 Hello and welcome to the session. In this session, we are going to discuss the following question and the question says that calculate the mode from the following data. Here the distribution is given such that the class size is given in the range of 0 to 5, 5 to 10, 10 to 20, 20 to 25, 25 to 30, 30 to 40, 40 to 44, 44 to 50 and 50 to 60 which we call the corresponding frequency given by 2, 5, 8, 10, 12, 24, 12, 11 and 7. If the class intervals are unequal then make them equal by combining two or more classes or by breaking one class into two or more classes. After calculating the modal group the value of mode is found by applying the formula m0 or z is equal to l1 plus fm minus f1 upon twice of fm minus f1 minus f2 into i with l1 is the lower limit of the modal class is the frequency of the modal class f1 is the frequency of the class preceding the modal class f2 is the frequency of the class succeeding the modal class and i is the width of the modal class. With this key idea we shall proceed with the solution since the given class intervals are unequal we will make them equal by combining two or more classes. So first we make the class intervals equal by combining two classes wherever required. Now in each case the width of the class interval is 10. Now we'll find the modal class by grouping method as the difference between the highest frequency and the preceding frequency is very small also the difference between the highest frequency and the succeeding frequency is small. For grouping method we'll prepare the grouping table. It consists of six columns. Now we write the frequencies in column 1 grouped by 2s that is 7 and 8, 22 and 24, 23 and 7 and write their total in column 2 that is 15, 46, 30 lead to first frequency that is 7 and group the remaining series by 2s and write their total in column 3 that is 8 plus 22 is 30, 24 plus 23 is 47 lead to last frequency 7 grouped by 3s that is 7, 8 and 22, 24, 23 and 7 and write their total in column 4 that is 37 and 54 lead to first frequency that is 7 and group the remaining by 3s that is 8, 22 and 24 and write their total in column 5 that is 54 lead to last two frequencies that is 23 and 7. Now leave the first two frequencies and group the remaining by 3s and write their total in column 6 that is 22 plus 24 plus 23 is 69. Leave the last frequency that is 7. Now circuit the highest frequency of each column that is 24 in column 1, 46 in column 2, 47 in column 3, 54 in column 4, 54 in column 5 and 69 in column 6. This is the required grouping table. Now we shall prepare the analysis table. In analysis table put the column numbers on the left hand side and the modal values of each column on the right hand side. Now enter into columns the highest frequency marked in the grouping table. In column 1 the highest frequency is 24 which lies in the class interval 30 to 30. So mark the value of x equal to 30 to 40 in column 1. For column 2 the highest frequency is 46 which lies in the class interval 20 to 30 and 30 to 40. So mark the values of x equal to 20 to 30 and 30 to 40 in column 2. For column 3 the highest frequency is 47 which lies in the class interval 30 to 40 and 40 to 50. So mark the value of x equal to 30 to 40 and 40 to 50 in column 3. For column 4 the highest frequency is 54 which lies in the class interval 30 to 40, 40 to 50 and 50 to 60. So mark the values of x equal to 30 to 40, 40 to 50 and 50 to 60 in column 4. For column 5 the highest frequency is 54 which lies in the class interval 10 to 20, 20 to 30 and 30 to 40. So mark the values of x equal to 10 to 20, 20 to 30 and 30 to 40 in column 5. For column 6 69 is the highest frequency which lies in the class interval 20 to 30, 30 to 40 and 30 to 50. So mark the values of x equal to 20 to 30, 30 to 40 and 40 to 50 in column 6. Now taking the total of each column we get the x value against which the total is the highest is the model class that is 30 to 30. Now from broken and the analysis table we find that the model class is 30 to 40. From the key idea we know that mode is given by the formula L1 plus fm minus f1 upon slice of fm minus f1 minus f2 into i where L1 is the lower limit of the model class that is 30, fm is the frequency of the model class that is 24, f1 is the frequency of the class preceding the model class that is 22, f2 is the frequency of the class succeeding the model class that is 23, i is the width of the model class given by 40 minus 30 that is 10. Now substituting all these values in this formula we get z is equal to L1 that is 30 plus fm minus f1 that is 24 minus 22 upon slice of fm that is 2 into 24 minus f1 minus f2 that is minus 22 minus 23 into i that is 10 which is equal to 30 plus 2 upon 48 minus 45 into 10 equal to 30 plus 20 upon 3 that is 30 plus 6.67 which is equal to 36.67 so z is equal to 36.67 therefore the mode for the given table is 36.67 which is the required answer this completes our session hope you enjoyed this session.