**Click on the link below : **

https://drive.google.com/drive/folders/0B5Rb5Vq6RKr6SS01Z2FVNmtVUnM

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**DIAGRAMATIC AND GRAPHICAL REPRESENTATION: **

**TYPES OF DIAGRAMS:**

**Three-dimensional Diagrams:**

Three-dimensional diagrams, also known as volume diagram, consist 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.

**Example 9:**

Represent the following data by volume diagram.

**Pictograms and Cartograms:**

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.

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 colours or dots or placing pictogram in each geographical unit.

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**TYPES OF DIAGRAMS:**

**Two-dimensional Diagrams:**

In one-dimensional diagrams, only length 9 is taken into account. But 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**

**1.Rectangles:**

Rectangles are used to represent the relative magnitude of two or more values. The area of the rectangle is kept in proportion to the values. Rectangles are placed side by side for comparison. When two sets of figures are to be represented by rectangles, either of the two methods may be adopted.

We may represent the figures as they are given or may convert them to percentages and then subdivide the length into various components. Thus the percentage sub-divided rectangular diagram is more popular than sub-divided rectangular since it enables comparison to be made on a percentage basis.

**Example 6:**

Represent the following data by sub-divided percentage rectangular diagram.

**2. 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.

**Example 7:**

Yield of rice in Kgs. per acre of five countries are

**3. Pie Diagram or Circular Diagram:**

Another way of preparing a two-dimensional diagram is in the form of circles. 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. In constructing a pie diagram the first step is to prepare the data so that various components values can be transposed into corresponding degrees on the circle.

The second step is to draw a circle of appropriate size with a compass. The size of the radius depends upon the available space and other factors of presentation. The third step is to measure points on the circle and representing the size of each sector with the help of a protractor.

**Example 8:**

Draw a Pie diagram for the following data of production of sugar in quintals of various countries.

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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.

Moreover even a layman who has nothing to do with numbers can also understands diagrams. Evidence of this can be found in newspapers, magazines, journals, advertisement, etc.

In this chapter some of the major types of diagrams and graphs frequently used in presenting statistical data are illustrated.

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 by all. It saves considerable amount of time and energy.

Diagrams and graphs are extremely **useful** because of the following reasons.

1. They are attractive and impressive.

2. They make data simple.

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.

The construction of diagrams is an art, which can be acquired through practice. However, observance of some general guidelines can help in making them more attractive and effective. The diagrammatic presentation of statistical facts will be advantageous provided the following rules are observed in drawing diagrams.

- The measurements of geometrical figures used in diagram should be accurate and proportional.
- A diagram should be neatly drawn and attractive.
- The size of the diagrams should match the size of the paper.
- Every diagram must have a suitable but short heading.
- The scale should be mentioned in the diagram.
- Diagrams should be neatly as well as accurately drawn with the help of drawing instruments.
- Index must be given for identification so that the reader can easily make out the meaning of the diagram.
- Footnote must be given at the bottom of the diagram.
- Economy in cost and energy should be exercised in drawing diagram.

For the sake of convenience and simplicity, they may be divided under the following heads:

1. One-dimensional diagrams

2. Two-dimensional diagrams

3. Three-dimensional diagrams

4. Pictograms and Cartograms

In such diagrams, only one-dimensional measurement, i.e height is used and the width is not considered.

1. Line Diagram

2. Simple Diagram

3. Multiple Bar Diagram

4. Sub-divided Bar Diagram

5. Percentage Bar Diagram

Line diagram makes comparison easy, but it is less attractive.

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.

**Example 1:**

Show the following data by a line chart:

Simple bar diagram can be drawn either on horizontal or vertical base, but bars on horizontal base more common. 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. 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 gender-wise distribution.

**Example 2:**

Represent the following data by a 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 dotting.

**Example 3:**

Draw a multiple bar diagram for the following data.

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 colours or crossings or dottings.

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.

**Example 4:**

Represent the following data by a sub-divided bar diagram.

This is another form of component bar diagram. Here 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.

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There are three methods of classifying the data according to class intervals –

a) Exclusive method

b) Inclusive method

c) Open-end classes

When the class intervals are so fixed that the upper limit of one class is the lower limit of the next class; it is known as the exclusive method of classification. The following data are classified on this basis.

It is clear that the exclusive method ensures continuity of data as much as the upper limit of one class is the lower limit of the next class. In the above example, there are so families whose expenditure is between Rs.0 and Rs.4999.99. A family whose expenditure is Rs.5000 would be included in the class interval 5000-10000. This method is widely used in practice.

In this method, the overlapping of the class intervals is avoided. Both the lower and upper limits are included in the class interval. This type of classification may be used for a grouped frequency distribution for discrete variable like members in a family, number of workers in a factory etc., where the variable may take only integral values. It cannot be used with fractional values like age, height, weight etc.

This method may be illustrated as follows:

Thus to decide whether to use the inclusive method or the exclusive method, it is important to determine whether the variable under observation in a continuous or discrete one. In case of continuous variables, the exclusive method must be used. The inclusive method should be used in case of discrete variable.

A class limit is missing either at the lower end of the first class interval or at the upper end of the last class interval or both are not specified. The necessity of open end classes arises in a number of practical situations, particularly relating to economic and medical data when there are few very high values or few very low values which are far apart from the majority of observations.

**The example for the open-end classes as follows :**

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Data in a frequency array is ungrouped data. To group the data setting up of a 'frequency distribution' is required. A frequency distribution classifies the data into groups. It is simply a table in which the data are grouped into classes and the number of cases which fall in each class are recorded. It shows the frequency of occurrence of different values of a single Phenomenon.

1. To facilitate the analysis of data.

2. To estimate frequencies of the unknown population distribution from the distribution of sample data.

3. To facilitate the computation of various statistical measures.

The statistical data collected are generally raw data or ungrouped data. Let us consider the daily wages (in Rs ) of 30 labourers in a factory.

The above figures are nothing but raw or ungrouped data and they are recorded as they occur without any pre consideration. This representation of data does not furnish any useful information and is rather confusing to mind. A better way to express the figures in an ascending or descending order of magnitude and is commonly known as array. But this does not reduce the bulk of the data. The above data when formed into an array is in the following form:

The array helps us to see at once the maximum and minimum values. It also gives a rough idea of the distribution of the items over the range . When we have a large number of items, the formation of an array is very difficult, tedious and cumbersome. The Condensation should be directed for better understanding and may be done in two ways, depending on the nature of the data.

In this form of distribution, the frequency refers to discrete value. Here the data are presented in a way that exact measurement of units are clearly indicated. There are definite difference between the variables of different groups of items. Each class is distinct and separate from the other class. Non-continuity from one class to another class exist. Data as such facts like the number of rooms in a house, the number of companies registered in a country, the number of children in a family, etc.

The process of preparing this type of distribution is very simple. We have just to count the number of times a particular value is repeated, which is called the frequency of that class. In order to facilitate counting prepare a column of tallies.

In another column, place all possible values of variable from the lowest to the highest. Then put a bar (Vertical line) opposite the particular value to which it relates.

To facilitate counting, blocks of five bars are prepared and some space is left in between each block. We finally count the number of bars and get frequency.

**b) Continuous frequency distribution:**

In this form of distribution refers to groups of values. This becomes necessary in the case of some variables which can take any fractional value and in which case an exact measurement is not possible. Hence a discrete variable can be presented in the form of a continuous frequency distribution.

**Wage distribution of 100 employees**

The following are some basic technical terms when a continuous frequency distribution is formed or data are classified according to class intervals.

The class limits are the lowest and the highest values that can be included in the class. For example, take the class 30-40. The lowest value of the class is 30 and highest class is 40. The two boundaries of class are known as the lower limits and the upper limit of the class. The lower limit of a class is the value below which there can be no item in the class. The upper limit of a class is the value above which there can be no item to that class. Of the class 60-79, 60 is the lower limit and 79 is the upper limit, i.e. in the case there can be no value which is less than 60 or more than 79. The way in which class limits are stated depends upon the nature of the data. In statistical calculations, lower class limit is denoted by L and upper class limit by U.

The class interval may be defined as the size of each grouping of data. For example, 50-75, 75-100, 100-125… are class intervals. Each grouping begins with the lower limit of a class interval and ends at the lower limit of the next succeeding class interval. It is also called the class width.

The central point of a class interval is called the mid value or mid-point. It is found out by adding the upper and lower limits of a class and dividing the sum by 2.

Number of observations falling within a particular class interval is called frequency of that class.

Let us consider the frequency distribution of weights if persons working in a company.