diagrammatic and graphical representation of data
- 2. One of the most convincing and appealing ways in which statistical results may be presented is through diagrams and graphs. Just one diagram is enough to represent a given data more effectively than thousand words. It easy to understand diagrams even for ordinary people. Introduction
- 3. A diagram is a visual form for presentation of statistical data, highlighting their basic facts and relationship. If we draw diagrams on the basis of the data collected they will easily be understood and appreciated by all. It is readily intelligible and save a considerable amount of time and energy. Diagrams
- 4. Significance of Diagrams and Graphs 1. They are attractive and impressive. 2. They make data simple and intelligible. 3. They make comparison possible 4. They save time and labour. 5. They have universal utility. 6. They give more information. 7. They have a great memorizing effect.
- 5. General rules for constructing diagrams 1. A diagram should be neatly drawn and attractive. 2. The measurements of geometrical figures used in diagram should be accurate and proportional. 3. The size of the diagrams should match the size of the paper. 4. Every diagram must have a suitable but short heading. 5. The scale should be mentioned in the diagram. 6. Diagrams should be neatly as well as accurately drawn with the help of drawing instruments. 7. Index must be given for identification so that the reader can easily make out the meaning of the diagram. 8. Footnote must be given at the bottom of the diagram. 9. Economy in cost and energy should be exercised in drawing diagram.
- 7. In such diagrams, only one-dimensional measurement, i.e. height is used and the width is not considered. Line diagram Simple diagram Multiple bar diagram Sub-divided bar diagram Percentage bar diagram
- 8. Line Diagram • Line diagram is used in case where there are many items to be shown and there is not much of difference in their values. • Such diagram is prepared by drawing a vertical line for each item according to the scale. • The distance between lines is kept uniform. • Line diagram makes comparison easy, but it is less attractive.
- 9. Simple Bar Diagram • Simple bar diagram can be drawn either on horizontal or vertical base. • Bars must be uniform width and intervening space between bars must be equal. • While constructing a simple bar diagram, the scale is determined on the basis of the highest value in the series. • To make the diagram attractive, the bars can be coloured. • Bar diagram are used in business and economics.
- 10. Simple Bar Diagram (Cont) • However, an important limitation of such diagrams is that they can present only one classification or one category of data. • For example, while presenting the population for the last five decades, one can only depict the total population in the simple bar diagrams, and not its sex-wise distribution.
- 11. Multiple Bar Diagram • Multiple bar diagram is used for comparing two or more sets of statistical data. • Bars are constructed side by side to represent the set of values for comparison. • In order to distinguish bars, they may be either differently coloured or there should be different types of crossings or dotting, etc. • An index is also prepared to identify the meaning of different colours or dottings.
- 13. Sub-divided Bar Diagram • In a sub-divided bar diagram, the bar is sub-divided into various parts in proportion to the values given in the data and the whole bar represent the total. • Such diagrams are also called Component Bar diagrams. • The sub divisions are distinguished by different colors or crossings or dotting. • The main defect of such a diagram is that all the parts do not have a common base to enable one to compare accurately the various components of the data.
- 14. Example 4 Represent the following data by a sub- divided bar diagram.
- 15. Percentage Bar Diagram • The components are not the actual values but percentages of the whole. • The main difference between the sub-divided bar diagram and percentage bar diagram is that in the former the bars are of different heights since their totals may be different whereas in the latter the bars are of equal height since each bar represents 100 percent. • In the case of data having sub-division, percentage bar diagram will be more appealing than sub-divided bar diagram.
- 18. In two-dimensional diagrams the area represent the data and so the length and breadth have both to be taken into account. Such diagrams are also called area diagrams or surface diagrams. The important types of area diagrams are: 1. Rectangles. 2. Squares. 3. Pie-diagrams
- 19. Rectangles • Rectangles are used to represent the relative magnitude of two or more values. • The area of the rectangles are kept in proportion to the values. • Rectangles are placed side by side for comparison. • We may represent the figures as they are given or may convert them to percentages and then subdivide the length into various components.
- 20. • Solution: The items of expenditure will be converted into percentage as shown below
- 22. Squares • The rectangular method of diagrammatic presentation is difficult to use where the values of items vary widely. • The method of drawing a square diagram is very simple. • One has to take the square root of the values of various item that are to be shown in the diagrams and then select a suitable scale to draw the squares.
- 24. Pie Diagram or Circular Diagram • In such diagrams, both the total and the component parts or sectors can be shown. The area of a circle is proportional to the square of its radius. • While making comparisons, pie diagrams should be used on a percentage basis and not on an absolute basis. • Example 8: Draw a Pie diagram for the following data of production of sugar in quintals of various countries.
- 26. It consists of cubes, cylinders, spheres, etc. In such diagrams three things, namely length, width and height have to be taken into account. Of all the figures, making of cubes is easy. Side of a cube is drawn in proportion to the cube root of the magnitude of data. Cubes of figures can be ascertained with the help of logarithms. The logarithm of the figures can be divided by 3 and the antilog of that value will be the cube-root.
- 28. Pictograms are not abstract presentation such as lines or bars but really depict the kind of data we are dealing with. Pictures are attractive and easy to comprehend and as such this method is particularly useful in presenting statistics to the layman. When Pictograms are used, data are represented through a pictorial symbol that is carefully selected.
- 29. Cartograms or statistical maps are used to give quantitative information as a geographical basis. They are used to represent spatial distributions. The quantities on the map can be shown in many ways such as through shades or colors or dots or placing pictogram in each geographical unit.
- 30. Graphs A graph is a visual form of presentation of statistical data. A graph is more attractive than a table of figure. Even a common man can understand the message of data from the graph. Comparisons can be made between two or more phenomena very easily with the help of a graph.
- 32. A. Histogram A histogram is a bar chart or graph showing the frequency of occurrence of each value of the variable being analyzed. In histogram, data are plotted as a series of rectangles. Class intervals are shown on the ‘X-axis’ and the frequencies on the ‘Y-axis’ . The height of each rectangle represents the frequency of the class interval. Each rectangle is formed with the other so as to give a continuous picture. Such a graph is also called staircase or block diagram. we cannot construct a histogram for distribution with open- end classes. It is also quite misleading if the distribution has unequal intervals and suitable adjustments in frequencies are not made.
- 34. B. Frequency Polygon If we mark the midpoints of the top horizontal sides of the rectangles in a histogram and join them by a straight line, the figure so formed is called a Frequency Polygon. This is done under the assumption that the frequencies in a class interval are evenly distributed throughout the class. The area of the polygon is equal to the area of the histogram, because the area left outside is just equal to the area included in it.
- 36. C. Frequency Curve If the middle point of the upper boundaries of the rectangles of a histogram is corrected by a smooth freehand curve, then that diagram is called frequency curve. The curve should begin and end at the base line.
- 38. D. Ogives For a set of observations, we know how to construct a frequency distribution. In some cases we may require the number of observations less than a given value or more than a given value. This is obtained by a accumulating (adding) the frequencies upto (or above) the give value. This accumulated frequency is called cumulative frequency. These cumulative frequencies are then listed in a table is called cumulative frequency table. The curve table is obtained by plotting cumulative frequencies is called a cumulative frequency curve or an ogive.
- 39. Ogive The ‘ less than ogive’ method The ‘more than ogive’ method.
- 40. • In less than ogive method we start with the upper limits of the classes and go adding the frequencies. When these frequencies are plotted, we get a rising curve. • In more than ogive method, we start with the lower limits of the classes and from the total frequencies we subtract the frequency of each class. When these frequencies are plotted we get a declining curve.
- 42. Class limit Less than ogive More than ogive 20 0 110 30 4 106 40 10 100 50 23 87 60 48 62 70 80 30 80 99 11 90 107 3 100 110 0
- 44. E. Lorenz Curve Lorenz curve is a graphical method of studying dispersion. It was introduced by Max.O.Lorenz, a great Economist and a statistician, to study the distribution of wealth and income. It is also used to study the variability in the distribution of profits, wages, revenue, etc. It is specially used to study the degree of inequality in the distribution of income and wealth between countries or between different periods. It is a percentage of cumulative values of one variable in combined with the percentage of cumulative values in other variable and then Lorenz curve is drawn.
- 45. E. Lorenz Curve (Cont) The curve starts from the origin (0,0) and ends at (100,100). If the wealth, revenue, land etc are equally distributed among the people of the country, then the Lorenz curve will be the diagonal of the square. But this is highly impossible. The deviation of the Lorenz curve from the diagonal, shows how the wealth, revenue, land etc are not equally distributed among people.