 Hello and welcome to the session. In this session we will discuss presentation of bivariate data. Now we will see what is bivariate data. It consists of the values of the two different variables that are obtained from the same population element. Each of the two variables may be qualitative or quantitative. As a result there are three combinations of variable types and these are both variables are qualitative i.e. both are attributes or one variable is qualitative i.e. attribute and the other is quantitative i.e. numerical or both variables are quantitative i.e. both are numerical. Now we will discuss two qualitative variables. Now then bivariate data results from two qualitative i.e. attribute or categorical variables. The data is often arranged on a cross tabulation or contingency table. Now let us consider the following example. A survey was conducted to investigate the favourite subject of students between maths and science. The results are shown below. Now here the table shows that 20 boys like maths and 25 like science. Also 10 girls like maths and 30 like science. Now this table can be extended to display the marginal totals or marginals. The total of the marginal totals is the grand total. Now here we see that the row total for boys will be 45 and for girls will be 40 and here the column total for the subject maths will be 30 and for science will be 55. Now here we see that the grand total will be 45 plus 40 that is 85 or we can calculate it as 30 plus 55 which is also 85. Here we should note that contingency tables often show percentages i.e. related frequencies. These percentages are based on the entire sample or on the sub-sample i.e. row or column classifications. Now we are going to discuss percentages based on grand total that is the entire sample. Now this contingency table can be converted to percentages of the grand total by dividing each frequency by the grand total and multiplying by 100 and hence we get this table. It shows that 23.5% of the boys like maths and 29.4% like science. Similarly 11.8% of the girls like maths and 35.3% like science. These same statistics that is the numerical values describing sample results can be shown in a side by side bar graph. Here we should see that this bar graph also shows the same statistics. Now we are going to discuss the percentages based on row totals or column totals. The entries in the contingency table can also be expressed as percentages of the row or column total by dividing each row or column entry by that rows or columns total and multiplying by 100. The entries in the contingency table below are expressed as percentages of the row total. Now we are going to discuss the case of one qualitative and one quantitative variable. When bivariate data results from one qualitative and one quantitative variable, the quantitative values are viewed as separate samples. Here each set is identified by levels of the qualitative variable. Each sample is described using summary statistics and the results are displayed for side by side comparison. Here statistics for comparison are measures of central tendency and measures of variation. Also graphs for comparison are dot plot, box plot. Let us consider the following example. A storekeeper randomly keeps the records of his sales per day in dollars for the month of May in different stores in the city. He recorded the following observations. In the north region, the sales are given as 50, 25, 10, 50, 50, 10, 25, 10, 50 and 15 in dollars. And in the west region the sales are 25, 35, 30, 50, 45, 25, 50, 27, 10 and 17 in dollars. Here part of the city is the qualitative variable and sales is the quantitative variable. Here sales can be compared with numerical and graphical techniques. Here we will see its comparison using dot plot. Now here this is the dot plot for the sales in north region and this is the dot plot for the sales in west region. From these dot plots we can conclude that sales in west region are more than sales in north region. Now we are going to discuss two quantitative variables. Here these variables are expressed as ordered pairs that is in the form of the ordered pair XY where X is the input variable which is independent and Y is the output variable that is the dependent variable. Now here we are going to discuss scatter diagram which is a plot of all the ordered pairs of bivariate data on a coordinate axis system. The input variable X is plotted on the horizontal axis and the output variable Y is plotted on the vertical axis. Let us consider the following example. The given table shows weights in kilograms and ages in years of students. We have to construct a scatter diagram for this data. Now this is the required scatter diagram for the given data. Here we have taken age along the horizontal axis and weight along the vertical axis. Thus in this session we have discussed presentation of bivariate data. This completes our session. Hope you enjoyed this session.