 Hello and welcome to the session. In this session, we shall study the concept of how a data collected can be represented in a histogram. The histogram is a graphical representation of data in form of rectangles to show the frequency of data items in successive intervals of equal size. In a histogram, frequency and class intervals are represented in form of rectangles and the successive class intervals are of equal size. Now let us see how to form a histogram. But before that, we must understand the method of mapping a data. Now consider the following data in which these numbers are given to us. Now we will group the data in intervals. For this, we will make a frequency distribution table for the given data in which in the first column we will write the class intervals and in the second column we will write the frequency of the particular class interval. Now where the numbers start from 3 and go on till 49, we will take intervals of equal size 10 like 0 to 10, 10 to 20, 20 to 30, 30 to 40, 40 to 50 and lower limits of these last intervals. That is, for the class interval 0 to 10, the lower limit it means for the class interval 0 to 10, the numbers greater than equal to 0 and less than 10 will be included. For the class interval 10 to 20, the numbers greater than equal to 10 and less than 20 will be included. Now from the data, we will find the numbers which are lying in these intervals in the interval 0 to 10. Only one number which is lying in the interval 0 to 10, the frequency of the interval 0 to 10 is 1. Then there are 4 numbers lying in the interval 10 to 20. So the frequency of the class interval 10 to 20 is 4. We have completed the frequency distribution table. Now there is a point to remember and that is in some histograms the intervals are taken as 0 to 9, 10 to 19 and so on in intervals 0 to 9, 0 to 9 only when integers are given only. In integers cannot be like 9.4, 9.5, etc. In case of failed numbers, we take the intervals as 0 to 10, 10 to 20 and so on in these intervals the numbers like 9.5, 0.6, etc. Lying in this interval in which all the numbers less than 10 are included. Now let us discuss an example. The following observations give the marks of students in class test here. The marks of 30 students are given to us as a histogram for this data but before that we will make frequency distribution table by the method of preparing the data. Now in this frequency distribution table in the first bracket the class interval that is the marks, the frequency of a particular class interval that is the number of students. Now here in the data there is 72 and the maximum value is 99. We will take an interval the intervals as 70 to 70, 75 to 80, 80 to 85 to 90, 90 to 95 and 95 to 100. Now there are two numbers lying in the interval 70 to 75 so the frequency for the class interval 70 to 75 is 2. Now there is only one number which is lying in the interval 75 to 80 so the frequency of the class interval 75 to 80 is 1. Now the number 80 will be included in the interval 80 to 85 the frequency for the class interval 80 to 85 we have completed this frequency distribution table for the given data. Now by using this frequency distribution table we will construct a histogram for this data. Now in the first step we have drawn the horizontal and vertical axis and here horizontal axis shows the marks of students and vertical axis shows the number of students. And now we will show the intervals from this frequency distribution table on the horizontal axis. So here we have shown the intervals on the horizontal axis and here you can see a pink or a zigzag curve near the origin which indicates that the scale along the horizontal axis does not start at the origin. Now on the vertical axis we will choose the scale of frequency. Here we have frequencies ranging from 1 to 10 the difference between the numbers on the vertical axis. You can see that we have taken the numbers as 0 to 4 on the vertical axis which interval we will draw a bar which is given by its frequency. For the class interval the frequency is 2. So here the first bar shows the class interval 70 to 75 with height of class intervals also. So here we have drawn the bars for each class interval. Here you can see the second bar shows the class interval 75 to 80 with height 1. Similarly we have third bar at height of 5 and the sixth bar at height is the required histogram for the given data. And you can answer this question which says percent of students got marks below 90. Now here the total number of students are the sum of the frequencies of all the intervals that is students is equal to 2 plus 1 plus 5 plus 8 plus 10 plus 30 also. Number of students below 90 marks are included in the intervals 70 to 75, 75 to 80, 80 to 85 and 85 to the frequencies of these class intervals to get the number of students below 90 marks. So the number of students below 90 marks is equal to, so the required percentage is equal to number of students below 90 marks that is 16 upon total number of students that is 30 into 100 which is equal to 53.3%. You should note one point that there cannot be gap or distance between the intervals in a frequency table or between the corresponding bars in a histogram. The bars of the successive intervals are joined together. If there is no data for a given interval as here you can see the frequency is 0. Then also we have included this interval in the histogram although the height of the bar of this interval is session you have learnt the concept of how to display numerical data in a histogram. And this completes our session. Hope you all have enjoyed the session.