1. Virtual University of Pakistan Lecture No. 3 Statistics and Probability By: Miss Saleha Naghmi Habibullah

2. IN THE LAST LECTURE, YOU LEARNT:  Concept of sampling  Random versus non-random sampling  Simple random sampling  A brief introduction to other types of random sampling  Methods of data collection

3. TOPICS FOR TODAY Data Representation  Tabulation  Simple bar chart  Component bar chart  Multiple bar chart  Pie chart

4. The tree-diagram below presents an outline of the various techniques TYPES OF DATA QuantitativeQualitative Univariate Frequency Table Percentages Pie Chart Bar Chart Bivariate Frequency Table Multiple Bar Chart Discrete Frequency Distribution Line Chart Continuous Frequency Distribution Histogram Frequency Polygon Frequency Curve Component Bar Chart

5. In today’s lecture, we will be dealing with various techniques for summarizing and describing qualitative data. Qualitative Univariate Frequency Table Percentages Pie Chart Bar Chart Bivariate Frequency Table Multiple Bar Chart Component Bar Chart We will begin with the univariate situation, and will proceed to the bivariate situation.

6. Suppose that we are carrying out a survey of the students of first year studying in a co- education. Suppose that in all there are 1200 students of first year in this large college. We wish to determine  What proportion of students have come from Urdu medium schools?  What proportion has come from English medium schools? Example

7. Interview Results We will have an array of observations as follows: U, U, E, U, E, E, E, U, …… (U : URDU MEDIUM) (E : ENGLISH MEDIUM) Question:   What should we do with this data? Obviously, the first thing that comes to mind is to count the number of students who said “Urdu medium” as well as the number of students who said “English medium”.

8. This will result in the following table: Medium of Institution No. of Students (f) Urdu719 English481 Total1200 Important: The technical term for the numbers given in the second column of this table is “frequency”. It means “how frequently something happens?” Out of the 1200 students, 719 stated that they had come from Urdu medium schools.

9. Dividing the cell frequencies by the total frequency and multiplying by 100 we obtain the following: Medium of Institutionf% Urdu719 59.9 = 60% English481 40.1 = 40% 1200

10. Diagrammatical Representation of Data A pie chart consists of a circle which is divided into two or more parts in accordance with the number of distinct categories that we have in our data. Medium of InstitutionfAngle Urdu719 215.7 0 ENGLISH481 144.3 0 1200

11. For the example that we have just considered, the circle is divided into two sectors, the larger sector pertaining to students coming from Urdu medium schools and the smaller sector pertaining to students coming from English medium schools. How do we decide where to cut the circle? The answer is very simple! All we have to do is to divide the cell frequency by the total frequency and multiply by 360. This process will give us the exact value of the angle at which we should cut the circle.

12. Diagrammatical Representation of Data SIMPLE BAR CHART A simple bar chart consists of horizontal or vertical bars of equal width and lengths proportional to values they represent.

13. Example Suppose we have available to us information regarding the turnover of a company for 5 years as given in the table below: Years19651966196719681969 Turnover(Rupees)35,00042,00043,50048,00048,500

14. In order to represent the above information in the form of a bar chart, all we have to do is to take the year along the x-axis and construct a scale for turnover along the y-axis. 0 10,000 20,000 30,000 40,000 50,000 19651966196719681969 Next, against each year, we will draw vertical bars of equal width and different heights in accordance with the turn-over figures that we have in our table.

15. As a result we obtain a simple and attractive diagram as shown below. When our values do not relate to time, they should be arranged in ascending or descending order before-charting.

16. BIVARIATE FREQUENCY TABLE What we have just considered was the univariate situation. In each of the two examples, we were dealing with one single variable. In the example of the first year students of a college, our alone variable of interest was ‘medium of schooling’. And in the second example, our one single variable of interest was turnover.

17. Example Suppose that along with the enquiry about the Medium of Institution we are also recording the sex of the student.

18. Student No. MediumGender 1UF 2UM 3EM 4UF 5EM 6EF 7UM 8EM ::: ::: Now this is a bivariate situation; we have two variables, medium of schooling and sex of the student.

19. Bivariate Frequency Table In order to summarize the above information, we will construct a table called Bivariate Frequency Table, containing a boxhead and a stub as shown below: Sex SexMed.MaleFemaleTotal Urdu English Total Box Head Stub

20. Next, we will count the number of students falling in each of the following four categories: Male student coming from an Urdu medium school. Female student coming from an Urdu medium school. Male student coming from an English medium school. Female student coming from an English medium school.

21. As a result, suppose we obtain the following figures: Sex SexMed.MaleFemaleTotal Urdu202517719 English350131481 Total5526481200 Bivariate Frequency Table pertaining to two qualitative variables.

22. Let us now consider how we will depict the above information diagrammatically

23. component This can be accomplish by constructing the component bar chart COMPONENT BAR CHART component bar chart is also known as the subdivided bar chart.

24. In the above figure, each bar has been divided into two parts. The first bar represents the total number of male students whereas the second bar represents the total number of female students. As far as the medium of schooling is concerned, the lower part of each bar represents the students coming from English medium schools. Whereas the upper part of each bar represents the students coming from the Urdu medium schools. The advantage of this kind of a diagram is that we are able to ascertain the situation of both the variables at a glance. We can compare the number of male students in the college with the number of female students, and at the same time we can compare the number of English medium students among the males with the number of English medium students among the females.

25. The next diagram to be considered is the Multiple Bar Chart

26. MULTIPLE BAR CHART Used in a situation where we have two or more related sets of data. Example: Suppose we have information regarding the imports and exports of Pakistan for the years 1970-71 to 1974-75 as shown in the table below: YearsImports (Crores of Rs.) Exports 1970-71370200 1971-72350337 1972-73840855 1973-7414381016 1974-7520921029 Source: State Bank of Pakistan

27. A multiple bar chart is a very useful and effective way of presenting this kind of information. This kind of a chart consists of a set of grouped bars, the lengths of which are proportionate to the values of our variables, and each of which is shaded or colored differently in order to aid identification. With reference to the above example, we With reference to the above example, we obtain the multiple bar chart shown ahead:

28. Multiple Bar Chart representing Imports & Exports of Pakistan ( 1970 - 71 to 1974 - 75)

29. Difference between Component Bar Chart and Multiple Bar Chart Information available regarding Totals and their components For Example: Total no. of male students i.e. English Medium and Urdu Medium No Information regarding Totals For example: Imports and Exports do not add up to give you the totality of some one thing. Component Bar ChartMultiple Bar Chart

30. Quantitative Variable Quantitative Variable Discrete Variable Frequency Distribution Line Chart Continuous Variable Frequency Distribution Histogram Frequency Polygon Ogive

31. Example Suppose we walk in the nursery class of a school and we count the no. of Books and copies that students have in their bags. Suppose we walk in the nursery class of a school and we count the no. of Books and copies that students have in their bags. Suppose the no. of books and copies are 3, 5, 7, 9 and so on.

32. Representation of Data in a Discrete Frequency Distribution XTallyFrequency 3|1 4|||3 5 |||| |||| 9 6 |||| |||| ||| 13 7 |||| |||| 10 8|||3 9 |||| | 6 Total45

33. Graphical Representation of Discrete Data 8 10 12 2 4 6 0 3 45678 X 14 9 No. of books and copies No. of students

34. Relative Frequency Distribution Relative Frequency Distribution XFrequency Relative Frequency 31 1/45 x 100 = 2.22% 43 3/45 x 100 = 6.67% 59 9/45 x 100 = 20% 613 13/45 x 100 = 28.89% 710 10/45 x 100 = 22.22% 83 3/45 x 100 = 6.67% 96 6/45 x 100 = 13.33% Total45

35. Cumulative Frequency Distribution XFrequency Cumulative Frequency 311 43 1+3 = 4 59 4+9 = 13 613 13+13 = 26 710 26+10 = 36 83 36+3 = 39 96 39+6 = 45 Total45

36. IN TODAY’S LECTURE, YOU LEARNT   Tabular and diagrammatic representation of Quantitative data   univariate   Bivariate   Tabular and diagrammatic representation of Discrete Quantitative variable

37. IN THE NEXT TWO LECTURES, YOU WILL LEARN Tabular and Diagrammatic representation of a Continuous Quantitative Variable.  Continuous Frequency Distribution  Histogram  Frequency polygon  Frequency curve  Cumulative frequency distribution (continuous)  Cumulative frequency polygon (Ogive)