 Hello and welcome to the session. In this session we will discuss two variables using frequency and relative frequency. Now suppose Jenny conducts a survey in her class to know how many students have a computer or a video at home. Now there are five possibilities of getting the response when the first possibility is a student has both video and a computer. Second possibility is the student has a computer but not a video. Then third possibility is the student has a video but not a computer and the first possibility is the student neither has a video nor a computer. Now if the total number of students surveyed are fourteen then the information gathered by Jenny can written in the form of a table. Now to read the information easily we construct a two way table. It means both rows and columns together are giving the information. Now here let us take computer response in row and video response in column. Now suppose she gets the following table. Now let us see the method of reading this table. Here the number of students for each response is written in the cells. Now in cell 1 the number is 22. Now here see the corresponding row and column. Number of students having a computer and a video is 22. Now from this table we can find how many students have video and not a computer. For this first of all we will see the second row which represents the number of students who do not have a computer. But here we have to find the number of students who have a video and not a computer. For this we will check the column which represents the number of students who have a video. So here this cell tells us about the number of students having video and not a computer. So the answer for this question is 8. This means there are 8 students who have video but not a computer. Similarly we can find other number of students. Now here we can see that 6 students have computer but they do not have a video and 4 students neither have video nor computer. Now here you should know one thing that these numbers are the frequencies of the collected data. Also here the sum of all the numbers should be equal to total number of students. Now let us see the sum of all these numbers which is equal to 22 plus 6 plus 8 plus 4 which is equal to 40. And we know that here the total number of students are 40. So sum of these frequencies is equal to total number of students. Now from this table we can also find the number of students who have computer irrespective of having a video or not. Now let us see this table again. Here if we add the frequencies given in the first row it is 22 plus 6 which is equal to 28. It means 28 students have computer. Similarly adding the entries in second row we get 8 plus 4 which is equal to 12. So here 12 students do not have computer. Similarly if we add the frequencies in first column then it will be 22 plus 8 which is 30. And if we add the frequencies in the second column then it will be 6 plus 4 which is equal to 10. It means 30 students have video and 30 students do not have video. Now here you can see that 28 plus 12 is 40 and 30 plus 10 is again 40. So here we can see that total number of students are 40 and out of these 40 students 28 students have computer. 12 students do not have computer and out of these 40 students 30 students have video and 10 students do not have a video. Now let us see related frequencies or percentages using two way table. Now suppose we have to find what percent of students have computer and a video. Now from the table we know that 22 students out of 40 have computer and a video. So related frequency is equal to number of students having a computer and a video which is 22 upon total number of students which is 40. And this is equal to 0.55. Now let us see where and column related frequency or percentage. Now suppose we have to find what percent of students having computer do not have a video. Now this statement is not same as percent of students having computer and not video. Here first of all we have to find the number of students who do not have a video. Out of those students who have computer. Now here we know that the total number of students who have computer is 38. Now in the first row see the number of students who do not have a video and they are 6. So the required percentage is 6 upon 28 into 100 which is equal to 21 approximately. So here 0.21 is a row related frequency. Now let us find what percent of students not having video have a computer. For this we see only those students who do not have a video and they are 10 in number. Now in this column see the number of students having computer and they are. So the required percentage is equal to 6 upon 10 into 100 and this is equal to 60%. So here the column related frequency is 0.6. Relative frequency or percentage of each cell then we get this relative frequency table. Now we can find the relationship between two variables whenever we are given. Now for categorical data we mean the variables can be divided into categories like colour of girls, wet, green, blue. Now we can find the relationship by making the percentage or relative frequency table. For example let us study this two-way table. Now here for each age group what percent of students in the survey write the bus to the school and what percent of students do not write the bus to the school. Organize the results in a two-way table. Double show any relationship between the age group and the students writing bus or not. Explain. Now let us start with solution. Now where we have given the age group and we have to find percentage of students writing or not writing. It means here we have to find the column relative frequency and hence percentage for each cell. Now we have age group in columns and write by bus or not in rows. Now number of students writing bus in the age group of 12 to 13 is 24. Now in cell 1 the frequency is 24 and the total number of students in this age group are 40. So the required percentage will be equal to 24 upon 40 into 100 which is equal to 60%. Similarly 16 upon 40 into 100 is equal to 40%. So the percentage of students writing bus in the age group of 12 to 13 is 60% and the percentage of not writing the bus in the age group of 12 to 13 is 40%. Now in the second column the total number of students are 25 in the age group of 14 to 15. So percentage for this column is 12 upon 25 into 100 which is equal to 48% and 13 upon 25 into 100 which is equal to 52%. We can find the percentage for the third column. So we have the following table that as the age group increases the number of students travelling by bus decreases. Now here see the percentage that is 16, 48 and 40% they are decreasing for each age group. So the number of students not writing the bus increases with increase in age group. So as the age group increases students are less likely to write by bus. So in this session we have learnt about two-way tables and this completes our session. Hope you all have enjoyed the session.