Graphical Representation of Cumulative Frequency Distribution
Real Life Applications for Ogives
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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.
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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 |
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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 â€“ 200 |
Less than 200 |
10 |
40 + 10 =50 |
The above distribution is called â€˜less thanâ€™ cumulative frequency distribution.
To represent the data graphically,
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1) Mark the upper limits of the class interval on theÂ x â€“ axis and the corresponding cumulative frequencies on the y âˆ’ axis choosing suitable scale.
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2) Plot the points with coordinates having abscissa as upper limits and ordinates as the cumulative frequencies.
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3) Join the points by a free hand smooth curve
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4) The curve we get is called â€˜Cumulative frequency curveâ€™ or â€˜less than ogiveâ€™
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This graphical representation of the frequency distribution is called Ogive.
Now we can see the â€˜more thanâ€™ 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 â€“ 14 = 24 |
160 - 180 |
More than or equal to 160 |
6 |
24 â€“ 8 = 16 |
180 â€“ 200 |
More than or equal to 180 |
10 |
16 â€“ 6 = 10 |
To represent the data graphically,
1) Mark the lower limits of the class interval on theÂ x â€“ axis and the corresponding cumulative frequencies on the y âˆ’ 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 â€˜Cumulative frequency curveâ€™ or â€˜more than ogiveâ€™.
Relation of Ogive and Median
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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 theÂ x-axis and cumulative frequencies on the y-axis. The middle value I = where N = Î£ f is then marked on the y-axis .From the marked point draw a line parallel to x â€“ axis till it cuts the curve. At that point drop a perpendicular . The point where the perpendicular meets the x-axis is the median.
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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 |
HereÂ N = 95
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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:
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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.
Mean marks: 52 Mode marks: 42
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If the class interval is unequal andÂ are large then how can we find the mean of the given data?
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 d_{i}Â â€˜s.
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Can we find the median of grouped data with unequal class sizes?
Yes.
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What is the intersection of two ogives?
Median.
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