Page 1 260 MATHEMA TICS 14 There are lies, damned lies and statistics. â€” by Disraeli 14.1 Introduction In Class IX, you have studied the classification of given data into ungrouped as well as grouped frequency distributions. You have also learnt to represent the data pictorially in the form of various graphs such as bar graphs, histograms (including those of varying widths) and frequency polygons. In fact, you went a step further by studying certain numerical representatives of the ungrouped data, also called measures of central tendency, namely, mean, median and mode. In this chapter, we shall extend the study of these three measures, i.e., mean, median and mode from ungrouped data to that of grouped data. We shall also discuss the concept of cumulative frequency, the cumulative frequency distribution and how to draw cumulative frequency curves, called ogives. 14.2 Mean of Grouped Data The mean (or average) of observations, as we know, is the sum of the values of all the observations divided by the total number of observations. From Class IX, recall that if x 1 , x 2 ,. . ., x n are observations with respective frequencies f 1 , f 2 , . . ., f n , then this means observation x 1 occurs f 1 times, x 2 occurs f 2 times, and so on. Now, the sum of the values of all the observations = f 1 x 1 + f 2 x 2 + . . . + f n x n , and the number of observations = f 1 + f 2 + . . . + f n . So, the mean x of the data is given by x = 11 2 2 12 ++ + ++ + L L nn n fxfx fx ff f Recall that we can write this in short form by using the Greek letter S (capital sigma) which means summation. That is, ST A TISTICS Page 2 260 MATHEMA TICS 14 There are lies, damned lies and statistics. â€” by Disraeli 14.1 Introduction In Class IX, you have studied the classification of given data into ungrouped as well as grouped frequency distributions. You have also learnt to represent the data pictorially in the form of various graphs such as bar graphs, histograms (including those of varying widths) and frequency polygons. In fact, you went a step further by studying certain numerical representatives of the ungrouped data, also called measures of central tendency, namely, mean, median and mode. In this chapter, we shall extend the study of these three measures, i.e., mean, median and mode from ungrouped data to that of grouped data. We shall also discuss the concept of cumulative frequency, the cumulative frequency distribution and how to draw cumulative frequency curves, called ogives. 14.2 Mean of Grouped Data The mean (or average) of observations, as we know, is the sum of the values of all the observations divided by the total number of observations. From Class IX, recall that if x 1 , x 2 ,. . ., x n are observations with respective frequencies f 1 , f 2 , . . ., f n , then this means observation x 1 occurs f 1 times, x 2 occurs f 2 times, and so on. Now, the sum of the values of all the observations = f 1 x 1 + f 2 x 2 + . . . + f n x n , and the number of observations = f 1 + f 2 + . . . + f n . So, the mean x of the data is given by x = 11 2 2 12 ++ + ++ + L L nn n fxfx fx ff f Recall that we can write this in short form by using the Greek letter S (capital sigma) which means summation. That is, ST A TISTICS ST A TISTICS 261 x = 1 1 n ii i n i i fx f = = ? ? which, more briefly, is written as x = S S ii i f x f , if it is understood that i varies from 1 to n. Let us apply this formula to find the mean in the following example. Example 1 : The marks obtained by 30 students of Class X of a certain school in a Mathematics paper consisting of 100 marks are presented in table below. Find the mean of the marks obtained by the students. Marks obtained 10 20 36 40 50 56 60 70 72 80 88 92 95 (x i ) Number of 113 4324 4112 3 1 student ( f i ) Solution: Recall that to find the mean marks, we require the product of each x i with the corresponding frequency f i . So, let us put them in a column as shown in Table 14.1. Table 14.1 Marks obtained (x i ) Number of students ( f i ) f i x i 10 1 10 20 1 20 . 36 3 108 40 4 160 50 3 150 56 2 112 60 4 240 70 4 280 72 1 72 80 1 80 88 2 176 92 3 276 95 1 95 Total Sf i = 30 Sf i x i = 1779 Page 3 260 MATHEMA TICS 14 There are lies, damned lies and statistics. â€” by Disraeli 14.1 Introduction In Class IX, you have studied the classification of given data into ungrouped as well as grouped frequency distributions. You have also learnt to represent the data pictorially in the form of various graphs such as bar graphs, histograms (including those of varying widths) and frequency polygons. In fact, you went a step further by studying certain numerical representatives of the ungrouped data, also called measures of central tendency, namely, mean, median and mode. In this chapter, we shall extend the study of these three measures, i.e., mean, median and mode from ungrouped data to that of grouped data. We shall also discuss the concept of cumulative frequency, the cumulative frequency distribution and how to draw cumulative frequency curves, called ogives. 14.2 Mean of Grouped Data The mean (or average) of observations, as we know, is the sum of the values of all the observations divided by the total number of observations. From Class IX, recall that if x 1 , x 2 ,. . ., x n are observations with respective frequencies f 1 , f 2 , . . ., f n , then this means observation x 1 occurs f 1 times, x 2 occurs f 2 times, and so on. Now, the sum of the values of all the observations = f 1 x 1 + f 2 x 2 + . . . + f n x n , and the number of observations = f 1 + f 2 + . . . + f n . So, the mean x of the data is given by x = 11 2 2 12 ++ + ++ + L L nn n fxfx fx ff f Recall that we can write this in short form by using the Greek letter S (capital sigma) which means summation. That is, ST A TISTICS ST A TISTICS 261 x = 1 1 n ii i n i i fx f = = ? ? which, more briefly, is written as x = S S ii i f x f , if it is understood that i varies from 1 to n. Let us apply this formula to find the mean in the following example. Example 1 : The marks obtained by 30 students of Class X of a certain school in a Mathematics paper consisting of 100 marks are presented in table below. Find the mean of the marks obtained by the students. Marks obtained 10 20 36 40 50 56 60 70 72 80 88 92 95 (x i ) Number of 113 4324 4112 3 1 student ( f i ) Solution: Recall that to find the mean marks, we require the product of each x i with the corresponding frequency f i . So, let us put them in a column as shown in Table 14.1. Table 14.1 Marks obtained (x i ) Number of students ( f i ) f i x i 10 1 10 20 1 20 . 36 3 108 40 4 160 50 3 150 56 2 112 60 4 240 70 4 280 72 1 72 80 1 80 88 2 176 92 3 276 95 1 95 Total Sf i = 30 Sf i x i = 1779 262 MATHEMA TICS Now, S = S ii i f x x f = 1779 30 = 59.3 Therefore, the mean marks obtained is 59.3. In most of our real life situations, data is usually so large that to make a meaningful study it needs to be condensed as grouped data. So, we need to convert given ungrouped data into grouped data and devise some method to find its mean. Let us convert the ungrouped data of Example 1 into grouped data by forming class-intervals of width, say 15. Remember that, while allocating frequencies to each class-interval, students falling in any upper class-limit would be considered in the next class, e.g., 4 students who have obtained 40 marks would be considered in the class- interval 40-55 and not in 25-40. With this convention in our mind, let us form a grouped frequency distribution table (see Table 14.2). Table 14.2 Class interval 10 - 25 25 - 40 40 - 55 55 - 70 70 - 85 85 - 100 Number of students 23766 6 Now, for each class-interval, we require a point which would serve as the representative of the whole class. It is assumed that the frequency of each class- interval is centred around its mid-point. So the mid-point (or class mark) of each class can be chosen to represent the observations falling in the class. Recall that we find the mid-point of a class (or its class mark) by finding the average of its upper and lower limits. That is, Class mark = Upper class limit + Lower class limit 2 With reference to Table 14.2, for the class 10-25, the class mark is 10 25 2 , i.e., 17.5. Similarly, we can find the class marks of the remaining class intervals. We put them in Table 14.3. These class marks serve as our x i â€™s. Now, in general, for the ith class interval, we have the frequency f i corresponding to the class mark x i . We can now proceed to compute the mean in the same manner as in Example 1. Page 4 260 MATHEMA TICS 14 There are lies, damned lies and statistics. â€” by Disraeli 14.1 Introduction In Class IX, you have studied the classification of given data into ungrouped as well as grouped frequency distributions. You have also learnt to represent the data pictorially in the form of various graphs such as bar graphs, histograms (including those of varying widths) and frequency polygons. In fact, you went a step further by studying certain numerical representatives of the ungrouped data, also called measures of central tendency, namely, mean, median and mode. In this chapter, we shall extend the study of these three measures, i.e., mean, median and mode from ungrouped data to that of grouped data. We shall also discuss the concept of cumulative frequency, the cumulative frequency distribution and how to draw cumulative frequency curves, called ogives. 14.2 Mean of Grouped Data The mean (or average) of observations, as we know, is the sum of the values of all the observations divided by the total number of observations. From Class IX, recall that if x 1 , x 2 ,. . ., x n are observations with respective frequencies f 1 , f 2 , . . ., f n , then this means observation x 1 occurs f 1 times, x 2 occurs f 2 times, and so on. Now, the sum of the values of all the observations = f 1 x 1 + f 2 x 2 + . . . + f n x n , and the number of observations = f 1 + f 2 + . . . + f n . So, the mean x of the data is given by x = 11 2 2 12 ++ + ++ + L L nn n fxfx fx ff f Recall that we can write this in short form by using the Greek letter S (capital sigma) which means summation. That is, ST A TISTICS ST A TISTICS 261 x = 1 1 n ii i n i i fx f = = ? ? which, more briefly, is written as x = S S ii i f x f , if it is understood that i varies from 1 to n. Let us apply this formula to find the mean in the following example. Example 1 : The marks obtained by 30 students of Class X of a certain school in a Mathematics paper consisting of 100 marks are presented in table below. Find the mean of the marks obtained by the students. Marks obtained 10 20 36 40 50 56 60 70 72 80 88 92 95 (x i ) Number of 113 4324 4112 3 1 student ( f i ) Solution: Recall that to find the mean marks, we require the product of each x i with the corresponding frequency f i . So, let us put them in a column as shown in Table 14.1. Table 14.1 Marks obtained (x i ) Number of students ( f i ) f i x i 10 1 10 20 1 20 . 36 3 108 40 4 160 50 3 150 56 2 112 60 4 240 70 4 280 72 1 72 80 1 80 88 2 176 92 3 276 95 1 95 Total Sf i = 30 Sf i x i = 1779 262 MATHEMA TICS Now, S = S ii i f x x f = 1779 30 = 59.3 Therefore, the mean marks obtained is 59.3. In most of our real life situations, data is usually so large that to make a meaningful study it needs to be condensed as grouped data. So, we need to convert given ungrouped data into grouped data and devise some method to find its mean. Let us convert the ungrouped data of Example 1 into grouped data by forming class-intervals of width, say 15. Remember that, while allocating frequencies to each class-interval, students falling in any upper class-limit would be considered in the next class, e.g., 4 students who have obtained 40 marks would be considered in the class- interval 40-55 and not in 25-40. With this convention in our mind, let us form a grouped frequency distribution table (see Table 14.2). Table 14.2 Class interval 10 - 25 25 - 40 40 - 55 55 - 70 70 - 85 85 - 100 Number of students 23766 6 Now, for each class-interval, we require a point which would serve as the representative of the whole class. It is assumed that the frequency of each class- interval is centred around its mid-point. So the mid-point (or class mark) of each class can be chosen to represent the observations falling in the class. Recall that we find the mid-point of a class (or its class mark) by finding the average of its upper and lower limits. That is, Class mark = Upper class limit + Lower class limit 2 With reference to Table 14.2, for the class 10-25, the class mark is 10 25 2 , i.e., 17.5. Similarly, we can find the class marks of the remaining class intervals. We put them in Table 14.3. These class marks serve as our x i â€™s. Now, in general, for the ith class interval, we have the frequency f i corresponding to the class mark x i . We can now proceed to compute the mean in the same manner as in Example 1. ST A TISTICS 263 Table 14.3 Class interval Number of students ( f i ) Class mark (x i ) f i x i 10 - 25 2 17.5 35.0 25 - 40 3 32.5 97.5 40 - 55 7 47.5 332.5 55 - 70 6 62.5 375.0 70 - 85 6 77.5 465.0 85 - 100 6 92.5 555.0 Total S f i = 30 S f i x i = 1860.0 The sum of the values in the last column gives us S f i x i . So, the mean x of the given data is given by x = 1860.0 62 30 ii i fx f S == S This new method of finding the mean is known as the Direct Method. We observe that Tables 14.1 and 14.3 are using the same data and employing the same formula for the calculation of the mean but the results obtained are different. Can you think why this is so, and which one is more accurate? The difference in the two values is because of the mid-point assumption in Table 14.3, 59.3 being the exact mean, while 62 an approximate mean. Sometimes when the numerical values of x i and f i are large, finding the product of x i and f i becomes tedious and time consuming. So, for such situations, let us think of a method of reducing these calculations. We can do nothing with the f i â€™s, but we can change each x i to a smaller number so that our calculations become easy. How do we do this? What about subtracting a fixed number from each of these x i â€™s? Let us try this method. The first step is to choose one among the x i â€™s as the assumed mean, and denote it by â€˜aâ€™. Also, to further reduce our calculation work, we may take â€˜aâ€™ to be that x i which lies in the centre of x 1 , x 2 , . . ., x n . So, we can choose a = 47.5 or a = 62.5. Let us choose a = 47.5. The next step is to find the difference d i between a and each of the x i â€™s, that is, the deviation of â€˜aâ€™ from each of the x i â€™s. i.e., d i = x i â€“ a = x i â€“ 47.5 Page 5 260 MATHEMA TICS 14 There are lies, damned lies and statistics. â€” by Disraeli 14.1 Introduction In Class IX, you have studied the classification of given data into ungrouped as well as grouped frequency distributions. You have also learnt to represent the data pictorially in the form of various graphs such as bar graphs, histograms (including those of varying widths) and frequency polygons. In fact, you went a step further by studying certain numerical representatives of the ungrouped data, also called measures of central tendency, namely, mean, median and mode. In this chapter, we shall extend the study of these three measures, i.e., mean, median and mode from ungrouped data to that of grouped data. We shall also discuss the concept of cumulative frequency, the cumulative frequency distribution and how to draw cumulative frequency curves, called ogives. 14.2 Mean of Grouped Data The mean (or average) of observations, as we know, is the sum of the values of all the observations divided by the total number of observations. From Class IX, recall that if x 1 , x 2 ,. . ., x n are observations with respective frequencies f 1 , f 2 , . . ., f n , then this means observation x 1 occurs f 1 times, x 2 occurs f 2 times, and so on. Now, the sum of the values of all the observations = f 1 x 1 + f 2 x 2 + . . . + f n x n , and the number of observations = f 1 + f 2 + . . . + f n . So, the mean x of the data is given by x = 11 2 2 12 ++ + ++ + L L nn n fxfx fx ff f Recall that we can write this in short form by using the Greek letter S (capital sigma) which means summation. That is, ST A TISTICS ST A TISTICS 261 x = 1 1 n ii i n i i fx f = = ? ? which, more briefly, is written as x = S S ii i f x f , if it is understood that i varies from 1 to n. Let us apply this formula to find the mean in the following example. Example 1 : The marks obtained by 30 students of Class X of a certain school in a Mathematics paper consisting of 100 marks are presented in table below. Find the mean of the marks obtained by the students. Marks obtained 10 20 36 40 50 56 60 70 72 80 88 92 95 (x i ) Number of 113 4324 4112 3 1 student ( f i ) Solution: Recall that to find the mean marks, we require the product of each x i with the corresponding frequency f i . So, let us put them in a column as shown in Table 14.1. Table 14.1 Marks obtained (x i ) Number of students ( f i ) f i x i 10 1 10 20 1 20 . 36 3 108 40 4 160 50 3 150 56 2 112 60 4 240 70 4 280 72 1 72 80 1 80 88 2 176 92 3 276 95 1 95 Total Sf i = 30 Sf i x i = 1779 262 MATHEMA TICS Now, S = S ii i f x x f = 1779 30 = 59.3 Therefore, the mean marks obtained is 59.3. In most of our real life situations, data is usually so large that to make a meaningful study it needs to be condensed as grouped data. So, we need to convert given ungrouped data into grouped data and devise some method to find its mean. Let us convert the ungrouped data of Example 1 into grouped data by forming class-intervals of width, say 15. Remember that, while allocating frequencies to each class-interval, students falling in any upper class-limit would be considered in the next class, e.g., 4 students who have obtained 40 marks would be considered in the class- interval 40-55 and not in 25-40. With this convention in our mind, let us form a grouped frequency distribution table (see Table 14.2). Table 14.2 Class interval 10 - 25 25 - 40 40 - 55 55 - 70 70 - 85 85 - 100 Number of students 23766 6 Now, for each class-interval, we require a point which would serve as the representative of the whole class. It is assumed that the frequency of each class- interval is centred around its mid-point. So the mid-point (or class mark) of each class can be chosen to represent the observations falling in the class. Recall that we find the mid-point of a class (or its class mark) by finding the average of its upper and lower limits. That is, Class mark = Upper class limit + Lower class limit 2 With reference to Table 14.2, for the class 10-25, the class mark is 10 25 2 , i.e., 17.5. Similarly, we can find the class marks of the remaining class intervals. We put them in Table 14.3. These class marks serve as our x i â€™s. Now, in general, for the ith class interval, we have the frequency f i corresponding to the class mark x i . We can now proceed to compute the mean in the same manner as in Example 1. ST A TISTICS 263 Table 14.3 Class interval Number of students ( f i ) Class mark (x i ) f i x i 10 - 25 2 17.5 35.0 25 - 40 3 32.5 97.5 40 - 55 7 47.5 332.5 55 - 70 6 62.5 375.0 70 - 85 6 77.5 465.0 85 - 100 6 92.5 555.0 Total S f i = 30 S f i x i = 1860.0 The sum of the values in the last column gives us S f i x i . So, the mean x of the given data is given by x = 1860.0 62 30 ii i fx f S == S This new method of finding the mean is known as the Direct Method. We observe that Tables 14.1 and 14.3 are using the same data and employing the same formula for the calculation of the mean but the results obtained are different. Can you think why this is so, and which one is more accurate? The difference in the two values is because of the mid-point assumption in Table 14.3, 59.3 being the exact mean, while 62 an approximate mean. Sometimes when the numerical values of x i and f i are large, finding the product of x i and f i becomes tedious and time consuming. So, for such situations, let us think of a method of reducing these calculations. We can do nothing with the f i â€™s, but we can change each x i to a smaller number so that our calculations become easy. How do we do this? What about subtracting a fixed number from each of these x i â€™s? Let us try this method. The first step is to choose one among the x i â€™s as the assumed mean, and denote it by â€˜aâ€™. Also, to further reduce our calculation work, we may take â€˜aâ€™ to be that x i which lies in the centre of x 1 , x 2 , . . ., x n . So, we can choose a = 47.5 or a = 62.5. Let us choose a = 47.5. The next step is to find the difference d i between a and each of the x i â€™s, that is, the deviation of â€˜aâ€™ from each of the x i â€™s. i.e., d i = x i â€“ a = x i â€“ 47.5 264 MATHEMA TICS The third step is to find the product of d i with the corresponding f i , and take the sum of all the f i d i â€™s. The calculations are shown in Table 14.4. Table 14.4 Class interval Number of Class mark d i = x i â€“ 47.5 f i d i students ( f i ) (x i ) 10 - 25 2 17.5 â€“30 â€“60 25 - 40 3 32.5 â€“15 â€“45 40 - 55 7 47.5 0 0 55 - 70 6 62.5 15 90 70 - 85 6 77.5 30 180 85 - 100 6 92.5 45 270 Total Sf i = 30 Sf i d i = 435 So, from Table 14.4, the mean of the deviations, d = ii i f d f S S . Now, let us find the relation between d and x . Since in obtaining d i , we subtracted â€˜aâ€™ from each x i , so, in order to get the mean x , we need to add â€˜aâ€™ to d . This can be explained mathematically as: Mean of deviations, d = ii i f d f S S So, d = () ii i f xa f S- S = ii i ii fxfa f f SS - SS = i i f xa f S - S = x a - So, x = a + d i.e., x = ii i f d a f S + SRead More

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