Page 1 1. INTRODUCTION In the previous chapter, you have studied how to sum up the data into a single representative value. However, that value does not reveal the variability present in the data. In this chapter you will study those measures, which seek to quantify variability of the data. Three friends, Ram, Rahim and Maria are chatting over a cup of tea. During the course of their conversation, they start talking about their family incomes. Ram tells them that there are four members in his family and the average income per member is Rs 15,000. Rahim says that the average income is the same in his family, though the number of members is six. Maria says that there are five members in her family, out of which one is not working. She calculates that the average income in her family too, is Rs 15,000. They are a little surprised since they know that Mariaâ€™s father is earning a huge salary. They go into details and gather the following data: Measures of Dispersion Studying this chapter should enable you to: â€¢ know the limitations of averages; ? appreciate the need of measures of dispersion; ? enumerate various measures of dispersion; ? calculate the measures and compare them; ? distinguish between absolute and relative measures. CHAPTER Page 2 1. INTRODUCTION In the previous chapter, you have studied how to sum up the data into a single representative value. However, that value does not reveal the variability present in the data. In this chapter you will study those measures, which seek to quantify variability of the data. Three friends, Ram, Rahim and Maria are chatting over a cup of tea. During the course of their conversation, they start talking about their family incomes. Ram tells them that there are four members in his family and the average income per member is Rs 15,000. Rahim says that the average income is the same in his family, though the number of members is six. Maria says that there are five members in her family, out of which one is not working. She calculates that the average income in her family too, is Rs 15,000. They are a little surprised since they know that Mariaâ€™s father is earning a huge salary. They go into details and gather the following data: Measures of Dispersion Studying this chapter should enable you to: â€¢ know the limitations of averages; ? appreciate the need of measures of dispersion; ? enumerate various measures of dispersion; ? calculate the measures and compare them; ? distinguish between absolute and relative measures. CHAPTER MEASURES OF DISPERSION 75 Family Incomes Sl. No. Ram Rahim Maria 1. 12,000 7,000 0 2. 14,000 10,000 7,000 3. 16,000 14,000 8,000 4. 18,000 17,000 10,000 5. ----- 20,000 50,000 6. ----- 22,000 ------ Total income 60,000 90,000 75,000 Average income 15,000 15,000 15,000 Do you notice that although the average is the same, there are considerable differences in individual incomes? It is quite obvious that averages try to tell only one aspect of a distribution i.e. a representative size of the values. To understand it better, you need to know the spread of values also. You can see that in Ramâ€™s family., differences in incomes are comparatively lower. In Rahimâ€™s family, differences are higher and in Mariaâ€™s family are the highest. Knowledge of only average is insufficient. If you have another value which reflects the quantum of variation in values, your understan- ding of a distribution improves considerably. For example, per capita income gives only the average income. A measure of dispersion can tell you about income inequalities, thereby improving the understanding of the relative standards of living enjoyed by different strata of society. Dispersion is the extent to which values in a distribution differ from the average of the distribution. To quantify the extent of the variation, there are certain measures namely: (i) Range (ii) Quartile Deviation (iii)Mean Deviation (iv)Standard Deviation Apart from these measures which give a numerical value, there is a graphic method for estimating dispersion. Range and Quartile Deviation measure the dispersion by calculating the spread within which the values lie. Mean Deviation and Standard Deviation calculate the extent to which the values differ from the average. 2. MEASURES BASED UPON SPREAD OF VALUES Range Range (R) is the difference between the largest (L) and the smallest value (S) in a distribution. Thus, R = L â€“ S Higher value of Range implies higher dispersion and vice-versa. Page 3 1. INTRODUCTION In the previous chapter, you have studied how to sum up the data into a single representative value. However, that value does not reveal the variability present in the data. In this chapter you will study those measures, which seek to quantify variability of the data. Three friends, Ram, Rahim and Maria are chatting over a cup of tea. During the course of their conversation, they start talking about their family incomes. Ram tells them that there are four members in his family and the average income per member is Rs 15,000. Rahim says that the average income is the same in his family, though the number of members is six. Maria says that there are five members in her family, out of which one is not working. She calculates that the average income in her family too, is Rs 15,000. They are a little surprised since they know that Mariaâ€™s father is earning a huge salary. They go into details and gather the following data: Measures of Dispersion Studying this chapter should enable you to: â€¢ know the limitations of averages; ? appreciate the need of measures of dispersion; ? enumerate various measures of dispersion; ? calculate the measures and compare them; ? distinguish between absolute and relative measures. CHAPTER MEASURES OF DISPERSION 75 Family Incomes Sl. No. Ram Rahim Maria 1. 12,000 7,000 0 2. 14,000 10,000 7,000 3. 16,000 14,000 8,000 4. 18,000 17,000 10,000 5. ----- 20,000 50,000 6. ----- 22,000 ------ Total income 60,000 90,000 75,000 Average income 15,000 15,000 15,000 Do you notice that although the average is the same, there are considerable differences in individual incomes? It is quite obvious that averages try to tell only one aspect of a distribution i.e. a representative size of the values. To understand it better, you need to know the spread of values also. You can see that in Ramâ€™s family., differences in incomes are comparatively lower. In Rahimâ€™s family, differences are higher and in Mariaâ€™s family are the highest. Knowledge of only average is insufficient. If you have another value which reflects the quantum of variation in values, your understan- ding of a distribution improves considerably. For example, per capita income gives only the average income. A measure of dispersion can tell you about income inequalities, thereby improving the understanding of the relative standards of living enjoyed by different strata of society. Dispersion is the extent to which values in a distribution differ from the average of the distribution. To quantify the extent of the variation, there are certain measures namely: (i) Range (ii) Quartile Deviation (iii)Mean Deviation (iv)Standard Deviation Apart from these measures which give a numerical value, there is a graphic method for estimating dispersion. Range and Quartile Deviation measure the dispersion by calculating the spread within which the values lie. Mean Deviation and Standard Deviation calculate the extent to which the values differ from the average. 2. MEASURES BASED UPON SPREAD OF VALUES Range Range (R) is the difference between the largest (L) and the smallest value (S) in a distribution. Thus, R = L â€“ S Higher value of Range implies higher dispersion and vice-versa. 76 STATISTICS FOR ECONOMICS Activities Look at the following values: 20, 30, 40, 50, 200 ? Calculate the Range. ? What is the Range if the value 200 is not present in the data set? ? If 50 is replaced by 150, what will be the Range? Range: Comments Range is unduly affected by extreme values. It is not based on all the values. As long as the minimum and maximum values remain unaltered, any change in other values does not affect range. It can not be calculated for open-ended frequency distri- bution. Notwithstanding some limitations, Range is understood and used frequently because of its simplicity. For example, we see the maximum and minimum temperatures of different cities almost daily on our TV screens and form judgments about the temperature variations in them. Open-ended distributions are those in which either the lower limit of the lowest class or the upper limit of the highest class or both are not specified. Activity ? Collect data about 52-week high/low of 10 shares from a newspaper. Calculate the range of share prices. Which stock is most volatile and which is the most stable? Quartile Deviation The presence of even one extremely high or low value in a distribution can reduce the utility of range as a measure of dispersion. Thus, you may need a measure which is not unduly affected by the outliers. In such a situation, if the entire data is divided into four equal parts, each containing 25% of the values, we get the values of Quartiles and Median. (You have already read about these in Chapter 5). The upper and lower quartiles (Q 3 and Q 1 , respectively) are used to calculate Inter Quartile Range which is Q 3 â€“ Q 1 . Inter-Quartile Range is based upon middle 50% of the values in a distribution and is, therefore, not affected by extreme values. Half of the Inter-Quartile Range is called Quartile Deviation. Thus: Q.D . = Q - Q 2 31 Q.D. is therefore also called Semi- Inter Quartile Range. Calculation of Range and Q.D. for ungrouped data Example 1 Calculate Range and Q.D. of the following observations: 20, 25, 29, 30, 35, 39, 41, 48, 51, 60 and 70 Range is clearly 70 â€“ 20 = 50 For Q.D., we need to calculate values of Q 3 and Q 1 . Page 4 1. INTRODUCTION In the previous chapter, you have studied how to sum up the data into a single representative value. However, that value does not reveal the variability present in the data. In this chapter you will study those measures, which seek to quantify variability of the data. Three friends, Ram, Rahim and Maria are chatting over a cup of tea. During the course of their conversation, they start talking about their family incomes. Ram tells them that there are four members in his family and the average income per member is Rs 15,000. Rahim says that the average income is the same in his family, though the number of members is six. Maria says that there are five members in her family, out of which one is not working. She calculates that the average income in her family too, is Rs 15,000. They are a little surprised since they know that Mariaâ€™s father is earning a huge salary. They go into details and gather the following data: Measures of Dispersion Studying this chapter should enable you to: â€¢ know the limitations of averages; ? appreciate the need of measures of dispersion; ? enumerate various measures of dispersion; ? calculate the measures and compare them; ? distinguish between absolute and relative measures. CHAPTER MEASURES OF DISPERSION 75 Family Incomes Sl. No. Ram Rahim Maria 1. 12,000 7,000 0 2. 14,000 10,000 7,000 3. 16,000 14,000 8,000 4. 18,000 17,000 10,000 5. ----- 20,000 50,000 6. ----- 22,000 ------ Total income 60,000 90,000 75,000 Average income 15,000 15,000 15,000 Do you notice that although the average is the same, there are considerable differences in individual incomes? It is quite obvious that averages try to tell only one aspect of a distribution i.e. a representative size of the values. To understand it better, you need to know the spread of values also. You can see that in Ramâ€™s family., differences in incomes are comparatively lower. In Rahimâ€™s family, differences are higher and in Mariaâ€™s family are the highest. Knowledge of only average is insufficient. If you have another value which reflects the quantum of variation in values, your understan- ding of a distribution improves considerably. For example, per capita income gives only the average income. A measure of dispersion can tell you about income inequalities, thereby improving the understanding of the relative standards of living enjoyed by different strata of society. Dispersion is the extent to which values in a distribution differ from the average of the distribution. To quantify the extent of the variation, there are certain measures namely: (i) Range (ii) Quartile Deviation (iii)Mean Deviation (iv)Standard Deviation Apart from these measures which give a numerical value, there is a graphic method for estimating dispersion. Range and Quartile Deviation measure the dispersion by calculating the spread within which the values lie. Mean Deviation and Standard Deviation calculate the extent to which the values differ from the average. 2. MEASURES BASED UPON SPREAD OF VALUES Range Range (R) is the difference between the largest (L) and the smallest value (S) in a distribution. Thus, R = L â€“ S Higher value of Range implies higher dispersion and vice-versa. 76 STATISTICS FOR ECONOMICS Activities Look at the following values: 20, 30, 40, 50, 200 ? Calculate the Range. ? What is the Range if the value 200 is not present in the data set? ? If 50 is replaced by 150, what will be the Range? Range: Comments Range is unduly affected by extreme values. It is not based on all the values. As long as the minimum and maximum values remain unaltered, any change in other values does not affect range. It can not be calculated for open-ended frequency distri- bution. Notwithstanding some limitations, Range is understood and used frequently because of its simplicity. For example, we see the maximum and minimum temperatures of different cities almost daily on our TV screens and form judgments about the temperature variations in them. Open-ended distributions are those in which either the lower limit of the lowest class or the upper limit of the highest class or both are not specified. Activity ? Collect data about 52-week high/low of 10 shares from a newspaper. Calculate the range of share prices. Which stock is most volatile and which is the most stable? Quartile Deviation The presence of even one extremely high or low value in a distribution can reduce the utility of range as a measure of dispersion. Thus, you may need a measure which is not unduly affected by the outliers. In such a situation, if the entire data is divided into four equal parts, each containing 25% of the values, we get the values of Quartiles and Median. (You have already read about these in Chapter 5). The upper and lower quartiles (Q 3 and Q 1 , respectively) are used to calculate Inter Quartile Range which is Q 3 â€“ Q 1 . Inter-Quartile Range is based upon middle 50% of the values in a distribution and is, therefore, not affected by extreme values. Half of the Inter-Quartile Range is called Quartile Deviation. Thus: Q.D . = Q - Q 2 31 Q.D. is therefore also called Semi- Inter Quartile Range. Calculation of Range and Q.D. for ungrouped data Example 1 Calculate Range and Q.D. of the following observations: 20, 25, 29, 30, 35, 39, 41, 48, 51, 60 and 70 Range is clearly 70 â€“ 20 = 50 For Q.D., we need to calculate values of Q 3 and Q 1 . MEASURES OF DISPERSION 77 Q 1 is the size of n th +1 4 value. n being 11, Q 1 is the size of 3rd value. As the values are already arranged in ascending order, it can be seen that Q 1 , the 3rd value is 29. [What will you do if these values are not in an order?] Similarly, Q 3 is size of 31 4 () n th + value; i.e. 9th value which is 51. Hence Q 3 = 51 Q.D . = Q - Q 2 31 = 51 29 2 11 - = Do you notice that Q.D. is the average difference of the Quartiles from the median. Activity ? Calculate the median and check whether the above statement is correct. Calculation of Range and Q.D. for a frequency distribution. Example 2 For the following distribution of marks scored by a class of 40 students, calculate the Range and Q.D. TABLE 6.1 Class intervals No. of students C I (f) 0â€“10 5 10â€“20 8 20â€“40 16 40â€“60 7 60â€“90 4 40 Range is just the difference between the upper limit of the highest class and the lower limit of the lowest class. So Range is 90 â€“ 0 = 90. For Q.D., first calculate cumulative frequencies as follows: Class- Frequencies Cumulative Intervals Frequencies CI f c. f. 0â€“10 5 05 10â€“20 8 13 20â€“40 16 29 40â€“60 7 36 60â€“90 4 40 n = 40 Q 1 is the size of nth 4 value in a continuous series. Thus it is the size of the 10th value. The class containing the 10th value is 10â€“20. Hence Q 1 lies in class 10â€“20. Now, to calculate the exact value of Q 1 , the following formula is used: QL n cf f i 1 4 =+ · Where L = 10 (lower limit of the relevant Quartile class) c.f. = 5 (Value of c.f. for the class preceding the Quartile class) i = 10 (interval of the Quartile class), and f = 8 (frequency of the Quartile class) Thus, Q 1 10 10 5 8 10 16 25 =+ - ·= . Similarly, Q 3 is the size of 3 4 nth Page 5 1. INTRODUCTION In the previous chapter, you have studied how to sum up the data into a single representative value. However, that value does not reveal the variability present in the data. In this chapter you will study those measures, which seek to quantify variability of the data. Three friends, Ram, Rahim and Maria are chatting over a cup of tea. During the course of their conversation, they start talking about their family incomes. Ram tells them that there are four members in his family and the average income per member is Rs 15,000. Rahim says that the average income is the same in his family, though the number of members is six. Maria says that there are five members in her family, out of which one is not working. She calculates that the average income in her family too, is Rs 15,000. They are a little surprised since they know that Mariaâ€™s father is earning a huge salary. They go into details and gather the following data: Measures of Dispersion Studying this chapter should enable you to: â€¢ know the limitations of averages; ? appreciate the need of measures of dispersion; ? enumerate various measures of dispersion; ? calculate the measures and compare them; ? distinguish between absolute and relative measures. CHAPTER MEASURES OF DISPERSION 75 Family Incomes Sl. No. Ram Rahim Maria 1. 12,000 7,000 0 2. 14,000 10,000 7,000 3. 16,000 14,000 8,000 4. 18,000 17,000 10,000 5. ----- 20,000 50,000 6. ----- 22,000 ------ Total income 60,000 90,000 75,000 Average income 15,000 15,000 15,000 Do you notice that although the average is the same, there are considerable differences in individual incomes? It is quite obvious that averages try to tell only one aspect of a distribution i.e. a representative size of the values. To understand it better, you need to know the spread of values also. You can see that in Ramâ€™s family., differences in incomes are comparatively lower. In Rahimâ€™s family, differences are higher and in Mariaâ€™s family are the highest. Knowledge of only average is insufficient. If you have another value which reflects the quantum of variation in values, your understan- ding of a distribution improves considerably. For example, per capita income gives only the average income. A measure of dispersion can tell you about income inequalities, thereby improving the understanding of the relative standards of living enjoyed by different strata of society. Dispersion is the extent to which values in a distribution differ from the average of the distribution. To quantify the extent of the variation, there are certain measures namely: (i) Range (ii) Quartile Deviation (iii)Mean Deviation (iv)Standard Deviation Apart from these measures which give a numerical value, there is a graphic method for estimating dispersion. Range and Quartile Deviation measure the dispersion by calculating the spread within which the values lie. Mean Deviation and Standard Deviation calculate the extent to which the values differ from the average. 2. MEASURES BASED UPON SPREAD OF VALUES Range Range (R) is the difference between the largest (L) and the smallest value (S) in a distribution. Thus, R = L â€“ S Higher value of Range implies higher dispersion and vice-versa. 76 STATISTICS FOR ECONOMICS Activities Look at the following values: 20, 30, 40, 50, 200 ? Calculate the Range. ? What is the Range if the value 200 is not present in the data set? ? If 50 is replaced by 150, what will be the Range? Range: Comments Range is unduly affected by extreme values. It is not based on all the values. As long as the minimum and maximum values remain unaltered, any change in other values does not affect range. It can not be calculated for open-ended frequency distri- bution. Notwithstanding some limitations, Range is understood and used frequently because of its simplicity. For example, we see the maximum and minimum temperatures of different cities almost daily on our TV screens and form judgments about the temperature variations in them. Open-ended distributions are those in which either the lower limit of the lowest class or the upper limit of the highest class or both are not specified. Activity ? Collect data about 52-week high/low of 10 shares from a newspaper. Calculate the range of share prices. Which stock is most volatile and which is the most stable? Quartile Deviation The presence of even one extremely high or low value in a distribution can reduce the utility of range as a measure of dispersion. Thus, you may need a measure which is not unduly affected by the outliers. In such a situation, if the entire data is divided into four equal parts, each containing 25% of the values, we get the values of Quartiles and Median. (You have already read about these in Chapter 5). The upper and lower quartiles (Q 3 and Q 1 , respectively) are used to calculate Inter Quartile Range which is Q 3 â€“ Q 1 . Inter-Quartile Range is based upon middle 50% of the values in a distribution and is, therefore, not affected by extreme values. Half of the Inter-Quartile Range is called Quartile Deviation. Thus: Q.D . = Q - Q 2 31 Q.D. is therefore also called Semi- Inter Quartile Range. Calculation of Range and Q.D. for ungrouped data Example 1 Calculate Range and Q.D. of the following observations: 20, 25, 29, 30, 35, 39, 41, 48, 51, 60 and 70 Range is clearly 70 â€“ 20 = 50 For Q.D., we need to calculate values of Q 3 and Q 1 . MEASURES OF DISPERSION 77 Q 1 is the size of n th +1 4 value. n being 11, Q 1 is the size of 3rd value. As the values are already arranged in ascending order, it can be seen that Q 1 , the 3rd value is 29. [What will you do if these values are not in an order?] Similarly, Q 3 is size of 31 4 () n th + value; i.e. 9th value which is 51. Hence Q 3 = 51 Q.D . = Q - Q 2 31 = 51 29 2 11 - = Do you notice that Q.D. is the average difference of the Quartiles from the median. Activity ? Calculate the median and check whether the above statement is correct. Calculation of Range and Q.D. for a frequency distribution. Example 2 For the following distribution of marks scored by a class of 40 students, calculate the Range and Q.D. TABLE 6.1 Class intervals No. of students C I (f) 0â€“10 5 10â€“20 8 20â€“40 16 40â€“60 7 60â€“90 4 40 Range is just the difference between the upper limit of the highest class and the lower limit of the lowest class. So Range is 90 â€“ 0 = 90. For Q.D., first calculate cumulative frequencies as follows: Class- Frequencies Cumulative Intervals Frequencies CI f c. f. 0â€“10 5 05 10â€“20 8 13 20â€“40 16 29 40â€“60 7 36 60â€“90 4 40 n = 40 Q 1 is the size of nth 4 value in a continuous series. Thus it is the size of the 10th value. The class containing the 10th value is 10â€“20. Hence Q 1 lies in class 10â€“20. Now, to calculate the exact value of Q 1 , the following formula is used: QL n cf f i 1 4 =+ · Where L = 10 (lower limit of the relevant Quartile class) c.f. = 5 (Value of c.f. for the class preceding the Quartile class) i = 10 (interval of the Quartile class), and f = 8 (frequency of the Quartile class) Thus, Q 1 10 10 5 8 10 16 25 =+ - ·= . Similarly, Q 3 is the size of 3 4 nth 78 STATISTICS FOR ECONOMICS value; i.e., 30th value, which lies in class 40â€“60. Now using the formula for Q 3 , its value can be calculated as follows: Q = L + 3n 4 - c.f. f i 3 Q = 40 + 30 - 29 7 20 3 Q = 42.87 Q.D. = 42.87 - 16.25 2 = 13.31 3 In individual and discrete series, Q 1 is the size of nth +1 4 value, but in a continuous distribution, it is the size of nth 4 value. Similarly, for Q 3 and median also, n is used in place of n+1. If the entire group is divided into two equal halves and the median calculated for each half, you will have the median of better students and the median of weak students. These medians differ from the median of the entire group by 13.31 on an average. Similarly, suppose you have data about incomes of people of a town. Median income of all people can be calculated. Now if all people are divided into two equal groups of rich and poor, medians of both groups can be calculated. Quartile Deviation will tell you the average difference between medians of these two groups belonging to rich and poor, from the median of the entire group. Quartile Deviation can generally be calculated for open-ended distribu- tions and is not unduly affected by extreme values. 3. MEASURES OF DISPERSION FROM AVERAGE Recall that dispersion was defined as the extent to which values differ from their average. Range and Quartile Deviation do not attempt to calculate, how far the values are, from their average. Yet, by calculating the spread of values, they do give a good idea about the dispersion. Two measures which are based upon deviation of the values from their average are Mean Deviation and Standard Deviation. Since the average is a central value, some deviations are positive and some are negative. If these are added as they are, the sum will not reveal anything. In fact, the sum of deviations from Arithmetic Mean is always zero. Look at the following two sets of values. Set A : 5, 9, 16 Set B : 1, 9, 20 You can see that values in Set B are farther from the average and hence more dispersed than values in Set A. Calculate the deviations from Arithmetic Mean amd sum them up. What do you notice? Repeat the same with Median. Can you comment upon the quantum of variation from the calculated values?Read More

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### Quartile Deviation

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