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Graphical Representation of Cumulative Frequency Distribution

Real Life Applications for Ogives
 


In aerodynamic, an ogives is a pointed, curved surface used to form the streamlined nose of a bullet shell.


In Gothic architecture, a unique combination of existing technologies established the emergence of a new building style. Those technologies were the ogival or pointed arch. In Gothic architecture, ogives are the intersecting transverse ribs of arches that establish the surface of a Gothic arch .Ogival arch is a decorative arch with a pointed head, formed of two ogee, or S-shaped curves.
 


In glaciology, ogives are three dimensional wave bulges of glaciers that have experienced extreme topographical changes. Forbes bands are light and dark bands that appear down glacier, resulting from different ice densities.

 

There are different types of graphical representation of statistical data.

a) Bar graphs

b) Histogram

c) Frequency polygon

d) Cumulative frequency Distribution


Now we can learn in detail about Cumulative frequency distribution as you have learned all other graphical representations

Cumulative frequency distribution

Cumulative frequency is obtained by adding the frequency of a class interval and the frequencies of its preceding intervals upto that class interval.


This is explained by an example below.

Daily income (in Rs)

100-120

120-140

140-160

160-180

180-200

Number of workers

12

14

8

6

10

 

Daily Income (in Rs)

Daily Income (in Rs)

(Upper Limit)

Number of Workers

Cumulative Frequency

100 - 120

Less than 120

12

12

120 - 140

Less than 140

14

12 + 14 = 26

140 - 160

Less than 160

8

26 + 8 = 34

160 - 180

Less than 180

6

34 +6 = 40

180 – 200

Less than 200

10

40 + 10 =50


The above distribution is called ‘less than’ cumulative frequency distribution.

To represent the data graphically,
 

1) Mark the upper limits of the class interval on the x – axis and the corresponding cumulative frequencies on the y − axis choosing suitable scale.
 

2) Plot the points with coordinates having abscissa as upper limits and ordinates as the cumulative frequencies.
 

3) Join the points by a free hand smooth curve
 

4) The curve we get is called ‘Cumulative frequency curve’ or ‘less than ogive’

 

This graphical representation of the frequency distribution is called Ogive.


Now we can see the ‘more than’ cumulative frequency distribution

Daily Income (in Rs)

Daily Income (in Rs)

(Lower Limit)

Number of Workers

Cumulative Frequency

100 - 120

More than or equal to 100

12

50

120 - 140

More than or equal to 120

14

50 - 12 = 38

140 - 160

More than or equal to 140

8

38 – 14 = 24

160 - 180

More than or equal to 160

6

24 – 8 = 16

180 – 200

More than or equal to 180

10

16 – 6 = 10


To represent the data graphically,

1) Mark the lower limits of the class interval on the x – axis and the corresponding cumulative frequencies on the y − axis choosing suitable scale.


2) Plot the points with coordinates having abscissa as lower limits and ordinates as the cumulative frequencies.


3) Join the points by a free hand smooth curve.


4) The curve we get is called ‘Cumulative frequency curve’ or ‘more than ogive’.

Relation of Ogive and Median

 
 


These graphical representation of the frequency distribution are called Ogives.


Relation between median and Ogive

Graphical Representation of Cumulative Frequency Distribution

1. By an Ogive

Actual limits are on the x-axis and cumulative frequencies on the y-axis. The middle value I = where N = Σ f is then marked on the y-axis .From the marked point draw a line parallel to x – axis till it cuts the curve. At that point drop a perpendicular . The point where the perpendicular meets the x-axis is the median.
 

Class

Less than c.f.

10 5
20 9
30 17
40 29
50 45
60 70
70 80
80 88
90 93
100 95


Here N = 95


Through 47.5 on the y-axis draw a horizontal line parallel to x-axis meeting the less than Ogive at P. Draw PQ perpendicular to x-axis. Q gives the value of the median.

 

Class

Greater than c.f.

0 95
10 90
20 86
30 78
40 66
50 50
60 25
70 15
80 7
90 2




Conceptual Questions:
 

Question 1 : 

On an average, 30 students can score 52 marks each and maximum number of students has scored 42 marks. What is the mean marks and mode marks.

Answer :

Mean marks: 52 Mode marks: 42


 

Question 2:

If the class interval is unequal and are large then how can we find the mean of the given data?

Answer :

We can apply step-deviation method to find the mean of the given data. Here we have to select h in such a way that is the divisor of all di ‘s.

 

 

Question 3:

Can we find the median of grouped data with unequal class sizes?

Answer :

Yes.


 

Question 4:

What is the intersection of two ogives?

Answer :

Median.

 






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