## Saturday, January 30, 2010

### Graphical Distribution of Frequency Distribution

Frequency distribution can be presented graphically in any one of the following ways:
1. Histogram
2. Frequency Polygon
3. Smooth Frequency Curve
4. Cumulative Frequency Curve of Ogive Curve
5. Pie-Chart

Histogram: - A histogram is an area diagram in which the frequencies corresponding to each class interval of frequency distribution are by the area of a rectangle without leaving no gap between the cosective rectangles.

Frequency Polygon: - This is one kind of histogram which is represented by joining the straight lines of the mid points of the upper horizontal side of each rectangle with adjacent rectangles.

Smooth Frequency Curve: - This is one kind of histogram which is represented by joining the mid points by free hand of the upper horizontal side of each rectangle with adjacent rectangles.

Comulative Frequency Curve or Ogive Curve: -
The total frequency of all values less then the upper class boundary of a given class interval is called the Cumulative Frequency up to and including that class interval. The graph of such a distribution is called a Cumulative Frequency or Ogive.

There are two methods of constructing Ogive, namely

1. The less than method and
2. The more than method

The less than method: - In the “less than” method, we start with the upper boundary of each class interval and cumulative frequencies; when the frequencies are plotted, we get a rising curve.

The more than method: - In the “more than” method, we start with the lower boundary-rise of each class interval and from the total frequencies we substrate the frequency of each class when these frequencies are plotted we get a declining curve.

Draw a 'less than' ogive curve for the following data:

### To Plot an Ogive:

(i) We plot the points with coordinates having abscissae as actual limits and ordinates as the cumulative frequencies, (10, 2), (20, 10), (30, 22), (40, 40), (50, 68), (60, 90), (70, 96) and (80, 100) are the coordinates of the points.

(ii) Join the points plotted by a smooth curve.

(iii) An Ogive is connected to a point on the X-axis representing the actual lower limit of the first class.

Scale:

X -axis 1 cm = 10 marks, Y -axis 1cm = 10 c.f.

Using the data given below, construct a 'more than' cumulative frequency table and draw the Ogive.

### To Plot an Ogive

(i) We plot the points with coordinates having abscissae as actual lower limits and ordinates as the cumulative frequencies,

(70.5, 2), (60.5, 7), (50.5, 13), (40.5, 23), (30.5, 37), (20.5, 49),

(10.5, 57), (0.5, 60) are the coordinates of the points.

(ii) Join the points by a smooth curve.

(iii) An Ogive is connected to a point on the X-axis representing the actual upper limit of the last class [in this case) i.e., point (80.5, 0)].

### Scale:

X-axis 1 cm = 10 marks

Y-axis 2 cm = 10 c.f

To reconstruct frequency distribution from cumulative frequency distribution.

Pie-Chart: -Represent the following distribution by Pie-Chart: -

Problem: -

Distribution of 100 students classified according to the marks that they securated in a class for an equal class interval.

 Class Intervals Frequency 60-65 8 65-70 20 70-75 27 75-80 15 80-90 10 90-100 20