CBSE Class 11 Applied Mathematics Interpretation of Data Notes Set 01

Diagrammatic Presentation of Data

Introduction

To understand the various trends of the data at a glance and to facilitate the comparison of various situations, the data are presented in the form of diagrams and graphs.
There are a large number of diagrams which can be used for presentation of data. The selection of a particular diagram depends upon the nature of the data, the object of presentation, ability and experience.

Simple Bar Diagram

A simple bar diagram comprises a group of rectangular bars of equal width, one for each class or category of data.
1. In this type of diagram, one bar represents only one figure. There will be as many bars as the number of figures.
2. These diagrams depict only one characteristic of the data.
3. The distance between the bars should be equal.
4. The height (or length) of the bar reads the magnitude of the data. The lower end of the bar touches the base line such that the height of a bar starts from the zero unit.
5. These diagrams can be either vertical or horizontal.
6. Bars of a bar diagram can be visually compared by their relative heights and accordingly the data are comprehended quickly.

Pie Diagram

Similar to a sub-divided bar diagram, a circle can also be partitioned into sections to show the proportions of various components. Such a diagram is known as a pie diagram. The circle is divided into as many parts as there are components by drawing straight lines from the centre to the circumference.
1. The size of each section indicates the proportion of each component part to the whole.
2. Different sectors representing the various component parts should be distinguished from one another by using different shades or colours, or by giving explanatory or descriptive labels.
3. A pie diagram is also known as an ‘Angular Circle Diagram’.

Steps to Draw a Pie Diagram

1. Construct a circle of an appropriate size with a compass, considering the size of the paper and the data to be represented.
2. Convert the values of the various components into percentages of the total.
3. The percentage values are converted into corresponding degrees in the circle. Since one circle contains 360° and the percentage values of all the items add up to 100, each percentage is multiplied by \( \frac{360^\circ}{100} = 3.6^\circ \) to get the size of the angles.
4. Take a radius (preferably horizontal) as the base line to draw the angle representing the first component. The new line will become the base for the angular representation of the second component. Repeat this procedure till all the components are represented.
5. Represent the various components by different shades, designs or colours for proper identification.

Graphic Presentation of Data

Graphic Presentation

From the statistical point of view, graphic presentation is much more accurate and appropriate than diagrammatic presentation. Like diagrammatic presentation, graphic presentation also provides a quick and easier way of understanding various trends of data. It also facilitates the comparison of two or more situations.

Construction of Graphs

Graphs are generally drawn on a specially designed paper, known as graph paper. Graph paper is a fine network of horizontal and vertical lines, dividing every inch or centimetre into 10 equal parts.

How to Construct a Graph

Step 1: Draw two straight lines which intersect each other at right angles. These lines are called axes.
Step 2: The horizontal line is known as the abscissa or x-axis and the vertical line as the ordinate or y-axis.
Step 3: The point at which they intersect each other is called the point of origin, denoted by O and represented by zero.
Step 4: By convention, the independent variable is normally measured along the x-axis and the dependent variable along the y-axis. The horizontal scale (scale of the x-axis) need not begin with 0. The y-axis, however, should have a scale beginning with 0.
Step 5: Select an appropriate scale. A scale indicates the unit of a variable that a fixed length of the axis would represent. It may be different for the two axes, but it should be taken in such a way as to accommodate the whole data on the given graph paper in a lucid and attractive style.
Step 6: Plot the data on the graph paper on the basis of the values of the X and Y variables. It is essential to have two values, one representing variable X and the second representing variable Y, to plot a point on a graph.
Step 7: Join the points (pairs of values) on the graph paper.
Step 8: Use an index if different kinds of lines are drawn. The index should clearly show which line represents which variable.
Step 9: Assign an appropriate title. The title should indicate the facts presented by the graph in a comprehensive and unambiguous manner.

Kinked Line

With the y-axis starting from zero, we can also start the x-axis from zero. In order to reduce the distance between ‘zero’ and the ‘minimum value’ on the horizontal axis (x-axis), we make use of a ‘kinked line’ (false base line), as shown in the figure below (scale: 1 cm = 10 marks on the x-axis, 1 cm = 4 students on the y-axis).

Graphs of Frequency Distributions

Frequency Distribution Graphs

In case of a frequency distribution, data is expressed in terms of items or class intervals. The class limits or unit values are taken along the x-axis and the frequencies along the y-axis.
The most common forms of graphs of a frequency distribution are:
1. Line frequency graph
2. Histogram
3. Frequency polygon
4. Frequency curve
5. Ogive or cumulative frequency curve

Histogram

A histogram is a graph of a frequency distribution in which the class intervals are plotted on the x-axis and their respective frequencies on the y-axis. It is a two-dimensional diagram.
(a) A histogram is a very popular method of presenting continuous series or class-interval frequency distributions. A histogram is never drawn for a discrete variable.
(b) The height of each rectangle shows the frequency of the class and the breadth indicates the size of the class interval. Therefore, the total area covered by the histogram represents the total frequency.
(c) Each rectangle is adjacent to the other to give a continuous picture.

Histogram when Equal Class Intervals are Given

In case of equal class intervals, the height of each rectangle is taken to be equal to the frequency of the corresponding class. In this case:
(a) On the x-axis, each class interval is drawn, in which the width of the rectangle is equal to the magnitude of the class interval.
(b) On the y-axis, we plot the frequencies.

Histogram when Unequal Class Intervals are Given

In case of unequal class intervals, frequencies are first adjusted before constructing the histogram. The frequency distribution is adjusted in accordance with equal class width.
Steps needed to adjust frequencies:
Step 1: Determine the width of the lowest class interval and take this width as the standard one.
Step 2: Determine the width of the other class intervals and calculate the ‘adjustment factor’ for each class.
\( \text{Adjustment factor for any class} = \frac{\text{Width of the class}}{\text{Width of the lowest class}} \)
Step 3: The frequencies of the classes with the lowest width are not changed and the frequencies of all other classes are adjusted by dividing them by the adjustment factor.

Frequency Polygon

A frequency polygon is another method of representing a frequency distribution on a graph. A frequency polygon is an alternative to a histogram and can also be derived from the histogram itself.

Frequency Polygon in a Discrete Series

In case of a discrete frequency distribution, the following steps are followed:
Step 1: Take the values of the variable on the x-axis and the corresponding frequencies on the y-axis.
Step 2: Plot the various frequencies and join the points by straight lines.
Step 3: The figure so obtained is extended to the base line at both ends by joining the extreme points (first and last point) to the two hypothetical values of the variable before the first value and after the last value, assumed to have zero frequencies.

Frequency Polygon in a Continuous Series

In case of a continuous frequency distribution, there are two ways of making a frequency polygon:
(i) With the help of a histogram
(ii) Without the help of a histogram
(i) Frequency polygon with the help of a histogram:
Step 1: Draw the histogram of the given frequency distribution.
Step 2: Join the mid-points of the tops (upper horizontal sides) of the adjacent rectangles of the histogram by straight lines.
Step 3: The figure so obtained is closed by extending the two ends to the base line. In doing this, two hypothetical classes, one at each end, have to be included, each with zero frequency.

(ii) Frequency polygon without the help of a histogram:
Step 1: Take the mid-points of the class intervals on the x-axis and the frequencies on the y-axis.
Step 2: Plot the various frequencies and join the points by straight lines.
Step 3: The figure so obtained is extended to the base at both ends by joining the extreme points (first and last point) to the mid-points of the two hypothetical classes (before the first class and after the last class), assuming that their frequencies are zero.

Frequency Curve

When the vertices of a frequency polygon are joined by a smooth curve, the resulting figure is known as a frequency curve.
(a) A frequency curve may not necessarily pass through all the points of the frequency polygon, but it passes through them as closely as possible.
(b) The curve is drawn free hand in such a manner that the area included under the curve is approximately the same as that of the polygon.
(c) The curve should start and end at the base line. As a general rule, it may be extended to the mid-points of the class intervals just outside the histogram.
(d) The area under the curve should represent the total frequency of the whole distribution.
Frequency polygon vs frequency curve: The only difference between the two is that in a frequency polygon the points are joined by straight lines, whereas in a frequency curve we join these points free hand.
Steps of a frequency curve:
Step 1: Draw a histogram with the given data.
Step 2: Obtain the mid-points of the upper horizontal side of each rectangle.
Step 3: Join these mid-points of the adjacent rectangles of the histogram by a smooth free hand curve to get the frequency curve.
Step 4: Extend the ends of the frequency curve to the base line free hand.

Ogive or Cumulative Frequency Curve

The curve obtained by representing a cumulative frequency distribution on a graph is known as a cumulative frequency curve or ogive.
As there are two types of cumulative frequency distributions, viz. ‘less than’ and ‘more than’ cumulative frequencies, we have accordingly two types of ogives:
(i) ‘Less than’ ogive
(ii) ‘More than’ ogive

‘Less than’ Ogive

Step 1: In the ‘less than’ method, the frequencies of all the preceding class intervals are added to the frequency of a class.
Step 2: Plot the ‘less than’ cumulative frequencies against the upper limits of the corresponding class intervals.
Step 3: The points so obtained are joined by a smooth free hand curve to give the ‘less than’ ogive. The ‘less than’ ogive so obtained is an increasing curve, sloping upwards from left to right.

‘More than’ Ogive

Step 1: In the ‘more than’ method, the frequencies of all the succeeding class intervals are added to the frequency of a class, i.e. start with the total frequency at the lower limit of the first class and go on subtracting the frequency of each class.
Step 2: Plot the ‘more than’ cumulative frequencies against the lower limits of the corresponding class intervals.
Step 3: The points so obtained are joined by a smooth free hand curve to give the ‘more than’ ogive. The ‘more than’ ogive so obtained slopes downwards from left to right.