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Graphical Representation of Data as Histograms

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description

  • Construction of a histogram for continuous frequency distribution
  • Construction of histogram for discontinuous frequency distribution.

definition

Histogram: Histogram is a type of bar diagram, where the class intervals are shown on the horizontal axis and the heights of the bars show the frequency of the class interval. Also, there is no gap between the bars as there is no gap between the class intervals.

notes

Graphical Representation of Data as Histograms:

  • Grouped data can be presented using a histogram.

  • Histogram is a type of bar diagram, where the class intervals are shown on the horizontal axis and the heights of the bars show the frequency of the class interval. Also, there is no gap between the bars as there is no gap between the class intervals.

  • A Histogram is a bar graph that shows data in intervals. It has adjacent bars over the intervals.

  • This is a form of representation like the bar graph, but it is used for continuous class intervals.

  • There are no gaps in between consecutive rectangles, the resultant graph appears like a solid figure. This is called a histogram, which is a graphical representation of a grouped frequency distribution with continuous classes.

  • Unlike a bar graph, the width of the bar plays a significant role in its construction. The widths of the rectangles are all equal and the lengths of the rectangles are proportional to the frequencies.

Construction of Histogram:

For instance, consider the frequency distribution Table, representing the weights of 36 students of a class:

Weights (in kg) Number of students
30.5 - 35.5 9
35.5 - 40.5 6
40.5 - 45.5 15
45.5 - 50.5 3
50.5 - 55.5 1
55.5 - 60.5 2
Total 36

Let us represent the data given above graphically as follows:

(i) We represent the weights on the horizontal axis on a suitable scale. We can choose the scale as 1 cm = 5 kg. Also, since the first class interval is starting from 30.5 and not zero, we show it on the graph by marking a kink or a break on the axis.

(ii) We represent the number of students (frequency) on the vertical axis on a suitable scale. Since the maximum frequency is 15, we need to choose the scale to accommodate this maximum frequency.

(iii) We now draw rectangles (or rectangular bars) of width equal to the class-size and lengths according to the frequencies of the corresponding class intervals. For example, the rectangle for the class interval 30.5 - 35.5 will be of width 1 cm and length 4.5 cm.

2) A teacher wanted to analyse the performance of two sections of students in a mathematics test of 100 marks. Looking at their performances, she found that a
few students got under 20 marks and a few got 70 marks or above. So she decided to group them into intervals of varying sizes as follows: 0 - 20, 20 - 30, ..., 60 - 70, 70 - 100. Then she formed the following table:
 
Marks Number of students
0 - 20 7
20 - 30 10
30 - 40 10
40 - 50 20
50 - 60  20
60 - 70 15
70 - above 8
Total 90
 
It shows a greater frequency in the interval 70 - 100, than in 60 - 70, which is not the case. So, we need to make certain modifications in the lengths of the rectangles so that the areas are again proportional to the frequencies.
 
The steps to be followed are as given below:
  1. Select a class interval with the minimum class size which is 10.
  2. The lengths of the rectangles are then modified to be proportionate to the class-size 10. For instance, when the class-size is 20, the length of the rectangle is 7.
    So when the class-size is 10, the length of the rectangle will be `7/20 xx 10 = 3.5`.
Similarly, proceeding in this manner, we get the following table:
Marks Frequency Width of the class Length of the rectangle
0 - 20 7 20 `7/20 xx 10` = 3.5
20 - 30 10 10 `10/10 xx 10` = 10
30 - 40 10 10 `10/10 xx 10` = 10
40 - 50 20 10 `20/10 xx 10` = 20
50 - 60 20 10 `20/10 xx 10` = 20
60 - 70 15 10 `15/10 xx 10` = 15
70 - 100 8 30 `8/30 xx 10` = 2.67
 
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