 Hello and welcome to the session. In this session, we will discuss graphical representation of data. To understand the actual data, we need graphs so that we understand the data better. We have three graphical representation of data. The first one is bar graphs. A bar graph is basically a pictorial representation of data in which usually bars of uniform width are drawn with equal spacing between them on one axis, say X axis. On the X axis basically we depict the variable and on the other axis that is on the Y axis we depict the values of the variable and the heights of the bars depend on the values of the variable. Consider this table which shows 70 students from a locality use different modes of transport like as you can see that 4 students use car, 27 students use bus, 11 students use moped, 20 students use bicycle, 8 students use rickshaw. Let's draw the bar graph representing this data. We have represented the mode of transport on the X axis that is on the horizontal axis. The scale on X axis that we have taken is one unit represents one mode of transport and as you can see on the Y axis we have represented the number of students and the scale on Y axis that we have taken is one unit represents five students. Now let's draw the bar graph for this one. To represent the first mode of transport that is car we draw a rectangular bar with width one unit and height four units. This rectangular bar is of width one unit and height four units. This represents the mode of transport car and it shows that four students use car as the mode of transport. In the same way we will draw the rectangular bars for different modes of transports and all these modes of transport would be represented on the X axis leaving a gap of one unit in between two consecutive bars. So this is how we represent different rectangular bars for different modes of transport. This is the bar graph for the given data. Now next way of representing the data graphically is by a histogram. Now let's see what is a histogram. A histogram is basically a graphical representation of a grouped frequency distribution with continuous classes. Now let's see how we can draw histogram of uniform width. Consider this data in which we are given the marks and the number of students. Now this is a grouped frequency distribution in which the classes that we are given are continuous. Let's represent this data by histogram. Histogram is basically like bar graph only but the difference is that it is used for continuous class intervals. We represent the marks on the X axis that is on the horizontal axis and the scale that we have taken on X axis is given by one unit represents ten marks and on the Y axis as you can see we have represented the number of students and the scale on the Y axis is taken as one unit represents five students. Now let's represent this data by histogram. We shall now draw the rectangular bars of width equal to the class size and the lengths of the rectangular bars should be according to the frequencies of the corresponding class intervals. This is the histogram for the given data as you can see there are no gaps in between consecutive rectangles so this resultant graph appears like a solid figure. We get one more thing that the area of the rectangles that we have formed would be proportional to the corresponding frequencies and since the widths of the rectangles are all equal so we have that the lengths of the rectangles are proportional to the frequencies. So this is how we can draw histogram for the given data when the class intervals are of uniform width. Next we see how we can draw histogram of varying width. Consider this data in which we are given the class interval and the corresponding frequencies. As you can see that the class intervals are different sizes that is the class intervals are of varying width. We need to make certain modifications in the lengths of the rectangles or you can say in the frequencies so that we get the areas of the rectangle that we form proportional to the frequencies. Now let's write the width of the class for each class interval for the first one it's 5 5 10 20 and 30. Now next we shall select a class interval with the minimum class size. Now as you can see the minimum class sizes 5 so what we do is the lengths of the rectangles are modified to be proportionate to the class size 5. So now we will find the length of the rectangles that is the adjusted frequencies or you can say the new frequency. This is given by the frequency of the given class interval that is for the first class interval that is 10 to 15 the frequency 6 upon the class size that is 5 multiplied by the minimum class size which is 5 this is equal to 6. Now for the next one it would be frequency that is 10 upon the class size that is 5 multiplied by the minimum class size this gives us 10. Now for the next one it is the frequency which is 10 upon the class size which is 10 multiplied by the minimum class size which is 5 this comes out to be equal to 5. Now for the next class interval that is 30 to 50 we have the adjusted frequencies given by the frequency which is 8 upon the class size that is 20 multiplied by the minimum class size which is 5 this gives us 2. Now for the next one it is frequency that is 18 upon the class size that is 30 multiplied by the minimum class size that is 5 and this is equal to 3. To draw the histogram of varying width we take the class intervals on the x-axis by taking the scale on x-axis as 1 unit equal to 10 and the scale on y-axis is given by 1 unit equal to 2 and on the y-axis we have represented the adjusted frequency that we have just calculated. Now we shall draw the rectangular bars this is the histogram of varying width representing the given data. Now next way of representing the data graphically is by frequency polygon let's see how we do this for this we need the histogram of uniform width this is the histogram of uniform width that we had already drawn for the given data. Now to draw frequency polygon using this histogram what we do is we first mark the midpoints of the upper sides of the adjacent rectangles of this histogram we mark these midpoints as B, C, D, E, F, G, H. Now we take this interval 0 to 10 before the interval 10 to 20 as the interval with zero frequency let's mark the midpoint of this interval as A. We also take the midpoint of the interval 80 to 90 now the interval 80 to 90 that is the interval after the interval 70 to 80 has zero frequency we take its midpoint as I. Now we join these points A, B, C, D, E, F, G, H, I by line segments on joining these points by line segments we get this and this is the frequency polygon that is we say the figure A, B, C, D, E, F, G, H, I is the frequency polygon. Frequency polygons can also be drawn independently that is without drawing the histograms for this we need the midpoints of the class intervals given to us and the midpoints of the class intervals are called class marks that is class mark is equal to the upper limit plus the lower limit total upon 2. Consider this data given to us in which we have the marks and the number of students. Now we will draw frequency polygons for this data without drawing the histogram. So for this as you know that we need the class marks now the class mark for the first interval that is 10 to 20 is given by upper limit that is 20 plus 10 upon 2 that gives us 15 for the next one also in the same way we get 25 then 35 45 55 65 and 75. Now we have taken the marks on the x-axis and the number of students on the y-axis. Now we mark these class marks also on the x-axis. Now the points that we need to plot on this graph would be point B with coordinates 15 7 C 25 11 D 35 9 E 45 13 F 55 16 G 65 4 and H 75 2 that is the x coordinate that we have taken is the class mark and the y coordinate that we have taken is the frequency or the number of students. Now we shall plot all these points on this graph as you can see we have marked the points B C D E F G H. We also have interval 0 to 10 before the interval 10 to 20 with zero frequency. Now the class mark for this interval would be 5 and we mark this point as that is the class mark for this interval 0 to 10 as A. Now next we have the interval 80 to 90 after the class interval 70 to 80 which has zero frequency. The class mark for this interval is 85 and we mark this point that is the class mark of this interval as I. Now we join the points A B C D E F G H I by line segments. So in joining these points we get this figure which is the frequency polygon so we say that A B C D E F G H I is a frequency polygon. So this is how we can draw a frequency polygon without drawing histogram. This completes the session hope you have understood all the three graphical representation of data that is biograph histogram and frequency polygon.