CU-BCA-SEM-III-PROBABILITY AND STATICS- Second Draft-converted

Median = Size of N+1th item = 122+ 1 = 61.5th item. 22 Size of 61.5th item = Rs. 5,060. Hence, the median income is Rs. 5,060. Example 4: Find the mean, median, mode, and range for the following list of values: 13, 18, 13, 14, 13, 16, 14, 21, 13 Solution: The mean is the usual average, so we’ll add and then divide: (13 + 18 + 13 + 14 + 13 + 16 + 14 + 21 + 13) ÷ 9 = 15 Note that the mean, in this case, isn’t a value from the original list. This is a common result. You should not assume that your mean will be one of your original numbers. The median is the middle value, so first we’ll have to rewrite the list in numerical order: 13, 13, 13, 13, 14, 14, 16, 18, 21 There are nine numbers in the list, so the middle one will be the (9 + 1) ÷ 2 = 10 ÷ 2 = 5th number: 13, 13, 13, 13, 14, 14, 16, 18, 21 So, the median is 14. The mode is the number that is repeated more often than any other, so 13 is the mode, since 13 is being repeated 4 times. The largest value in the list is 21, and the smallest is 13, so the range is 21 – 13 = 8. Example 5: The heights (in cm) of 11 players of a team are as follows: 160, 158, 158, 159, 160, 160, 162, 165, 166, 167, 170. Solution: Arranging the variates in the ascending order, we get 157, 158, 158, 159, 160, 160, 162, 165, 166, 167, 170. The number of variates = 11, which is odd. Therefore, median = 11+1211+12th variate = 6th variate = 160. Example 6: Find the median of the first five odd integers. If the sixth odd integer is also included, find the difference of medians in the two cases. Solution: Writing the first five odd integers in ascending order, we get 1, 3, 5, 7, 9. The number of variates = 5, which is odd. Therefore, median = 5+12th variate = 3th variate = 5. 51 CU IDOL SELF LEARNING MATERIAL (SLM)

When the sixth integer is included, we have (in ascending order) 1, 3, 5, 7, 9, 11. Now, the number of variates = 6, which is even. Therefore, median = mean of 62th and (62 + 1) th variates = Mean of 3rd and 4th variates = Mean of 5 and 7 = 5+72 = 6. Therefore, the difference of medians in the two cases = 6 - 5 = 1. Example 7: If the median of 17, 13, 10, 15, x happens to be the integer x then find x. Solution: There are five (odd) variates. So, 5+125+12th variate, i.e., 3rd variate when written in ascending order will the median x. So, the variates in ascending order should be 10, 13, x, 15, 17. Therefore, 13 < x < 15. But x is an integer. So, x = 14. Example 8: The marks obtained by 20 students in a class test are given below. Marks Obtained 6 7 8 9 10 Number of Students 58421 Find the median of marks obtained by the students. Solution: Arranging the variates in ascending order, we get 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 10. The number of variates = 20, which is even. Therefore, median = mean of 202th and (202 + 1) th variate = mean of 10th and 11th variate = mean of 7 and 7 = 7+72 = 7. 3.4 GROUPED DATA Grouped Data: When the data is continuous and in the form of a frequency distribution, the median is found as shown below: Step 1: Find the median class. 52 CU IDOL SELF LEARNING MATERIAL (SLM)

Let n = total number of observations i.e., ∑fi Note: Median Class is the class where n/2 lies. Step 2: Use the following formula to find the median. were, l=lower limit of median class c= cumulative frequency of the class preceding the median class f=frequency of the median class h=class size Example 1: Find the median marks for the following distribution: Classes 0-10 10-20 20-30 30-40 40-50 Frequency 2 12 22 8 6 Solution: We need to calculate the cumulative frequencies to find the median. Calculation table: Classes Number of students Cumulative frequency 0-10 2 2 10-20 12 2 + 12 = 14 20-30 22 14 + 22 = 36 30-40 8 36 + 8 = 44 40-50 6 44 + 6 = 50 h=10 N=50 N/2=50/2=25 Median Class =20−30 l=20, f=22, c. f=14, Using Median formula: 53 CU IDOL SELF LEARNING MATERIAL (SLM)

∴ Median = 25 Example 2: Calculate the median for the following frequency distribution: Marks No. of Students 45-50 10 40-45 15 35-40 26 30-35 30 25-30 42 20-25 31 15-20 24 10-15 15 5-10 7 Solution: First arrange the data in ascending order and then find out median. Calculation of Median Marks f c.f Marks f c.f. 5-10 77 30-35 30 149 54 CU IDOL SELF LEARNING MATERIAL (SLM)

10-15 15 22 35-40 26 175 15-20 24 46 40-45 15 190 20-25 31 T7 45-50 10 200 25-30 42 119 Med. = Size of N item = 200= 100th item No. of Apples 22 14 Median lies in the class 25-30 20 Median = L+N/2 – c.f x i 42 f L=25 N/2=100 c.f. =77 F=42 I=5 =25+100-77 x 5 42 =25+2.74 = 27.74 Example 3: Calculate the median from the following data: Weight (In gms.) 410-419 420-429 430-439 55 CU IDOL SELF LEARNING MATERIAL (SLM)

440-449 54 450-459 45 460-469 18 470-479 7 --------------------------------------------------------------------------------------------------- Solution: Since we are given inclusive class intervals, we should\" convert it to the exclusive one by deducting 0.5 from the lower limits and adding 0.5 to the upper limits. Weight f c.f. 409.5-419.5 14 14 419.5-429.5 20 34 429.5-439.5 42 76 439.5-449.5 54 130 449.5-459.5 45 175 459.5-469.5 18 193 469.5-479.5 7 200 N =200 L+ N/2 – c.f x i Median = f L = 439.5, N/2 = 100, c.f. = 76, f = 54, i = 10 Med. = 439.5 + 100-76 x 10 = 439.5+4.44 = 443.94 56 54 3.5 MERITS AND LIMITATIONS OF MEDIAN The merits and Demerits are as follows: CU IDOL SELF LEARNING MATERIAL (SLM)

Merits: • It is especially useful in case of open-end classes since only the position and not the values of items must be known. The median is also recommended if the distribution has unequal classes. Since it is easier to compute than the mean. • Extreme values do not affect the median as strongly as they do the mean. For example, the median of 10. 20. 30. 40 and 150 would be 30 whereas the mean 50. Hence very often when extreme values are present in a set of observations, the median is a more satisfactory measure of the central tendency than the mean. • In markedly skewed distributions such as income distributions or price distributions where the arithmetic mean would be distorted by extreme values, the median is especially useful. Consequently, the median income~ for some purposes be regarded as a more 'representative figure, for half the income earners must be receiving at least the median income. One can say as many receive the mediaI1 income and as many do not. • It is the most appropriate average in dealing with qualitative data, i.e., where ranks are given or there are other types of items that are not counted or measured but are scored. • The value of median can be determined graphically whereas the value of mean cannot be graphically ascertained. • Perhaps the greatest advantage of median is, however, the fact that the median actually does indicate what many people incorrectly believe the arithmetic mean indicates. The median indicates the value of the middle item in the distribution. This is a clear-cut meaning and makes the median a measure that can be easily explained. 3.6 LIMITATIONS • For Calculating median, it is necessary to arrange the data; other averages do not need any arrangement. • Since it is a positional average, its value is not determined by each and every observation. • It is not capable of algebraic treatment. For example, median cannot be used for determining the combined median of two or more groups as is possible in case of mean. Similarly, the median wage of a skewed distribution times the number of workers will not give the total payroll. Because of this limitation the median is much less popular as compared to the arithmetic mean. 57 CU IDOL SELF LEARNING MATERIAL (SLM)

• The value of median is affected more by sampling fluctuations than the value of the arithmetic mean. • The median, in some cases, cannot be computed exactly as the mean. When the -number of items included in a series of data is even, the median is determined approximately as the mid-point of the two middle items. • It is erratic if the number of items is small. Usefulness: The median is useful for distributions containing open-end intervals since these intervals do not enter its computation. Also, since the median is affected by the number rather than the size of items, it is frequently used instead of the mean as a measure of central tendency in cases where such values are likely to distort the mean. 3.7 SUMMARY • The middle number; found by ordering all data points and picking out the one in the middle or if there are two middle numbers, taking the mean of those two numbers. For Example, the median of 4, 1, and 7 is 4 because when the numbers are put in order (1, 4, 7), the number 4 is in the middle. • Median is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. • The basic feature of the median in describing data compared to the mean is that it is not skewed by a small proportion of extremely large or small values, and therefore provides a better representation of a \"typical\" value. 3.8 KEYWORDS • Absolute frequency: A particular value appears during a trial or set of trials. • Median: The class where the cumulative frequency reaches half the sum of the absolute frequencies • Cumulative frequency: The total of a frequency and all frequencies so far in a frequency distribution. It is the 'running total' of frequencies • Interval data: It is always appearing in the form of numbers or numerical values where the distance between the two points is standardized and equal. 58 CU IDOL SELF LEARNING MATERIAL (SLM)

3.9 LEARNING ACTIVITY Math students are more engaged and participative learners when they are actively engaged with what they are learning. And one of the best ways to achieve this high level of engagement is through the use of hands-on activities. So, when it comes time to help students gain a deep conceptual understanding of measures of central tendency (mean, median, mode, and range), you'll need a fun and engaging hands -on group activity. 1. Find the median of 2.2, 10.2, 14.7, 5.9, 4.9, 11.1, 10.5. 2. Find the median of a series of all the even terms from 4 to 296. _______________________________________________________________________________ ________________________________________________________________ 3.10 UNIT END QUESTIONS A. Descriptive Questions Short Questions 1. Find the median of 2.2, 10.2, 14.7, 5.9, 4.9, 11.1, 10.5. 2. What is a real-life example of median? 3. What is mean median mode example? 4. The runs scored in a cricket match by 11 players is as follows: 7, 16, 121, 51, 101, 81, 1, 16, 9, 11, 16. What is the median of this data? 5. What is the median of 12, 11, 14, 10, 8, 13, 11, 9? 6. Find the median of the following data: 25, 34, 31, 23, 22, 26, 35, 29, 20, 32. Long Questions 1. A survey on the weight (in cm) of 50 girls of class X was conducted at a school and the following data was obtained: Weight (in cm) 20-30 30-40 40-50 50-60 60-70 Total 59 CU IDOL SELF LEARNING MATERIAL (SLM)

Number of girls 28 12 20 8 50 Find the median of the above data. 2. Find the median marks for the following distribution: Classes 0-10 10-20 20-30 30-40 40-50 8 6 Frequency 2 12 22 3. The following observations are arranged in ascending order. 17, x, 24, x + 7, 35, 36, 46. The median of the data is 25 find the value of x. 4. If the mean of five observations x, x + 2, x + 4, x + 6, x + 8 is 13, find the value of x and hence find the mean of the last three observations. 5. There are 3,500 people in group A and 5,000 people in group B: Car Type % in Group A % in Group B Who Own Who Own Motorbike 4 9 Sedan 35 25 Minivan 22 15 Van 9 12 Coupe 3 6 What is the median of the number of people in group B who own either a minivan, van, or coupe? B. Multiple choice Questions 60 1. The median of a frequency distribution is found graphically with the help of: a. Histogram b. Frequency curve CU IDOL SELF LEARNING MATERIAL (SLM)

c. Frequency polygon d. Ogive 2. Find the median of the given: a. 150 b. 200 c. 148 d. 175 3. The sum of squares of the deviations is minimum, when deviations are taken from a. Mode b. Zero c. Mean d. Median 4. Calculate mean deviation from mean for the following data: 100, 150, 200, 250, 360, 490, 500, 600, and 671 a. 172.56 b. 174.44 c. 178.23 d. 173.45 61 CU IDOL SELF LEARNING MATERIAL (SLM)

5. The positive square-root of the arithmetic mean of the Square of the deviations of the given observation from their arithmetic mean is called a. Mean deviation b. Standard deviation c. Quartile deviation d. Variance Answers 1.d, 2.d, 3.c, 4.b, 5.a 3.11 REFERENCES Reference Books: • Dr. B. Krishna Gandhi, Dr. T.K.V Iyengar, M.V.S.S.N. Prasad, Probability and Statistics. Chand Publishing Co. • Quantitative Techniques for Decision Making by A. Sharma, Publisher: HPH, 2007 Edition. • Statistical Methods by S.P Gupta, Publisher: Sultan Chand & Sons, 2008 Edition. • Research Methodology by C. R. Kothari, Publisher: Vikas Publishing House Textbooks: • S.C. Gupta, V.K. Kapoor, Fundamental of Mathematical Statistics, Sultan Chand and Company. • Research Methodology and Statistical Techniques by Santosh Gupta, Publisher: Deep and Deep Publication • Research Methodology by V. P. Pandey, Publisher: Himalaya Publication 62 CU IDOL SELF LEARNING MATERIAL (SLM)

UNIT 4: MODE AND MEAN DEVIATION Structure 4.0 Learning Objectives 4.1 Introduction 4.2 Mode 4.3 Ungrouped Data 4.4 Grouped Data 4.5 Relation between Mean, Median and Mode 4.6 Merits and Limitations of Mode 4.7 Mean Deviation 4.8 Mean Deviation for Frequency Distribution 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 • Explain the purposes of measures of central tendency. • Differentiate and calculate the measures of central tendency (mode, median, mean) for a set of data. • Identify the mode from a frequency distribution table or figure. • Explain why measures of dispersion must be reported in addition to measures of central tendency. • Evaluate how to apply the correct measure(s) of dispersion to any given variable based on that variable’s level of measurement. • Explain the normal curve and the concept behind two-thirds of cases being within one standard deviation of the mean. 63 CU IDOL SELF LEARNING MATERIAL (SLM)

4.1 INTRODUCTION The mode or the modal value is that value in a series of observations which occurs with the greatest frequency. For example, the mode of the series 3, 5, 8, 5. 4, 5, 9, 3 would be 5. Since this value occurs more frequently than any of the others. The mode is often said to be that value which. occurs most often in the data, that is, with the highest frequency. While this statement is quite helpful in interpreting the mode, it cannot safely be applied to any distribution because of the vagaries of sampling. Even fairly large samples drawn from a statistical population with a single well-defined mode may exhibit very erratic fluctuations in this average if the mode is defined as that exact value in the ungrouped data of each sample which occurs most frequently. Rather it should be thought as the value about which the terns are most closely concentrated. It is the value which has the greatest frequency density in its immediate neighbourhood. For this reason, mode is also called the most typical or fashionable value of a distribution. The following diagram shows the modal value: Fig 4.1 Mode 4.2 MODE The value of the variable at which the curve reaches a maximum is called the mode. It is the value around which the items tend to be most easily concentrated. Although mode is that value which occurs most frequently, yet it does to follow that its frequency represents a majority out of all the total number of frequencies. For example, in the election of college president the votes obtained by three candidates contesting for president ship out of total of 816 votes polled are as follows: Mr. X 268; Mr. Y 278; Mr. Z 270: Total 816. 64 CU IDOL SELF LEARNING MATERIAL (SLM)

Mr. Y will be elected as president because he has obtained highest votes. But it will be wrong to say that he represents majority because there are more votes against him (268 + 270 = 538) than those for him. There are many situations in which arithmetic mean and median fail to reveal the true characteristic of data. For example, when we talk of most common wage, most common income, most common height, most common size of shoe or ready-made garments we have in mind mode and not the arithmetic mean or median discussed earlier. The mean does not always provide an accurate reflection of the data due to the presence of extreme items. Median may also prove to be quite unrepresentative of the data owing to an uneven distribution of the series. For example, the values in the lower half of a distribution range from, say, Rs. 10 to Rs. 100, while the same numbers of items in the upper half of the series range from Rs. 100 to Rs. 6,000 with most of them near the higher limit. In such a distribution the median value of Rs. 100 will provide little indication of the true nature of the data. Both these shortcomings may be overcome by the use of mode which refers to the value which occurs most frequently in a distribution. Moreover, mode is the easiest to compute since it is the value corresponding to the highest frequency. For example, if the data are: Size of shoes 5 6 7 8 9 10 11 No. of persons 10 20 25 40 22 15 6 The modal size is '8' since it appears maximum number of times in the series. 4.3 UNGROUPED DATA For ungrouped data, we just need to identify the observation which occurs maximum times. Mode = Observation with maximum frequency For example, in the data: 6, 8, 9, 3, 4, 6, 7, 6, 3 the value 6 appears the greatest number of times. Thus, mode = 6. An easy way to remember mode is: Most Often Data Entered. Note: A data may have no mode, 1 mode or more than 1 mode. Depending upon the number of modes the data has, it can be called unimodal, bimodal, trimodal or multimodal. The example discussed above has only 1 mode, so it is unimodal. 65 CU IDOL SELF LEARNING MATERIAL (SLM)

Example 1: Find the mode of the given data set: 3, 3, 6, 9, 15, 15, 15, 27, 27, 37, 48. Solution: In the following list of numbers, 3, 3, 6, 9, 15, 15, 15, 27, 27, 37, 48 15 is the mode since it is appearing a greater number of times in the set compared to other numbers. Example 2: Find the mode of 4, 4, 4, 9, 15, 15, 15, 27, 37, 48 data set. Solution: Given: 4, 4, 4, 9, 15, 15, 15, 27, 37, 48 is the data set. As we know, a data set or set of values can have more than one mode if more than one value occurs with equal frequency and number of times compared to the other values in the set. Hence, here both the number 4 and 15 are modes of the set. Example 3: Find the mode of 3, 6, 9, 16, 27, 37, 48. Solution: If no value or number in a data set appears more than once, then the set has no mode. Hence, for set 3, 6, 9, 16, 27, 37, 48, there is no mode available. 4.4 GROUPED DATA 66 When the data is continuous, the mode can be found using the following steps: Step 1: Find modal class i.e., the class with maximum frequency. CU IDOL SELF LEARNING MATERIAL (SLM)

Step 2: Find mode using the following formula: Where, l= lower limit of modal class, fm= frequency of modal class, f1= frequency of class preceding modal class, f2= frequency of class succeeding modal class, h= class width Example 1: Find the mode of the given data: Marks Obtained 0-20 20-40 40-60 60-80 80-100 Number of students 5 10 12 6 3 Solution: The highest frequency = 12, so the modal class is 40-60. l= lower limit of modal class = 40 fm= frequency of modal class =12 f1=frequency of class preceding modal class = 10 f2=frequency of class succeeding modal class = 6 h=class width = 20 Using the mode formula, 67 CU IDOL SELF LEARNING MATERIAL (SLM)

∴ Mode = 45 Example 2: In a class of 30 students marks obtained by students in mathematics out of 50 is tabulated as below. Calculate the mode of data given. Solution: The maximum class frequency is 12 and the class interval corresponding to this frequency is 20 – 30. Thus, the modal class is 20 – 30. Lower limit of the modal class (l) = 20 Size of the class interval (h) = 10 Frequency of the modal class (f1) = 12 Frequency of the class preceding the modal class (f0) = 5 Frequency of the class succeeding the modal class (f2) = 8 Substituting these values in the formula we get; 4.5 RELATION BETWEEN MEAN, MEDIAN AND MODE A distribution in which the values of mean, median and mode coincide (i.e., mean = median = mode) is known as a symmetrical distribution. Conversely stated, when the values of mean, median and mode are not equal the distribution is known as asymmetrical or skewed. In moderately skewed 68 CU IDOL SELF LEARNING MATERIAL (SLM)

or asymmetrical distributions, a very important relationship exists among mean, median and mode. In such distributions the distance between the mean and the median is about one-third the distance between the mean and the mode as will be clear from the diagram given below. Fig 4.2 Relation between Mean, Median and Mode Relationship among Mean, Median and Mode The three measures of central values i.e., mean, median and mode are closely connected by the following relations (called an empirical relationship). 2Mean + Mode =3Median Mode = Mean - 3 [Mean - Median] Mode = 3 Median - 2 Mean and Median = Mode + 2/3 [Mean - Mode] For instance, if we are asked to calculate the mean, median and mode of a continuous grouped data, then we can calculate mean and median using the formulae as discussed in the previous sections and then find mode using the empirical relation. For example, we have a data whose mode = 65 and median = 61.6. Then, we can find the mean using the above relation. 2Mean+Mode=3 Median ∴ 2Mean=3×61.6−65 ∴ 2Mean=119.8⇒Mean=119.8/2⇒Mean=59.9 Example 1 If the mean of the following data is 20.6, find the missing frequency (p). x 10 15 20 25 35 69 CU IDOL SELF LEARNING MATERIAL (SLM)

f3 10 p7 5 Solution: xifi 10 x 3 = 30 Let us make the calculation table for this: 15 x 10 = 150 20 x p = 20p xi fi 25 x 7= 175 35 x 5 = 175 10 3 ∑fixi = 530+20p 15 10 20 p 25 7 35 5 Total: ∑fi=25+p ∴ The missing frequency (p) =25 Example 2 The mean of 5 numbers is 18. If one number is excluded, their mean is 16. Find the excluded number. Solution: Given, n = 5, x¯ = 18 70 CU IDOL SELF LEARNING MATERIAL (SLM)

Thus, the total of 5 numbers = 90 Let the excluded number be a. Therefore, total of 4 numbers=90−a Mean of 4 numbers=(90−a)/4 16=(90−a)/4 90−a=64 ∴ a=26 Example 3 A survey on the heights (in cm) of 50 girls of class X was conducted at a school and the following data was obtained: Height (in cm) 120-130 130-140 140-150 150-160 160-170 Total Number of girls 2 8 12 20 8 50 Find the mode and median of the above data. Solution: Modal class=150−160 [as it has maximum frequency] To find the median, we need cumulative frequencies. Consider the table: Class Intervals No. of girls (fi) Cumulative frequency (c) 71 CU IDOL SELF LEARNING MATERIAL (SLM)

120-130 2 2 130-140 8 2+8=10 140-150 150-160 12 = f1 10+12=22 (c) 160-170 20 = fm 22+20=42 8 = f2 42+8=50 (n) ∴ Mode = 154, Median= 151.5 Example 4: a) In a moderately asymmetrical distribution, the mode and mean are 32.1 and 35.4 respectively. Find out the value of Median. Solution: (a) Mode = 3 Median - 2 Mean Given mean = 35.4, mode = 32.1 32.1 = 3 median - 2 x 35.4 3 medians = 32.1 + 70.8 = 102.9 or median = 102.9/3 = 34.3 b) Given median = 20.6, mode = 26, find mean. 72 CU IDOL SELF LEARNING MATERIAL (SLM)

Solution: (b) Mode = 3 Median - 2 Mean Given mean = 20.6, mode = 26 26 = 3 x 20.6 - 2 mean 26 = 61.8 - 2 mean. 2 mean = 61.8 - 26 = 35.8 or Mean = 17.9 Example 5: Calculate arithmetic mean, median and mode from the following frequency distribution: Variable Frequency 10-13 8 13-16 15 16-19 27 19-22 51 22-25 75 25-28 54 28-31 36 31-34 18 34-37 9 37-40 7 Solution: CALCULATION OF MEAN MEDIAN & MODE Variable m.p. (m-23.5)/3 F fd c.f. m D 10-13 11.5 -4 8 -32 8 13-16 14.5 -3 15 -45 23 16-19 17.5 -2 27 -54 50 73 CU IDOL SELF LEARNING MATERIAL (SLM)

19-22 20.5 -1 51 -51 101 22-25 23.5 0 75 0 176 25-28 26.5 +1 54 +54 230 28-31 29.5 +2 36 +72 266 31-34 32.5 +3 18 +54 284 34-37 35.5 +4 9 +36 293 37-40 38.5 +5 7 +35 300 N=300 ∑fd = 69 Mean X = A + ∑fd x i N = 23.5 + 69 x 3 = 24.19 300 Med. = Size of (N/2) th item = 300/2 = 150th item Median lies in the class 22 - 25. Med. = L + (N/2 - c.f.) x i f = 22 + 150-101 x 3 75 = 22 + 1.96 = 23.96 Difference between Mean and Average The term average is frequently used in everyday life to denote a value that is typical for a group of quantities. Average rainfall in a month or the average age of employees of an organization is typical examples. We might read an article stating \"People spend an average of 2 hours every day on social media”. We understand from the use of the term average that not everyone is spending 2 hours a day on social media but some spend more time and some less. However, we can understand from the term average that 2 hours is a good indicator of the amount of time spent on social media per day. Most people use average and mean interchangeably even though they are not the same. 74 CU IDOL SELF LEARNING MATERIAL (SLM)

• Average is the value which indicates what is most likely to be expected. • They help to summarise large data into a single value. An average tends to lie centrally with the values of the observations arranged in ascending order of magnitude. So, we call an average measure of the central tendency of the data. Averages are of different types. What we refer to as mean i.e., the arithmetic mean is one of the averages. Mean is called as the mathematical average whereas median and mode are positional averages. Fig 4.3 Central tendency of the data Difference between Mean and Median Mean is known as the mathematical average whereas median is known as the positional average. To understand the difference between the two, consider the following example. A department of an organization has 5 employees which include a supervisor and four executives. The executives draw a salary of Rs10,000 per month while the supervisor gets Rs 40,000. Mean, = (10000+10000+10000+10000+40000)/5 =80000/5 =16000 Thus, the mean salary is Rs 16,000. To find the median, we consider the ascending order: 10000, 10000, 10000, 10000, 40000. n=5, so (n+1)/2 = 3. Thus, the median is the 3rd observation. Median=10000Median=10000 75 CU IDOL SELF LEARNING MATERIAL (SLM)

Thus, the median is ₹10,000 per month. Now let us compare the two measures of central tendencies. We can observe that the mean salary of ₹16,000 does not give even an estimated salary of any of the employees whereas the median salary represents the data more effectively. One of the weaknesses of mean is that it gets affected by extreme values. Look at the following graph to understand how extreme values affect mean and median: So, mean is to be used when we don't have extremes in the data. If we have extreme points, then the median gives a better estimation. Here's a quick summary of the differences between the two. Mean Vs Median Mean Median Definition Average of given data Central value of data (Positional (Mathematical Average) Average) Calculation Add all values and divide by total Arrange data in ascending / descending number of observations order and find middle value Every value is considered for Every value is not considered Values of data calculation Effect of extreme Greatly affected by extreme points Doesn't get effected by extreme points points 76 CU IDOL SELF LEARNING MATERIAL (SLM)

4.6 MERITS AND LIMITATIONS OF MODE Merit: The main merits of mode are: • By definition mode is the most typical or representative value 01 a distribution. Hence, when we talk of modal wage, modal size of shoe or modal size of family it is this average that we refer to. The mode is a measure which actually does indicate what many people incorrectly believe the arithmetic mean indicates. The mode is the most frequently occurring value. If the modal wage in a factory is Rs. 916 then more workers receive Rs. 916 than any other wage. This is what many believe the \"average\" wage always indicates, but actually such a meaning is indicated only if the average used is the mode. • Like median, the mode is not unduly affected by extreme values. Even if the high values are very high and the low values are very low, we choose the most frequent value of the data to the modal value: for example, the mode of 10, 2, 5, 10, 5, 60, 5, 10, and 60 is 10 as this value. i.e., 10 has occurred most often in the data set. • Its value can be determined in open-end distributions without ascertaining the class limits. • It can be used to describe qualitative phenomenon. For example, if we want to compare the consumer preferences for different types of products, say, soap, toothpaste, etc., or different media of advertising we should complete the modal preferences expressed by different groups of people. • The value of mode can also be determined graphically whereas the value of mean cannot be graphically ascertained. Limitations: The important limitations of this average are: • The value of mode cannot always be determined. In some cases, we may have a bimodal series. • It is not capable of algebraic manipulations. For example, from the modes of two sets of data we cannot calculate the overall mode of the combined data. Similarly, the modal wage times the number of workers will not give the total payroll-except, of course, when the distribution is normal and then the mean, median and mode are all equal. • The value of mode is not based on each and every item of the series. It is not a rigidly defined measure. There are several formulae for calculating the mode, all of which usually 77 CU IDOL SELF LEARNING MATERIAL (SLM)

give somewhat different answers. In fact, mode is the most unstable average and its value is difficult to determine. • While dealing with quantitative data, the disadvantages of the mode outweigh its good features and hence it is seldom used. Usefulness: The mode is employed when the most typical value of a distribution is desired. It is the most meaningful measure of central tendency in case of highly skewed or non-normal distributions, as it provides the best indication of the point of maximum concentration. 4.7 MEAN DEVIATION Definition: The mean deviation is defined as a statistical measure which is used to calculate the average deviation from the mean value of the given data set. The mean deviation of the data values can be easily calculated using the below procedure. Step 1: Find the mean value for the given data values Step 2: Now, subtract mean value form each of the data value given (Note: Ignore the minus symbol) Step 3: Now, find the mean of those values obtained in step 2. Mean Deviation Formula: The formula to calculate the mean deviation for the given data set is given below. Mean Deviation = [Σ |X – µ|]/N where, Σ represents the addition of values X represents each value in the data set Μ represents the mean value of the data set N represents the number of data values || represents the absolute value, which ignores the “-” symbol 4.8 MEAN DEVIATION FOR FREQUENCY DISTRIBUTION To present the data in the more compressed form we group it and mention the frequency distribution of each such group. These groups are known as class intervals. Grouping of data is possible in two ways: 78 CU IDOL SELF LEARNING MATERIAL (SLM)

1. Discrete Frequency Distribution 2. Continuous Frequency Distribution In the upcoming discussion, we will be discussing mean absolute deviation in a discrete frequency distribution. Let us first know what is actually meant by the discrete distribution of frequency. Mean Deviation for Discrete Distribution Frequency As the name itself suggests, by discrete we mean distinct or non-continuous. In such a distribution the frequency (number of observations) given in the set of data is discrete in nature. If the data set consists of values x1, x2, x3………xn each occurring with a frequency of f1, f2… fn respectively then such a representation of data is known as the discrete distribution of frequency. To calculate the mean deviation for grouped data and particularly for discrete distribution data the following steps are followed: Step I: The measure of central tendency about which mean deviation is to be found out is calculated. Let this measure be a. If this measure is mean then it is calculated as, where N=∑ni If the measure is median then the given set of data is arranged in ascending order and then the cumulative frequency is calculated then the observations whose cumulative frequency is equal to or just greater than N/2 is taken as the median for the given discrete distribution of frequency and it is seen that this value lies in the middle of the frequency distribution. Step II: Calculate the absolute deviation of each observation from the measure of central tendency calculated in step (I) Step III: The mean absolute deviation around the measure of central tendency is then calculated by using the formula If the central tendency is mean then, 79 CU IDOL SELF LEARNING MATERIAL (SLM)

In case of median Example 1: Determine the mean deviation for the data values 5, 3,7, 8, 4, 9. Solution: Given data values are 5, 3, 7, 8, 4, 9. We know that the procedure to calculate the mean deviation. First, find the mean for the given data: Mean, µ = (5+3+7+8+4+9)/6 µ = 36/6 µ=6 Therefore, the mean value is 6. Now, subtract each mean from the data value, and ignore the minus symbol if any (Ignore”-”) 5–6=1 3–6=3 7–6=1 8–6=2 4–6=2 9–6=3 Now, the obtained data set is 1, 3, 1, 2, 2, 3. Finally, find the mean value for the obtained data set Therefore, the mean deviation is = (1+3 + 1+ 2+ 2+3) /6 = 12/6 =2 Hence, the mean deviation for 5, 3,7, 8, 4, 9 is 2. Example 2: In a foreign language class, there are 4 languages and the frequencies of students learning the language and the frequency of lectures per week is given as: 80 CU IDOL SELF LEARNING MATERIAL (SLM)

Language Sanskrit Spanish French English 6 5 9 12 No. of students(xi) 5 7 49 Frequency of lectures(fi) Calculate the mean deviation about the mean for the given data. Solution: The following table gives us a tabular representation of data and the calculations Mean Deviation for Continuous Distribution Frequency: In frequency distribution of continuous type, the class intervals or groups are arranged in such a way that there are no gaps between the classes and each class in the table has its respective frequency. The class intervals are chosen in such a way that they must be mutually exclusive and exhaustive. Example: The following table represents the age group of employees working in a certain company. Age Group Number of people 15-25 25 25-35 54 81 CU IDOL SELF LEARNING MATERIAL (SLM)

35-45 34 45-55 20 This representation is continuous in nature and the frequency is mentioned according to the class interval. Steps to Calculate Mean Deviation of Continuous Frequency Distribution To calculate the mean deviation for continuous frequency distribution, following steps are followed: Step i) Assume that the frequency in each class is centred at the mid-point. The mean is calculated for these mid-points. Considering the above example, the mid points are given as: Age Group xi Number of people(fi) 15-25 20 25 25-35 30 54 35-45 40 34 45-55 50 20 The mean is calculated by the formula Step ii) The mean absolute deviation about mean is given by: The above example can be tabulated as: Age Number of |xi − x¯| fi|xi − x¯| Group xi people fi fixi 82 CU IDOL SELF LEARNING MATERIAL (SLM)

15-25 20 25 500 13.684 324.1 25-35 30 54 1620 3.684 198.936 35-45 40 34 1360 6.316 214.744 45-55 50 20 1000 16.316 352.32 ∑ fi = 133 Now, 4.9 SUMMARY • The mode is the value that appears most often in a set of data values. • If X is a discrete random variable, the mode is the value x (i.e., X = x) at which the probability mass function takes its maximum value. • In other words, it is the value that is most likely to be sampled. • The term mean deviation is a measure that tells how many the observations in the data set deviates from the mean value of the observations in the data set. 83 CU IDOL SELF LEARNING MATERIAL (SLM)

4.10 KEYWORDS • Mode: is the value that appears most frequently in a data set. • Frequency: The number of occurrences of a repeating event per unit of time. • Interval data: It is always appearing in the form of numbers or numerical values where the distance between the two points is standardized and equal. • Frequency of modal class: The class with the highest frequency. • Mean deviation: A statistical measure which is used to calculate the average deviation from the mean value of the given data set. 4.11 LEARNING ACTIVITY 1. Find the median of the following data. (a) 27, 39, 49, 20, 21, 28, 38 (b) 10, 19, 54, 80, 15, 16 (c) 47, 41, 52, 43, 56, 35, 49, 55, 42 (d) 12, 17, 3, 14, 5, 8, 7, 15 ________________________________________________________________________________ _______________________________________________________________ 2. Find the mode of the following data. (a) 12, 8, 4, 8, 1, 8, 9, 11, 9, 10, 12, 8 (b) 15, 22, 17, 19, 22, 17, 29, 24, 17, 15 (c) 0, 3, 2, 1, 3, 5, 4, 3, 42, 1, 2, 0 (d) 1, 7, 2, 4, 5, 9, 8, 3 ________________________________________________________________________________ _______________________________________________________________ 4.12 UNIT END QUESTIONS A. Descriptive Questions Short Questions 84 CU IDOL SELF LEARNING MATERIAL (SLM)

1. For the following grouped frequency distribution find the mode: Class : 3-6 6-9 9-12 12-15 15-18 18-21 21-24 Frequency : 2 5 10 23 21 12 3 2. The runs scored in a cricket match by 11 players is as follows: 7, 16, 121, 51, 101, 81, 1, 16, 9, 11, 16 Find the mean, mode, median of this data. 3. The weights in kg of 10 students are given below: 39, 43, 36, 38, 46, 51, 33, 44, 44, 43 Find the mode of this data. Is there more than 1 mode? If yes, why? 4. How do you solve a mode question? 5. What is the formula of mode in statistics? 6. How do you find the mode in statistics examples? Long Questions 1. Compute the mode for the following frequency distribution: Size of items : 0-4 4-8 8-12 12-16 16-20 20-24 24-28 28-32 32-36 36-40 Frequency : 5 7 9 17 12 10 6 3 10 2. The marks obtained by 40 students out of 50 in a class are given below in the table. Marks (in $) 42 36 30 45 50 Number of Students 7 10 13 8 2 Find the mode of the above data. 3. The number of rupee notes of different denominations are given below in the table. Denominations (Rs) 10 20 5 50 100 Number of Notes 40 30 10 25 20 Find the mode of the above data. 85 CU IDOL SELF LEARNING MATERIAL (SLM)

4. The following observations are arranged in ascending order. The median of the data is 25 find the value of x. 17, x, 24, x + 7, 35, 36, 46 5. The mean of the following distribution is 26. Find the value of p and also the value of the observation. xi 0 1 2 3 4 5 fi 3 3 P 7 p - 1 4 Also, find the mode and the given data. B. Multiple choice Questions 1. Find the value of x, if the mode of the data is 25: 15, 20, 25, 18, 14, 15, 25, 15, 18, 16, 20, 25, 20, x, a. 15 b. 18 c. 25 d. 20 2. Compute the modal value for 175 x: 95 105 115 125 135 145 155 165 2 f: 4 2 18 22 21 19 10 3 a. 175 b. 125 c. 145 d. 165 3. One of the methods for determining mode is 86 a. Mode = 3 Median – 2 Mean b. Mode = 2 Median -3 Mean CU IDOL SELF LEARNING MATERIAL (SLM)

c. Mode = 2 Mean – 3 Median d. Mode = 3 Mean – 2 Median 4. Mode is the value of the variable which has a. minimum frequency b. maximum frequency c. mean frequency d. middle most frequency 5. Mode and mean of a data are 12k and 15A. Median of the data is a. 12k b. 13k c. 15k d. 14k Answers 1.c, 2.b, 3.a, 4.b, 5.d 4.13 REFERENCES Reference Books: • Dr. J. Ravichandran, Probability & Statistics for Eng., Willey Publications • Quantitative Techniques for Decision Making by A. Sharma, Publisher: HPH, 2007 Edition. • Statistical Methods by S.P Gupta, Publisher: Sultan Chand & Sons, 2008 Edition. • Research Methodology by C. R. Kothari, Publisher: Vikas Publishing House Textbooks: • Seymour Lipschutz, Jack Schiller, Jack Schiller S, Introduction to Probability & Statistics, McGraw-Hill Publishers. 87 CU IDOL SELF LEARNING MATERIAL (SLM)

• Research Methodology and Statistical Techniques by Santosh Gupta, Publisher: Deep and Deep Publication • Research Methodology and Statistical Methods by T. Subbi Reddy, Publisher: Reliance Publishing House • Introduction to Linear Optimization,\" by Dimitris Bertsimas and John 88 CU IDOL SELF LEARNING MATERIAL (SLM)

UNIT 5: MEASURES OF DISPERSION Structure 5.0 Learning Objectives 5.1 Introduction 5.2 Types of Measures of Dispersion 5.3 Various ‘Absolute Measures’ of Dispersion 5.4 Various ‘Relative Measures’ of Dispersion 5.5 Merits and Demerits 5.6 Quartile, Percentile, Deciles 5.7 Summary 5.8 Keywords 5.9 Learning activity 5.10 Unit End Questions 5.11 References 5.0 LEARNING OBJECTIVES After studying this Unit students will be able to: • Explain the difference between measures of dispersion and measures of central tendency. • Explain why measures of dispersion must be reported in addition to measures of central tendency. • Explain and know the relevant formulas for variation ratio, range, variance, and standard deviation. • Apply the correct measure(s) of dispersion to any given variable based on that variable’s level of measurement. • Explain the normal curve and the concept behind two-thirds of cases being within one standard deviation of the mean. • Calculate the upper and lower boundaries of the typical range of scores in a normal distribution based on the mean and the standard deviation. 89 CU IDOL SELF LEARNING MATERIAL (SLM)

5.1 INTRODUCTION The measure of dispersion indicates the scattering of data. It explains the disparity of data from one another, delivering a precise view of the distribution of data. The measure of dispersion displays and gives us an idea about the variation and central value of an individual item. In other words, Dispersion is the extent to which values in a distribution differ from the average of the distribution. It gives us an idea about the extent to which individual items vary from one another and from the central value. The variation can be measured in different numerical measures, namely: (i) Range – It is the simplest method of measurement of dispersion and defines the difference between the largest and the smallest item in a given distribution. Suppose Y max and Y min are the two ultimate items then Range = Y max – Y min (ii) Quartile Deviation – It is known as Semi-Inter-Quartile Range, i.e., half of the difference between the upper quartile and lower quartile. The first quartile is derived as (Q), the middle digit (Q1) connects the least number with the median of the data. The median of a data set is the (Q2) second quartile. Lastly, the number connecting the largest number and the median is the third quartile (Q3). Quartile deviation can be calculated by Q = ½ × (Q3 – Q1) (iii) Mean Deviation-Mean deviation is the arithmetic mean (average) of deviations ⎜D⎜of observations from a central value {Mean or Median}. Mean deviation can be evaluated by using the formula: A = 1⁄n [∑i|xi – A|] (iv) Standard Deviation- Standard deviation is the Square Root of the Arithmetic Average of the squared of the deviations measured from the mean. The standard deviation is given as σ = [(Σi (yi – ȳ) ⁄ n] ½ = [(Σ i yi 2 ⁄ n) – ȳ 2] ½ Apart from a numerical value, graphics method is also applied for estimating dispersion. 5.2 TYPES OF MEASURES OF DISPERSION There are two main types of dispersion methods in statistics which are: • Absolute Measure of Dispersion • Relative Measure of Dispersion Absolute Measures An absolute measure of dispersion contains the same unit as the original data set. Absolute dispersion method expresses the variations in terms of the average of deviations of observations like standard or means deviations. It includes range, standard deviation, quartile deviation, etc. 90 CU IDOL SELF LEARNING MATERIAL (SLM)

The types of absolute measures of dispersion are: 1. Range: It is simply the difference between the maximum value and the minimum value given in a data set. Example: 1, 3,5, 6, 7 => Range = 7 -1= 6 2. Variance: Deduct the mean from each data in the set then squaring each of them and adding each square and finally dividing them by the total no of values in the data set is the variance. Variance (σ2) =∑(X−μ)2/N 3. Standard Deviation: The square root of the variance is known as the standard deviation i.e., S.D. = √σ. 4. Quartiles and Quartile Deviation: The quartiles are values that divide a list of numbers into quarters. The quartile deviation is half of the distance between the third and the first quartile. 5. Mean and Mean Deviation: The average of numbers is known as the mean and the arithmetic mean of the absolute deviations of the observations from a measure of central tendency is known as the mean deviation (also called mean absolute deviation). Relative Measures • Relative measures of dispersion are obtained as ratios or percentages of the average. • These are also known as ‘Coefficient of dispersion.’ • These are pure numbers or percentages totally independent of the units of measurements. 5.3 VARIOUS ‘ABSOLUTE MEASURES’ OF DISPERSION Following Are the Different ‘absolute Measures’ of Dispersion: (1) Range • It is the simplest method of measurement of dispersion. • It is defined as the difference between the largest and the smallest item in a given distribution. • Range = Largest item (L) – Smallest item (S) (2) Interquartile Range • It is defined as the difference between the Upper Quartile and Lower Quartile of a given distribution. Interquartile Range = Upper Quartile (Q3)–Lower Quartile(Q1) (3) Quartile Deviation 91 CU IDOL SELF LEARNING MATERIAL (SLM)

• It is known as Semi-Inter-Quartile Range, i.e., half of the difference between the upper quartile and lower quartile. • Quartile Deviation = Upper Quartile(Q3) −Lower Quartile(Q1)/2 (4) Mean Deviation • Mean deviation is the arithmetic mean (average) of deviations ⎜D⎜of observations from a central value {Mean or Median}. (5) Standard Deviation • Standard deviation is the Square Root of the Arithmetic Average of the squared of the deviations measured from the mean. (6) Lorenz Curve • The Lorenz Curve is a graphic method of measuring estimated dispersion. • This curve is often used to measure the inequalities of income or wealth in a society. 5.4 VARIOUS ‘RELATIVE MEASURES’ OF DISPERSION Following Are the Relative Measures of Dispersion: (1) Coefficient of Range It refers to the ratio of the difference between two extreme items of the distribution to their sum. Coefficient of Range = (largest item – smallest item) (largest item + smallest item) (2) Coefficient of Quartile Deviation It refers to the ratio of the difference between Upper Quartile and Lower Quartile of a distribution to their sum. Coefficient of Quartile Deviation = Q3−Q1Q3+Q1 (3) Coefficient of Mean Deviation • Mean deviation is an absolute measure of dispersion. • In order to transform it into a relative measure, it is divided by the particular average, from which it has been calculated. • It is then known as the Coefficient of Mean Deviation. • Coefficient of Mean Deviation from Mean (X¯) =MDX¯X¯ • Coefficient of Mean Deviation from Median (ME)=MDMeMe (4) Coefficient of Standard Deviation • It is calculated by dividing the standard deviation (σ) by the mean (X¯) of the data. 92 CU IDOL SELF LEARNING MATERIAL (SLM)

• Coefficient of Standard Deviation =σX (5) Coefficient of Variation • It is used to compare two data with respect to stability (or uniformity or consistency or homogeneity). • It indicates the relationship between the standard deviation and the arithmetic mean expressed in terms of percentage. • Coefficient of Variation (C.V.) =σXX100 • Where, C.V. = Coefficient of Variation; σ= Standard Deviation; X¯ = Arithmetic Mean 5.5 MERITS AND DEMERITS Merits and Demerits of Range Merits: 1. It is very easy to calculate and simple to understand. 2. No special knowledge is needed while calculating range. 3. It takes the least time for computation. 4. It provides a broad picture of the data at a glance. Demerits: 1. It is a crude measure because it is only based on two extreme values (highest and lowest). 2. It cannot be calculated in the case of open-ended series. 3. Range is significantly affected by fluctuations of sampling, i.e., it varies widely from sample to sample. Merits and Demerits of Quartile Deviation Merits: 1. It is also quite easy to calculate and simple to understand. 2. It can be used even in case of open-end distribution. 3. It is less affected by extreme values so, it a superior to ‘Range’. 4. It is more useful when the dispersion of the middle 50% is to be computed. Demerits: 93 CU IDOL SELF LEARNING MATERIAL (SLM)

1. It is not based on all the observations. 2. It is not capable of further algebraic treatment or statistical analysis. 3. It is affected considerably by fluctuations of sampling. 4. It is not regarded as a very reliable measure of dispersion because it ignores 50% observations. Merits and Demerits of Mean Deviation Merits: 1. It is based on all the observations of the series and not only on the limits like Range and QD. 2. It is simple to calculate and easy to understand. 3. It is not much affected by extreme values. 4. For calculating mean deviation, deviations can be taken from any average. Demerits: 1. Ignoring + and – signs is bad from the mathematical viewpoint. 2. It is not capable of further mathematical treatment. 3. It is difficult to compute when the mean or median is in fraction. 4. It may not be possible to use this method in case of open-ended series. Example 1: 1. Find the range and coefficient of range of the following data. (i) 63, 89, 98, 125, 79, 108, 117, 68 (ii) 43.5, 13.6, 18.9, 38.4, 61.4, 29.8 94 CU IDOL SELF LEARNING MATERIAL (SLM)

Example 2: 95 CU IDOL SELF LEARNING MATERIAL (SLM)

If the range and the smallest value of a set of data are 36.8 and 13.4 respectively, then find the largest value. Example 3: Calculate the range of the following data. Example 4: A teacher asked the students to complete 60 pages of a record note book. Eight students have completed only 32, 35, 37, 30, 33, 36, 35 and 37 pages. Find the standard deviation of the pages yet to be completed by them. 96 CU IDOL SELF LEARNING MATERIAL (SLM)

Example 5: Find the variance and standard deviation of the wages of 9 workers given below: 310, 290, 320, 280, 300, 290, 320, 310, 280. 97 CU IDOL SELF LEARNING MATERIAL (SLM)

Example 6: A wall clock strikes the bell once at 1 o’ clock, 2 times at 2 o’ clock, 3 times at 3 o’ clock and so on. How many times will it strike in a particular day? Find the standard deviation of the number of strikes the bell make a day. 98 CU IDOL SELF LEARNING MATERIAL (SLM)

Example 7: If the standard deviation of a data is 3.6 and each value of the data is divided by 3, then find the new variance and new standard deviation 99 CU IDOL SELF LEARNING MATERIAL (SLM)

Example 8: The rainfall recorded in various places of five districts in a week are given below. Find its standard deviation. 100 CU IDOL SELF LEARNING MATERIAL (SLM)