Diagrammatic Presentation of a Data
This is the third method of presenting data. This method provides the quickest understanding of the actual situation to be explained by data in comparison to tabular or textual presentations. Diagrammatic presentation of data translates quite effectively the highly abstract ideas contained in numbers into more concrete and easily comprehensible form. Diagrams may be less accurate but are much more effective than tables in presenting the data. There are various kinds of diagrams in common use. Amongst them the important ones are the following:
 Geometric diagram
 Frequency diagram
 Arithmetic line graph
Geometric Diagram
Bar diagram and pie diagram come in the category of geometric diagram for presentation of data. The bar diagrams are of three types  simple, multiple and component bar diagrams.
Bar Diagram
Simple Bar Diagram Bar diagram comprises a group of equi spaced and equi width rectangular bars for each class or category of data. Height or length of the bar reads the magnitude of data. The lower end of the bar touches the base line such that the height of a bar starts from the zero unit. Bars of a bar diagram can be visually compared by their relative height and accordingly data are comprehended quickly. Data for this can be of frequency or nonfrequency type. In nonfrequency type data a particular characteristic, say production, yield, population, etc. at various points of time or of different states are noted and corresponding bars are made of the respective heights according to the values of the characteristic to construct
the diagram. The values of the characteristics (measured or counted) retain the identity of each value. Figure 4.1 is an example of a bar diagram.
Activity

You had constructed a table presenting the data about the students of your class. Draw a bar diagram for the same table. Different types of data may require different modes of diagrammatical representation. Bar diagrams are suitable both for frequency type and nonfrequency type variables and attributes. Discrete variables like family size, spots on a dice, grades in an examination, etc. and attributes such as gender, religion, caste, country, etc. can be represented by bar diagrams. Bar diagrams are more convenient for nonfrequency data such as income expenditure profile, export/imports over the years, etc.
TABLE 4.6
Literacy Rates of Major States of India
Fig. 4.1: Bar diagram showing literacy rates (person) of major states of India, 2001. rate, registered unemployed over the years, literacy rates, etc.) (Fig 4.2). Bar diagrams can have different forms such as multiple bar diagram and component bar diagram.Activities

How many states (among the major states of India) had higher female literacy rate than the national average in 2001?

Has the gap between maximum and minimum female literacy rates over the states in two consecutive census years 2001 and 1991 declined?
Multiple Bar Diagram
Multiple bar diagrams (Fig.4.2) are used for comparing two or more sets of data, for example income and expenditure or import and export for different years, marks obtained in different subjects in different classes, etc.
Component bar Diagram
Component bar diagrams or charts (Fig.4.3), also called subdiagrams, are very useful in comparing the sizes of different component parts (the elements or parts which a thing is made up of) and also for throwing light on the relationship among these integral parts.
For example, sales proceeds from different products, expenditure pattern in a typical Indian family (components being food, rent, medicine, education, power, etc.), budget outlay for receipts and expenditures, components of labour force, population etc. Component bar diagrams are usually shaded or coloured suitably.
Interpretation
It can be very easily derived from Figure 4.2 that female literacy rate over the years was on increase throughout the country. Similar other interpretations can be made from the figure like the state of Rajasthan experienced the sharpest rise in female literacy, etc.
Table 4.7
Enrolment by gender at schools (per cent) of children aged 614 years in a district of Bihar
Gender  Enrolled  Out of School (%) 
Boy  91.5  8.5 
Girl  58.6  41.4 
Total  78.0  22.0 
Data Source: Unpublished data
A component bar diagram shows the bar and its subdivisions into two or more components. For example, the bar might show the total population of children in the agegroup of 614 years. The components show the proportion of those who are enrolled and those who are not. A component bar diagram might also contain different component bars for boys, girls and the total of children in the given age group range, as shown in Figure 4.3. To construct a component bar diagram, first of all, a bar is constructed on the xaxis with its height equivalent to the total value of the bar [for per cent data the bar height is of 100 units (Figure 4.3)]. Otherwise the height is equated to total value of the bar and proportional heights of the components are worked out using unitary method. Smaller components are given priority in parting the bar.
Pie Diagram
A pie diagram is also a component Fig. 4.2: Multiple bar (column) diagram showing female literacy rates over two census years 1991 and 2001 by major states of India.
Interpretation: It can be very easily derived from Figure 4.2 that female literacy rate over the years was on increase throughout the country. Similar other interpretations can be made from the figure like the state of Rajasthan experienced the sharpest rise in female literacy, etc.
Fig. 4.3:
Enrolment at primary level in a district of Bihar (Component Bar Diagram) diagram, but unlike a component bar diagram, a circle whose area is proportionally divided among the components (Fig.4.4) it represents. It is also called a pie chart.
 The circle is divided into as many parts as there are components by drawing straight lines from the centre to the circumference.
 Pie charts usually are not drawn with absolute values of a category. The values of each category are first expressed as percentage of the total value of all the categories.
 â€‹A circle in a pie chart, irrespective of its value of radius, is thought of having 100 equal parts of 3.6Â° (360Â°/100) each. To find out the angle, the component shall subtend at the centre of the circle, each percentage figure of every component is multiplied by 3.6Â°.
 An example of this conversion of percentages of components into angular components of the circle is shown in Table 4.8.
 It may be interesting to note that data represented by a component bar diagram can also be represented equally well by a pie chart, the only requirement being that absolute values of the components have to be converted into percentages before they can be used for a pie diagram.
Distribution of Indian population by their working status (crore)
Status 
Population 
Percent 
Angular Component 
Marginal workers  9  8.8  32^{0} 
Main Worker  31  30.4  109^{0} 
NonWorkers  62  60.8  219^{0} 
Total  102  100  360^{0} 
Activities
 Represent data presented through Figure 4.4 by a component bar diagram.
 Does the area of a pie have any bearing on total value of the data to be represented by the pie diagram?
Frequency Diagram
Data in the form of grouped frequency distributions are generally represented by frequency diagrams like histogram, frequency polygon, frequency curve and ogive.
Histogram
 A histogram is a two dimensional diagram. It is a set of rectangles with bases as the intervals between class boundaries (along Xaxis) and with areas proportional to the class frequency (Fig.4.5).
 If the class intervals are of equal width, which they generally are, the area of the rectangles are proportional to their respective frequencies. However, in some type of data, it is convenient, at times necessary, to use varying width of class intervals.
 For example, when tabulating deaths by age at death, it would be very meaningful as well as useful too to have very short age intervals (0, 1, 2, ..., yrs/ 0, 7, 28, ..., days) at the beginning when death rates are very high compared to deaths at most other higher age segments of the population.
 For graphical representation of such data, height for area of a rectangle is the quotient of height (here frequency) and base (here width of the class interval). When intervals are equal, that is, when all rectangles have the same base, area can conveniently be represented by the frequency of any interval for purposes of comparison.
 When bases vary in their width, the heights of rectangles are to be adjusted to yield comparable measurements. The answer in such a situation is frequency density (class frequency divided by width of the class interval) instead of absolute frequency.
Distribution of daily wage earners in a locality of a town
Source: Unpublished data Since histograms are rectangles, a line parallel to the base line and of the same magnitude is to be drawn at a vertical distance equal to frequency (or frequency density) of the class interval.
 A histogram is never drawn for a discrete variable/data. Since in an interval or ratio scale the lower class boundary of a class interval fuses with the upper class boundary of the previous interval, equal or unequal, the rectangles are all adjacent and there is no open space between two consecutive rectangles.
 If the classes are not continuous they are first converted into continuous classes as discussed in Chapter 3. Sometimes the common portion between two adjacent rectangles (Fig.4.6) is omitted giving a better impression of continuity.
 The resulting figure gives the impression of a double staircase. A histogram looks similar to a bar diagram. But there are more differences than similarities between the two than it may appear at the first impression.
 The spacing and the width or the area of bars are all arbitrary. It is the height and not the width or the area of the bar that really matters. A single vertical line could have served the same purpose as a bar of same width. Moreover, in histogram no space is left in between two rectangles, but in a bar diagram some space must be left between consecutive bars (except in multiple bar or component bar diagram).
 Although the bars have the same width, the width of a bar is unimportant for the purpose of comparison. The width in a histogram is as important as its height.
 We can have a bar diagram both for discrete and continuous variables, but histogram is drawn only for a continuous variable. Histogram also gives value of mode of the frequency distribution graphically as shown in Figure 4.5 and the xcoordinate of the dotted vertical line gives the mode.
Frequency Polygon
 A frequency polygon is a plane bounded by straight lines, usually four or more lines. Frequency polygon is an alternative to histogram and is also derived from histogram itself.
 A frequency polygon can be fitted to a histogram for studying the shape of the curve. The simplest method of drawing a frequency polygon is to join the midpoints of the topside of the consecutive rectangles of the histogram.
 It leaves us with the two ends away from the base line, denying the calculation of the area under the curve.
Fig. 4.5: Histogram for the distribution of 85 daily wage earners in a locality of a town.
 The solution is to join the two endpoints thus obtained to the base line at the midvalues of the two classes with zero frequency immediately at each end of the distribution.
 Broken lines or dots may join the two ends with the base line. Now the total area under the curve, like the area in the histogram, represents the total frequency or sample size.
 Frequency polygon is the most common method of presenting grouped frequency distribution.
 Both class boundaries and classmarks can be used along the Xaxis, the distances between two consecutive class marks being proportional/equal to the width of the class intervals.
 Plotting of data becomes easier if the classmarks fall on the heavy lines of the graph paper. No matter whether class boundaries or midpoints are used in the Xaxis, frequencies (as ordinates) are always plotted against the midpoint of class intervals.
 When all the points have been plotted in the graph, they are carefully joined by a series of short straight lines.
 Broken lines join midpoints of two intervals, one in the beginning and the other at the end, with the two ends of the plotted curve (Fig.4.6).
 When comparing two or more distributions plotted on the same axes, frequency polygon is likely to be more useful since the vertical and horizontal lines of two or more distributions may coincide in a histogram.
Frequency Curve
The frequency curve is obtained by drawing a smooth free hand curve passing through the points of the frequency polygon as closely as possible.
Fig. 4.6: Frequency polygon drawn for the data given in Table 4.9
Fig. 4.7: Frequency curve for Table 4.9
It may not necessarily pass through all the points of the frequency polygon but it passes through them as closely as possible (Fig. 4.7).Ogivez
Ogive is also called cumulative frequency curve. As there are two types of cumulative frequencies, for example less than type and more than type, accordingly there are two ogives for any grouped frequency distribution data. Here in place of simple frequencies as in the case of frequency polygon, cumulative frequencies are plotted along yaxis against class limits of the frequency distribution.
For less than ogive the cumulative frequencies are plotted against the respective upper limits of the class intervals whereas for more than ogives the cumulative frequencies are plotted against the respective lower limits of the class interval. An interesting feature of the two ogives together is that their intersection point gives the median
Fig. 4.8 (b) of the frequency distribution.
As the shapes of the two ogives suggest, less than ogive is never decreasing and more than ogive is never increasing.
TABLE 4.10
Frequency distribution of marks obtained in mathematics
Fig. 4.8(a): 'Less than' and 'More than' ogive for data given in Table 4.10
Fig. 4.8(b): 'Less than' and 'More than' ogive the distribution it represents? for data given in Table 4.10Arithmetic Line Graph
An arithmetic line graph is also called time series graph and is a method of diagrammatic presentation of data. In it, time (hour, day/date, week, month, year, etc.) is plotted along xaxis and the value of the variable (time series data) along yaxis. A line graph by joining these plotted points, thus, obtained is called arithmetic line graph (time series graph). It helps in understanding the trend, periodicity, etc. in a long term time series data.
Activity
 Can the ogive be helpful in locating the partition values of the distribution it represents.
Value of Exports and Imports of India (Rs in 100 crores)
Year  Exports  Imports 
197778  54  60 
197879  57  68 
197980  64  91 
198081  67  125 
198182  88  143 
198283  98  158 
198384  117  171 
198485  109  197 
198586  125  201 
198687  157  222 
198889  202  282 
198990  277  353 
199091  326  432 
199192  440  479 
199293  532  634 
199394  698  731 
199596  827  900 
199697  1064  1227 
199798  1186  1369 
199899  1301  1542 
Here you can see from Fig. 4.9 that for the period 1978 to 1999, although the imports were more than the exports all through, the rate of acceleration went on increasing after 198889 and the gap between the two (imports and exports) was widened after 1995.