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Mode can be obtained from Histogram, and is obtained by joining the upper end points of the highest rectangle to the diagonal end points of adjacent rectangles, the intersection of diagonal end points a perpendicular drawn to the x-axis, which gives the mode.

Usually the two Ogives are drawn together with common axes. The point of intersection of two Ogives correspond to Median at the x-axis of the distribution.

(a) Histogram: – “A Histogram is a pictorial representation of graphs of frequency distribution by means of adjacent rectangles, whose areas are proportional to the frequencies represented” The Histogram can be constructed by taking variable (class intervals) on x-axis and class frequency (f) along y-axis. On each of the class intervals rectangles are erected. The width and height of the rectangles are proportional to the length of the class and class frequencies respectively.

The graph formed by series of such rectangles adjacent to one another is called Histogram. From the Histogram Mode(Z) can be obtained by joining the end point of the highest rectangle to the diagonal end point of the adjacent rectangles, and a perpendicular drawn to intersection of these lines to the x-axis, which gives the value of mode.

On the basis of Histogram, Frequency polygon and Frequency curve can be constructed. Frequency distribution with Inclusive class intervals should be converted into Exclusive and for unequal width of the class interval. Histogram is constructed with, width of the class interval against Frequency density (f/w).

(b) Frequency Polygon: – This graph is preferred when two or more frequency distributions are required to compare on the same graph. It is so called because of its resemblance with the plane geometrical figure polygon (many angled) representing frequency distribution. We can construct polygon in two ways- By drawing first Histogram and then joining the. mid-points of the upper horizontal lines of each rectangles. Thus obtained polygons ends are extended to touch the base line at a distance of half class interval.

Another method of drawing a polygon is that by taking all the midpoints of the class intervals and the corresponding frequencies areplotted. Thus obtained end points are joined by straight lines. And end points are extended to reach the base line at a distance of half class interval.

(c) Frequency Curve: – A frequency polygon obtained from the Histogram or direct by midpoints of the various classes, is not a smooth curve. Its boundaries are made up of straight lines and it has sharp corners, these sharp corners can be removed by a free hand curve drawn along the frequency polygon.

(d) Ogives/cumulative frequency curves: – Sometimes it is required to plot a graph of variables which is less than some value or more than a value. So, in such cases we are required to add up the frequencies lying below or above a given point of variable. Thus added frequencies are called cumulative frequency. The curve obtained by plotting cumulative frequency and the respective variable is called cumulative frequency curves or Ogives. There are two types of Ogives (i) Less than Ogive (ii) More than Ogive

• ‘If a curve is drawn for the cumulative frequency added from the top (l.c.f) and the upper limits of the class is called Less Than Ogive ’.
• Similarly ‘If a curve is drawn for the cumulative frequency from below (m.c.f) and lower limits of the classes then curve is called More Than Ogive’’.
• Here the variable is taken along x-axis and l.c.f / m.c.f. along y-axis. The corresponding points are joined by a smooth curve. The resulting graph is Less/More than Ogive

Frequency Polygon: – This graph is preferred when two or more frequency distributions are required to compare on the same graph. It is so called because of its resemblance with the plane geometrical figure polygon (many angled) representing frequency distribution. We can construct polygon in two ways- By drawing first Histogram and then joining the. mid-points of the upper horizontal lines of each rectangles. Thus obtained polygons ends are extended to touch the base line at a distance of half class interval.

Another method of drawing a polygon is that by taking all the midpoints of the class intervals and the corresponding frequencies areplotted. Thus obtained end points are joined by straight lines. And end points are extended to reach the base line at a distance of half class interval.

Graphs /diagrams are -only visual aids, and they cannot be considered as alternatives to numerical data.

• Graphs/diagrams are not accurate and gives only a rough idea of,the data.
• They cannot be used for further analysis of data. 
• They can be easily misled and can create wrong impression about the data.

Frequency distributions when they are presented in Tabular form becomes dull and uninteresting, moreover they require close reading of the figures to understand what is represented. Diagrams and graphs are the means of making the data easy to understand even by layman at a glance.

In one dimensional diagram only the height (length) is considered. They are mostly bar diagrams. Here, only one characteristic is considered to represent the data.

Some of the one dimensional diagrams are: 

• Simple Bar diagram 
• Multiple Bar diagram 
• Component/Subdivided Bar diagram 
• Percentage Bar diagram.

General Rules for constructing Diagrams and Graphs are: – 

• Every diagrams and should have a suitable Title and is written above it. 
• The proper scale according with the size of the paper should be selected. 
• It should be neat and clean.
• It should not be overloaded with more information
• To indicate different parts suitable shadings, colours, crossings should made use of.
• An Index indicating different shades colours, crossing etc. used should be clearly shown
• It should be complete in all respects
• It should be simple and self explanatory.

Some commonly used graphs are : 

• Histogram 
• Frequency Polygon 
• Frequency Curve 
• Ogives.

(a) Simple bars or thick lines aredrawn to represent the items of the data. The length of the bar is taken in proportion to the magnitude of the item in the data. The proper width of the bar is taken merely for attraction, but’it is nothing to do with the data. It represents only one character of the data. 

Ex- The figures of Imports, Exports, and Population etc. for few years are to be represented by simple bars.

(b) In subdivided bar diagrams, where in some problems it is required to represent more than one or two variables of the same kind, we use component/sub-divided bar diagrams. In this case each bar is subdivided to in to various components and different shades, crossings, colours are used, and an Index is given to that effect. These are useful in comparing the total magnitudes, along with the components.

Ex: The figures of expenditure of family, Exports or Imports of certain commodities, population according to sex,

1. Diagrams give only an approximate idea where as graphs give more accurate information. 

2. Diagrams are drawn on plain paper where as graphs need graph paper. 3. Diagrams can be easily understood than graphs. 

4. Time series and frequency distributions can be represented by graphs but not by diagrams. 

5. Diagrams are more suitable for publicity and advertisement, where as graphs are suitable for statistical analysis.

As in bar diagram, a bar is sub-divided to represent its components. Where as in the pie- diagram a circle is subdivided into sectors by subtending the angle at the circle. The total of the components are equated to 360° and each component is expressed in degrees. The area of the sector formed by the angle measured by the degrees of the component is proportional to the magnitude of that component. To distinguish between the different components shades, coloures are used. It is so called because, it resembles a pie (cake) and components resemble slices of the pie. It is also known as angular diagram or sector graphs.

Generally X and Y-axes begin from zero ie, the origin is at zero. When lowest value to be plotted is high and detailed scale needed to study all the variations in the data taking zero at the origin becomes impractical. Hence, if the fluctuations in the variable are too small, or if the lowest value of the variable is large, the False Base Line should be used, Here False Base Line can be shown by vertical wavy line between zero and first scale or the axes can be broken by kinked line.

A Histogram is a pictorial representation of graphs of frequency distribution by means of adjacent rectangles, whose areas are proportional to the frequencies represented”.

Where as frequency polygon is preferred when two or more frequency distributions are required to compare on the same graph. It is so called because of its resemblance with the plane geometrical figure polygon (many angled) representing frequency distribution. We can construct polygon in two ways- By drawing first Histogram and then joining the mid-points of the upper horizontal lines of each rectangles.