly sales of two products A and B were recorded as given below Find out which of the two shows greater fluctuations in sales. Solution: For comparing the fluctuations in sales of two products, we will prefer to calculate coefficient of variation for both the products. Product A: Let A = 56 be the assumed mean of sales for product A 128 CU IDOL SELF LEARNING MATERIAL (SLM)

129 CU IDOL SELF LEARNING MATERIAL (SLM)

Since the coefficient of variation for product A is more than that of product B, Therefore the fluctuation in sales of product A is higher than product B. 4.6 MOMENTS 4.6.1 Raw moments: Raw moments can be defined as the arithmetic mean of various powers of deviations taken from origin. The rth Raw moment is denoted by nrl , r = 1,2,3... Then the first raw moments are given by 130 CU IDOL SELF LEARNING MATERIAL (SLM)

4.6.2 Central Moments: Central moments can be defined as the arithmetic mean of various powers of deviation taken from the mean of the distribution. The r th central moment is denoted by nr, r = 1, 2, 3... 131 CU IDOL SELF LEARNING MATERIAL (SLM)

4.6.3 Relation between raw moments and central moments Example The first two moments of the distribution about the value 5 of the variable, are 2 and 20.fnd the mean and the variance. Solution: 4.7 SKEWNESS AND KURTOSIS There are two other comparable characteristics called skewness and kurtosis that help us to understand a distribution. 132 CU IDOL SELF LEARNING MATERIAL (SLM)

4.7.1 Skewness Skewness means ‘lack of symmetry’. We study skewness to have an idea about the shape of the curve drawn from the given data. When the data set is not a symmetrical distribution, it is called a skewed distribution and such a distribution could either be positively skewed or negatively skewed. The concept of skewness will be clear from the following three diagrams showing a symmetrical distribution, a positively skewed distribution and negatively skewed distribution. We can see the symmetricity from the following (a) Symmetrical Distribution Mean = Median = Mode It is clear from the diagram below that in a symmetrical distribution the values of mean, median and mode coincide. The spread of the frequencies is the same on both sides of the centre point of the curve. (b) Positively Skewed Distribution In the positively skewed distribution, the value of the mean is maximum and that of mode is least – the median lies in between the two. In the positively skewed distribution, the frequencies are spread out over a greater range of values on the high-value end of the curve (the right-hand side) than they are on the low – value end. For a positively skewed distribution, Mean>Median> Mode. 133 CU IDOL SELF LEARNING MATERIAL (SLM)

(c) Negatively skewed distribution in a negatively skewed distribution the value of mode is maximum and that of mean least-the median lies in between the two. In the negatively skewed distribution the position is reversed, i.e., the excess tail is on the left-hand side. It should be noted that in moderately symmetrical distribution the interval between the mean and the median is approximately one-third of the interval between the mean and the mode. It is this relationship which provides a means of measuring the degree of skewness. (d). Some important Measures of Skewness (i) Karl-Pearson coefficient of skewness (ii) Bowley’s coefficient of skewness (iii) Coefficient of skewness based on moments (i) Karl – Person coefficient of skewness According to Karl-Pearson the absolute measure of skewness = Mean – Mode. Karl- Pearson coefficient of skewness Example From the known data, mean = 7.35, mode = 8 and Variance = 1.69 then find the Karl- Pearson coefficient of skewness. Bowley’s coefficient of skewness In Karl Pearson method of measuring skewness the whole of the series is needed. Prof. Bowley has suggested a formula based on position of quartiles. In symmetric 134 CU IDOL SELF LEARNING MATERIAL (SLM)

distribution quartiles will be equidistance from the median. Q2 – Q1 = Q3 – Q2, but in skewed distributions it may not happen. Hence Example If Q1 = 40, Q2 = 50, Q3 = 60, find Bowley’s coefficient of skewness Solution: iii) Measure of skewness based on Moments The Measure of skewness based on moments is denoted by β1 and is given by Example Find β1 for the following data n1 = 0, n2 = 8.76, n3 = -2.91 Solution: 4.7.2 Kurtosis Kurtosis in Greek means ‘bulginess’. In statistics kurtosis refers to the degree of flatness or peakedness in the region about the mode of a frequency curve. The degree of kurtosis of distribution is measured relative to the peakedness of normal curve. In other words, measures of kurtosis tell us the extent of which a distribution is more peaked or flat-topped than the normal curve. The following diagram illustrates the shape of three different curves mentioned below: If a curve is more peaked than the normal curve, it is called ‘leptokurtic’. In such a case items are more closely bunched around the mode. On the other hand if a curve is more flat-topped than the normal curve, it is called ‘platykurtic’. The bell shaped normal curve itself is known as ‘mesokurtic’. We can find how much the frequency curve is flatter than the normal curve using measure of kurtosis. 135 CU IDOL SELF LEARNING MATERIAL (SLM)

Measures of Kurtosis The most important measure of kurtosis is the value of the coefficient. It is defined as: Example Find the value of β2 for the following data Solution: 4.8 BOX PLOT A box plot can be used to graphically represent the data set. These plots involve five specific values: These values are called a five- number summary of the data set. A box plot is a graph of a data set obtained by drawing a horizontal line from the minimum data value to Q1 and a horizontal line from Q3 to the maximum data value, and drawing a box by vertical lines passing through Q1 and Q3, with a vertical line inside the box passing through the median or Q2. 4.8.1 Description of box plot (i) If the median is near the center of the box, the distribution is approximately symmetric (ii) If the median falls to the left of the center of the box, the distribution is positively skewed. 136 CU IDOL SELF LEARNING MATERIAL (SLM)

(iii) If the median falls to the right of the center of the box, the distribution is negatively skewed. (iv) If the lines are about the same length, the distribution is approximately symmetric (v) If the right line is larger than the left line. the distribution is positively skewed. (vi) If the left line is larger than the right line. the distribution is negatively skewed. Example The following data gives the Number of students studying in XI standard in 10 different schools 89,47,164,296,30,215,138,78,48, 39 construct a boxplot for the data. Solution: 137 CU IDOL SELF LEARNING MATERIAL (SLM)

Example Construct a box –whisker plot for the following data 96, 151, 167, 185, 200, 220, 246, 269, 238, 252, 297, 105, 123, 178, 202 Solution: 4.9 SUMMARY • The range is the difference between the largest and smallest observations. • The inter quartile range (IQR) is the difference between the upper and lower quartiles. • The variance is the average of the squares of the values of x-x • The standard deviation (SD) is the square root of the variance and has the same units as x • If a population is approximately symmetric in a sample the mean and the median will have similar values. Typically their values will also be close to that of the mode of the population (if there is one!) • A population that is not symmetric is said to be skewed. A distribution with a long ‘tail’ of high values is said to be positively skewed, in which case the mean is usually greater than the mode or the median. If it has a long tail of low values it is said to be negatively skewed, then the mean is likely to be the lowest of the three location measures of the distribution • Box plots (Box-whisker diagrams): indicate the least and greatest values together with the quartiles and the median. 138 CU IDOL SELF LEARNING MATERIAL (SLM)

4.10 KEYWORDS • Symmetrical Distribution: The spread of the frequencies is the same on both sides of the centre point of the curve. • Positively Skewed Distribution: In the positively skewed distribution the value of the mean is maximum and that of mode is least – the median lies in between the two. • Negatively Skewed Distribution: In a negatively skewed distribution the Mean value of mode is maximum and that of mean least-the median lies in between the two. • Kurtosis: Kurtosis in Greek means ‘bulginess’. In statistics kurtosis refers to the degree of flatness or peakedness in the region about the mode of a frequency curve. • Box Plot: A box plot is a graph of a data set obtained by drawing a horizontal line from the minimum data value to Q1 and a horizontal line from Q3 to the maximum data value. 4.11 LEARNING ACTIVITY 1. Calculate the total distance to be travelled by students if the college is situated at town A, at town C, or town E and also if it is exactly half way between A and E. ___________________________________________________________________________ ____________________________________________________________________ 2. Decide where, in your opinion, the college should be established, if there is only one student in each town. Does it change your answer? ___________________________________________________________________________ ____________________________________________________________________ 4.12 UNIT END QUESTIONS A. Descriptive Questions: Short Questions 1. What is dispersion? What are various measures of dispersion? 2. What is meant by relative measure of dispersion? Describe its uses. 3. Define mean deviation. How does it differ from standard deviation? 4. What is standard deviation? Explain its important properties? What is variance? 5. What are the measures of skewness? Long Questions 1. Explain dispersion and write their uses? 139 CU IDOL SELF LEARNING MATERIAL (SLM)

2. What are the requisites of a good measure of variation? 3. Explain how measures of central tendency and measures of variations are complementary to each other in the context of analysis of data. 4. Distinguish between absolute and relative measures of variation. Give a broad classification of the measures of variation. 5. Explain and illustrate how the measures of variation afford a supplement to averages in frequency distribution. B. Multiple choice questions 1. When a distribution is symmetrical and has one mode, the highest point on the curve is called the a. Mode b. Median c. Mean d. All of these 2. When referring to a curve tails to the left end, you would call it. a. Symmetrical b. Negatively skewed c. Positively skewed d. All of these 3. Disadvantages of using the range as a measure of dispersion include all of the following except a. It is heavily influenced by extreme values b. It can change drastically from one sample to the next c. It is difficult to calculate d. It is determined by only two points in the data set. 4. Which of the following is true? 140 a. The variance can be calculated for grouped or ungrouped data. CU IDOL SELF LEARNING MATERIAL (SLM)

b. The standard deviation can be calculated for grouped or ungrouped data. c. The standard deviation can be calculated for grouped or ungrouped data but the variance can be calculated only for ungrouped data. d. (a) and (b), but not (c). 5. The square root of the variance of a distribution is the a. Standard deviation b. Mean c. Range d. Absolute deviation Answer 1) d 2) b 3) c 4) d 5) a 4.13 REFERENCES Textbooks / Reference Books • T1: Levine, D., Sazbat, K. and Stephan, D. 2013. Business Statistics, 7thEdition, Pearson Education, India, ISBN: 9780132807265. • T2; Gupta, C. and Gupta, V. 2004. An Introduction to Statistical Methods, 23rdEdition, Vikas Publications, India, ISBN: 9788125916543. • R1: Croucher, J. 2011. Statistics: Making Business Decisions, 13thEdition, Tata McGraw Hill, ISBN: 9780074710419. • R2 Gupta, S. 2011. Statistical Methods, 4thEdition, Sultan Chand & Sons, ISBN: 8180548627. 141 CU IDOL SELF LEARNING MATERIAL (SLM)

UNIT 5 CORRELATION ANALYSIS Structure 5.0 Learning Objectives 5.1 Introduction 5.2 Types of Correlation 5.3 Scatter Diagram 5.4 Karl Pearson’s Correlation Coefficient 5.5 Summary 5.6 Keywords 5.7 Learning Activity 5.8 Unit End Questions 5.9 References 5.0 LEARNING OBJECTIVES After studying this unit, students will be able to: • Learn the meaning, definition and the uses of correlation. • Identify the types of correlation. • State the correlation coefficient for different types of measurement scales. • Differentiate different types of correlation using scatter diagram. • Calculate Karl Pearson’s coefficient of correlation, • Interpret the given data with the help of coefficient of correlation. 5.1 INTRODUCTION Karl Pearson (1857-1936) was an English Mathematician and Biostatistician. He founded the world’s first university statistics department at University College, London in 1911. The linear correlation coefficient is also called Pearson product moment correlation coefficient. It was developed by Karl Pearson with a related idea by Francis Galton. It is the first of the correlation measures developed and commonly used. Charles Edward Spearman (1863-1945) was an English psychologist and, after serving 15 years in Army he joined to study PhD in Experimental Psychology and obtained his degree in 1906. Spearman was strongly influenced by the work of Galton and developed rank correlation in 1904. He also pioneered factor analysis in statistics. 142 CU IDOL SELF LEARNING MATERIAL (SLM)

“When the relationship is of a quantitative nature, the appropriate statistical tool for discovering the existence of relation and measuring the intensity of relationship is known as correlation” —CROXTON AND COWDEN “When the relationship is of a quantitative nature, the appropriate statistical tool for discovering the existence of relation and measuring the intensity of relationship is known as correlation” —CROXTON AND COWDEN The statistical techniques discussed so far are for only one variable. In many research situations one has to consider two variables simultaneously to know whether these two variables are related linearly. If so, what type of relationship that exists between them. This leads to bivariate (two variables) data analysis namely correlation analysis. If two quantities vary in such a way that movements ( upward or downward) in one are accompanied by the movements( upward or downward) in the other, these quantities are said to be co-related or correlated. The correlation concept will help to answer the following types of questions. • Whether study time in hours is related with marks scored in the examination? • Is it worth spending on advertisement for the promotion of sales? • Whether a woman’s age and her systolic blood pressure are related? • Is age of husband and age of wife related? • Whether price of a commodity and demand related? • Is there any relationship between rainfall and production of rice? Correlation is a statistical measure which helps in analyzing the interdependence of two or more variables. In this chapter the dependence between only two variables are considered. 1. A.M. Tuttle defines correlation as: “An analysis of the co-variation of two or more variables is usually called correlation” 2. Ya-kun-chou defines correlation as: “The attempts to determine the degree of relationship between variables”. Correlation analysis is the process of studying the strength of the relationship between two related variables. High correlation means that variables have a strong linear relationship with each other while a low correlation means that the variables are hardly related. The type and intensity of correlation is measured through the correlation analysis. The measure of correlation is the correlation coefficient or correlation index. It is an absolute measure. 143 CU IDOL SELF LEARNING MATERIAL (SLM)

Uses of correlation • Investigates the type and strength of the relationship that exists between the two variables. • Progressive development in the methods of science and philosophy has been characterized by the rich knowledge of relationship. • Correlation is very important in the field of Psychology and Education as a measure of relationship between test scores and other measures of performance. • With the help of correlation, it is possible to have a correct idea of the working capacity of a person. • With the help of it, it is also possible to have a knowledge of the various qualities of an individual. • After finding the correlation between the two qualities or different qualities of an individual, it is also possible to provide his vocational guidance. • In order to provide educational guidance to a student in selection of his subjects of study, correlation is also helpful and necessary. 5.2 TYPES OF CORRELATION 1. Simple (Linear) correlation (2 variables only): The correlation between the given two variables. It is denoted by rxy 2. Partial correlation (more than 2 variables): The correlation between any two variables while removing the effect of other variables. It is denoted by rxy.z … 3. Multiple correlation (more than 2 variables): The correlation between a group of variables and a variable which is not included in that group. It is denoted by Ry. (xz…) Here, we are dealing with data involving two related variables and generally we assign a symbol ‘x’ to scores of one variable and symbol ‘y’ to scores of the other variable. There are five types in simple correlation. They are, 1. Positive correlation (Direct correlation) 2. Negative correlation (Inverse correlation) 3. Uncorrelated 4. Perfect positive correlation 5. Perfect negative correlation 1) Positive correlation: (Direct correlation) 144 CU IDOL SELF LEARNING MATERIAL (SLM)

The variables are said to be positively correlated if larger values of x are associated with larger values of y and smaller values of x are associated with smaller values of y. In other words, if both the variables are varying in the same direction then the correlation is said to be positive. In other words, if one variable increases, the other variable (on an average) also increases or if one variable decreases, the other (on an average) variable also decreases. For example, i) Income and savings ii) Marks in Mathematics and Marks in Statistics. (i.e., Direct relationship pattern exists). 2) Negative correlation: (Inverse correlation) The variables are said to be negatively correlated if smaller values of x are associated with larger values of y or larger values x are associated with smaller values of y. That is the variables varying in the opposite directions is said to be negatively correlated. In other words, if one variable increases the other variable decreases and vice versa. For example, i) Price and demand ii) Unemployment and purchasing power 3) Uncorrelated: The variables are said to be uncorrelated if smaller values of x are associated with smaller or larger values of y and larger values of x are associated with larger or smaller values of y. If the two variables do not associate linearly, they are said to be uncorrelated. Here r = 0. 145 CU IDOL SELF LEARNING MATERIAL (SLM)

Important note: Uncorrelated does not imply independence. This means “do not interpret as the two variables are independent instead interpret as there is no specific linear pattern exists but there may be nonlinear relationship”. 4) Perfect Positive Correlation If the values of x and y increase or decrease proportionately then they are said to have perfect positive correlation. 5) Perfect Negative Correlation If x increases and y decreases proportionately or if x decreases and y increases proportionately, then they are said to have perfect negative correlation. Correlation Analysis The purpose of correlation analysis is to find the existence of linear relationship between the variables. However, the method of calculating correlation coefficient depends on the types of measurement scale, namely, ratio scale or ordinal scale or nominal scale. Methods to find correlation 1. Scatter diagram 2. Karl Pearson’s product moment correlation coefficient: ‘r’ 5.3 SCATTER DIAGRAM A scatter diagram is the simplest way of the diagrammatic representation of bivariate data. One variable is represented along the X-axis and the other variable is represented along the Y- axis. The pair of points are plotted on the two dimensional graph. The diagram of points so obtained is known as scatter diagram. The direction of flow of points shows the type of correlation that exists between the two given variables. 1) Positive correlation 146 CU IDOL SELF LEARNING MATERIAL (SLM)

If the plotted points in the plane form a band and they show the rising trend from the lower left hand corner to the upper right hand corner, the two variables are positively correlated. 2) Negative correlation If the plotted points in the plane form a band and they show the falling trend from the upper left hand corner to the lower right hand corner, the two variables are negatively correlated. 3) Uncorrelated If the plotted points spread over in the plane then the two variables are uncorrelated. 4) Perfect positive correlation If all the plotted points lie on a straight line from lower left hand corner to the upper right hand corner then the two variables have perfect positive correlation. 5) Perfect Negative correlation If all the plotted points lie on a straight line falling from upper left hand corner to lower right hand corner, the two variables have perfect negative correlation. 147 CU IDOL SELF LEARNING MATERIAL (SLM)

Merits of scatter diagram • It is a simple and non-mathematical method of studying correlation between the variables. • It is not influenced by the extreme items • It is the first step in investigating the relationship between two variables. • It gives a rough idea at a glance whether there is a positive correlation, negative correlation or uncorrelated. Demerits of scatter diagram • We get an idea about the direction of correlation but we cannot establish the exact strength of correlation between the variables. • No mathematical formula is involved. 5.4 KARL PEARSON’S CORRELATION COEFFICIENT When there exists some relationship between two measurable variables, we compute the degree of relationship using the correlation coefficient. Co-variance Let (X, Y) be a bivariable normal random variable where V(X) and V(Y) exists. Then, covariance between X and Y is defined as COV (X, Y) = E[(X-E(X)) (Y-E(Y))] = E(XY) – E(X)E(Y) If (xi, yet), i=1,2, ..., n is a set of n realisations of (X, Y), then the sample covariance between X and Y can be calculated from 148 CU IDOL SELF LEARNING MATERIAL (SLM)

Karl Pearson’s coefficient of correlation When X and Y are linearly related and (X, Y) has a bivariate normal distribution, the co-efficient of correlation between X and Y is defined as This is also called as product moment correlation co-efficient which was defined by Karl Pearson. Based on a given set of n paired observations (xi, yi), i=1,2, ... n the sample correlation co-efficient between X and Y can be calculated from Properties 1. The correlation coefficient between X and Y is same as the correlation coefficient between Y and X (i.e., rxy = ryx). 2. The correlation coefficient is free from the units of measurements of X and Y 3. The correlation coefficient is unaffected by change of scale and origin. 149 CU IDOL SELF LEARNING MATERIAL (SLM)